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<title>QAWS adaptive integration for singular functions - GNU Scientific Library -- Reference Manual</title>
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<h3 class="section">16.8 QAWS adaptive integration for singular functions</h3>
<p><a name="index-QAWS-quadrature-algorithm-1461"></a><a name="index-singular-functions_002c-numerical-integration-of-1462"></a>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.
<div class="defun">
— Function: gsl_integration_qaws_table * <b>gsl_integration_qaws_table_alloc</b> (<var>double alpha, double beta, int mu, int nu</var>)<var><a name="index-gsl_005fintegration_005fqaws_005ftable_005falloc-1463"></a></var><br>
<blockquote>
<p>This function allocates space for a <code>gsl_integration_qaws_table</code>
struct describing a singular weight function
W(x) with the parameters (\alpha, \beta, \mu, \nu),
<pre class="example"> W(x) = (x-a)^alpha (b-x)^beta log^mu (x-a) log^nu (b-x)
</pre>
<p class="noindent">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,
<pre class="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)
</pre>
<p class="noindent">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.
<p>The function returns a pointer to the newly allocated table
<code>gsl_integration_qaws_table</code> if no errors were detected, and 0 in
the case of error.
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_integration_qaws_table_set</b> (<var>gsl_integration_qaws_table * t, double alpha, double beta, int mu, int nu</var>)<var><a name="index-gsl_005fintegration_005fqaws_005ftable_005fset-1464"></a></var><br>
<blockquote><p>This function modifies the parameters (\alpha, \beta, \mu, \nu) of
an existing <code>gsl_integration_qaws_table</code> struct <var>t</var>.
</p></blockquote></div>
<div class="defun">
— Function: void <b>gsl_integration_qaws_table_free</b> (<var>gsl_integration_qaws_table * t</var>)<var><a name="index-gsl_005fintegration_005fqaws_005ftable_005ffree-1465"></a></var><br>
<blockquote><p>This function frees all the memory associated with the
<code>gsl_integration_qaws_table</code> struct <var>t</var>.
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_integration_qaws</b> (<var>gsl_function * f, const double a, const double b, gsl_integration_qaws_table * t, const double epsabs, const double epsrel, const size_t limit, gsl_integration_workspace * workspace, double * result, double * abserr</var>)<var><a name="index-gsl_005fintegration_005fqaws-1466"></a></var><br>
<blockquote>
<p>This function computes the integral of the function f(x) over the
interval (a,b) with the singular weight function
(x-a)^\alpha (b-x)^\beta \log^\mu (x-a) \log^\nu (b-x). The parameters
of the weight function (\alpha, \beta, \mu, \nu) are taken from the
table <var>t</var>. The integral is,
<pre class="example"> I = \int_a^b dx f(x) (x-a)^alpha (b-x)^beta log^mu (x-a) log^nu (b-x).
</pre>
<p class="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.
</blockquote></div>
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