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Variable transformations and Gauss-Legendre quadrature for integrals with endpoint singularities

Author: Avram Sidi
Journal: Math. Comp. 78 (2009), 1593-1612
MSC (2000): Primary 40A25, 41A60, 65B15, 65D30, 65D32
Published electronically: January 22, 2009
MathSciNet review: 2501065
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Abstract: Gauss–Legendre quadrature formulas have excellent convergence properties when applied to integrals $\int ^1_0f(x) dx$ with $f\in C^\infty [0,1]$. However, their performance deteriorates when the integrands $f(x)$ are in $C^\infty (0,1)$ but are singular at $x=0$ and/or $x=1$. One way of improving the performance of Gauss–Legendre quadrature in such cases is by combining it with a suitable variable transformation such that the transformed integrand has weaker singularities than those of $f(x)$. Thus, if $x=\psi (t)$ is a variable transformation that maps $[0,1]$ onto itself, we apply Gauss–Legendre quadrature to the transformed integral $\int ^1_{0}f(\psi (t))\psi ’(t) dt$, whose singularities at $t=0$ and/or $t=1$ are weaker than those of $f(x)$ at $x=0$ and/or $x=1$. In this work, we first define a new class of variable transformations we denote $\widetilde {\mathcal {S}}_{p,q}$, where $p$ and $q$ are two positive parameters that characterize it. We also give a simple and easily computable representative of this class. Next, by invoking some recent results by the author concerning asymptotic expansions of Gauss–Legendre quadrature approximations as the number of abscissas tends to infinity, we present a thorough study of convergence of the combined approximation procedure, with variable transformations from $\widetilde {\mathcal {S}}_{p,q}$. We show how optimal results can be obtained by adjusting the parameters $p$ and $q$ of the variable transformation in an appropriate fashion. We also give numerical examples that confirm the theoretical results.

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Additional Information

Avram Sidi
Affiliation: Computer Science Department, Technion, Israel Institute of Technology, Haifa 32000, Israel

Keywords: Variable transformations, Gauss–Legendre quadrature, singular integrals, endpoint singularities, asymptotic expansions, Euler–Maclaurin expansions.
Received by editor(s): March 3, 2008
Received by editor(s) in revised form: July 28, 2008
Published electronically: January 22, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.