Asymptotic expansions of Gauss-Legendre quadrature rules for integrals with endpoint singularities
Author: Avram Sidi
Journal: Math. Comp. 78 (2009), 241-253
MSC (2000): Primary 40A25, 41A55, 41A60, 65D30.
Published electronically: May 16, 2008
MathSciNet review: 2448705
Abstract: Let where , and let be the -point Gauss-Legendre quadrature approximation to . In this paper, we derive an asymptotic expansion as for the error when has general algebraic-logarithmic singularities at one or both endpoints. We assume that has asymptotic expansions of the forms
where and are some polynomials in . Here, and are, in general, complex and . An important special case is that in which and are constant polynomials; for this case, the asymptotic expansion of assumes the form
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J. Math. and Phys., 40:271-276, 1961. MR 0140876 (25:4290)
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Academic Press, New York, 1974. MR 0435697 (55:8655)
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Euler-Maclaurin expansions for integrals with endpoint singularities: a new perspective.
Numer. Math., 98:371-387, 2004. MR 2092747 (2005g:65012)
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Introduction to Numerical Analysis.
Springer-Verlag, New York, third edition, 2002. MR 1923481 (2003d:65001)
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Acceleration of Gauss-Legendre quadrature for an integrand with an endpoint singularity.
J. Comp. Appl. Math., 77:277-287, 1997. MR 1440013 (98f:65029)
Affiliation: Computer Science Department, Technion–Israel Institute of Technology, Haifa 32000, Israel
Keywords: Gauss--Legendre quadrature, singular integrals, endpoint singularities, asymptotic expansions, Euler--Maclaurin expansions
Received by editor(s): September 24, 2007
Received by editor(s) in revised form: January 10, 2008
Published electronically: May 16, 2008
Additional Notes: This research was supported in part by the United States–Israel Binational Science Foundation grant no. 2004353.
Dedicated: This paper is dedicated to the memory of Professor Philip Rabinowitz
Article copyright: © Copyright 2008 American Mathematical Society