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<title>GNU Scientific Library – Reference Manual: QAGS adaptive integration with singularities</title>
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<a name="QAGS-adaptive-integration-with-singularities"></a>
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Next: <a href="QAGP-adaptive-integration-with-known-singular-points.html#QAGP-adaptive-integration-with-known-singular-points" accesskey="n" rel="next">QAGP adaptive integration with known singular points</a>, Previous: <a href="QAG-adaptive-integration.html#QAG-adaptive-integration" accesskey="p" rel="previous">QAG adaptive integration</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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<a name="QAGS-adaptive-integration-with-singularities-1"></a>
<h3 class="section">17.4 QAGS adaptive integration with singularities</h3>
<a name="index-QAGS-quadrature-algorithm"></a>
<p>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.
</p>
<dl>
<dt><a name="index-gsl_005fintegration_005fqags"></a>Function: <em>int</em> <strong>gsl_integration_qags</strong> <em>(const gsl_function * <var>f</var>, double <var>a</var>, double <var>b</var>, double <var>epsabs</var>, double <var>epsrel</var>, 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 applies the Gauss-Kronrod 21-point integration rule
adaptively until an estimate of the integral of <em>f</em> over
<em>(a,b)</em> is achieved within the desired absolute and relative error
limits, <var>epsabs</var> and <var>epsrel</var>. 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</var>, and an estimate of the absolute error,
<var>abserr</var>. The subintervals and their results are stored in the
memory provided by <var>workspace</var>. The maximum number of subintervals
is given by <var>limit</var>, which may not exceed the allocated size of the
workspace.
</p>
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