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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.