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<title>GNU Scientific Library – Reference Manual: QAWS adaptive integration for singular functions</title>
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<a name="QAWS-adaptive-integration-for-singular-functions"></a>
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<p>
Next: <a href="QAWO-adaptive-integration-for-oscillatory-functions.html#QAWO-adaptive-integration-for-oscillatory-functions" accesskey="n" rel="next">QAWO adaptive integration for oscillatory functions</a>, Previous: <a href="QAWC-adaptive-integration-for-Cauchy-principal-values.html#QAWC-adaptive-integration-for-Cauchy-principal-values" accesskey="p" rel="previous">QAWC adaptive integration for Cauchy principal values</a>, Up: <a href="Numerical-Integration.html#Numerical-Integration" accesskey="u" rel="up">Numerical Integration</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="QAWS-adaptive-integration-for-singular-functions-1"></a>
<h3 class="section">17.8 QAWS adaptive integration for singular functions</h3>
<a name="index-QAWS-quadrature-algorithm"></a>
<a name="index-singular-functions_002c-numerical-integration-of"></a>
<p>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.
</p>
<dl>
<dt><a name="index-gsl_005fintegration_005fqaws_005ftable_005falloc"></a>Function: <em>gsl_integration_qaws_table *</em> <strong>gsl_integration_qaws_table_alloc</strong> <em>(double <var>alpha</var>, double <var>beta</var>, int <var>mu</var>, int <var>nu</var>)</em></dt>
<dd><a name="index-gsl_005fintegration_005fqaws_005ftable"></a>
<p>This function allocates space for a <code>gsl_integration_qaws_table</code>
struct describing a singular weight function
<em>W(x)</em> with the parameters <em>(\alpha, \beta, \mu, \nu)</em>,
</p>
<div class="example">
<pre class="example">W(x) = (x-a)^alpha (b-x)^beta log^mu (x-a) log^nu (b-x)
</pre></div>
<p>where <em>\alpha > -1</em>, <em>\beta > -1</em>, and <em>\mu = 0, 1</em>,
<em>\nu = 0, 1</em>. The weight function can take four different forms
depending on the values of <em>\mu</em> and <em>\nu</em>,
</p>
<div class="example">
<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></div>
<p>The singular points <em>(a,b)</em> do not have to be specified until the
integral is computed, where they are the endpoints of the integration
range.
</p>
<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></dd></dl>
<dl>
<dt><a name="index-gsl_005fintegration_005fqaws_005ftable_005fset"></a>Function: <em>int</em> <strong>gsl_integration_qaws_table_set</strong> <em>(gsl_integration_qaws_table * <var>t</var>, double <var>alpha</var>, double <var>beta</var>, int <var>mu</var>, int <var>nu</var>)</em></dt>
<dd><p>This function modifies the parameters <em>(\alpha, \beta, \mu, \nu)</em> of
an existing <code>gsl_integration_qaws_table</code> struct <var>t</var>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005fintegration_005fqaws_005ftable_005ffree"></a>Function: <em>void</em> <strong>gsl_integration_qaws_table_free</strong> <em>(gsl_integration_qaws_table * <var>t</var>)</em></dt>
<dd><p>This function frees all the memory associated with the
<code>gsl_integration_qaws_table</code> struct <var>t</var>.
</p></dd></dl>
<dl>
<dt><a name="index-gsl_005fintegration_005fqaws"></a>Function: <em>int</em> <strong>gsl_integration_qaws</strong> <em>(gsl_function * <var>f</var>, const double <var>a</var>, const double <var>b</var>, gsl_integration_qaws_table * <var>t</var>, const double <var>epsabs</var>, const double <var>epsrel</var>, const size_t <var>limit</var>, gsl_integration_workspace * <var>workspace</var>, double * <var>result</var>, double * <var>abserr</var>)</em></dt>
<dd>
<p>This function computes the integral of the function <em>f(x)</em> over the
interval <em>(a,b)</em> with the singular weight function
<em>(x-a)^\alpha (b-x)^\beta \log^\mu (x-a) \log^\nu (b-x)</em>. The parameters
of the weight function <em>(\alpha, \beta, \mu, \nu)</em> are taken from the
table <var>t</var>. The integral is,
</p>
<div class="example">
<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></div>
<p>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.
</p>
</dd></dl>
<hr>
<div class="header">
<p>
Next: <a href="QAWO-adaptive-integration-for-oscillatory-functions.html#QAWO-adaptive-integration-for-oscillatory-functions" accesskey="n" rel="next">QAWO adaptive integration for oscillatory functions</a>, Previous: <a href="QAWC-adaptive-integration-for-Cauchy-principal-values.html#QAWC-adaptive-integration-for-Cauchy-principal-values" accesskey="p" rel="previous">QAWC adaptive integration for Cauchy principal values</a>, Up: <a href="Numerical-Integration.html#Numerical-Integration" accesskey="u" rel="up">Numerical Integration</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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