Asymptotic expansions of Gauss-Legendre quadrature rules for integrals with endpoint singularities
Math. Comp. 78 (2009), 241-253
Primary 40A25, 41A55, 41A60, 65D30.
May 16, 2008
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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
where , and and are constants independent of .
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An extension of the Euler-Maclaurin summation formula to functions with a branch singularity.
J. Math. and Phys., 40:271-276, 1961. MR 0140876 (25:4290)
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Asymptotics and Special Functions.
Academic Press, New York, 1974. MR 0435697 (55:8655)
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Euler-Maclaurin expansions for integrals with endpoint singularities: a new perspective.
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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)
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Computer Science Department, Technion–Israel Institute of Technology, Haifa 32000, Israel
Received by editor(s):
September 24, 2007
Received by editor(s) in revised form:
January 10, 2008
May 16, 2008
This research was supported in part by the United States–Israel Binational Science Foundation grant no. 2004353.
This paper is dedicated to the memory of Professor Philip Rabinowitz
© Copyright 2008 American Mathematical Society