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<title>GNU Scientific Library – Reference Manual: Integrands with singular weight functions</title>
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Previous: <a href="Integrands-with-weight-functions.html#Integrands-with-weight-functions" accesskey="p" rel="previous">Integrands with weight functions</a>, Up: <a href="Numerical-Integration-Introduction.html#Numerical-Integration-Introduction" accesskey="u" rel="up">Numerical Integration Introduction</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<h4 class="subsection">17.1.3 Integrands with singular weight functions</h4>
<p>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 <small>QUADPACK</small> separate out several common
weight functions which cause slow convergence.
</p>
<p>These weight functions are integrated analytically against the Chebyshev
polynomials to precompute <em>modified Chebyshev moments</em>. 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.
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