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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
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Abstract: Let $ I[f]=\int_{-1}^1f(x)\,dx,$ where $ f\in C^\infty(-1,1)$, and let $ G_n[f]=\sum^n_{i=1}w_{ni}f(x_{ni})$ be the $ n$-point Gauss-Legendre quadrature approximation to $ I[f]$. In this paper, we derive an asymptotic expansion as $ n\to\infty$ for the error $ E_n[f]=I[f]-G_n[f]$ when $ f(x)$ has general algebraic-logarithmic singularities at one or both endpoints. We assume that $ f(x)$ has asymptotic expansions of the forms

$\displaystyle f(x)$ $\displaystyle \sim\sum^\infty_{s=0}U_s(\log (1-x))(1-x)^{\alpha_s}$   as $ x\to 1-$,    
$\displaystyle f(x)$ $\displaystyle \sim\sum^\infty_{s=0}V_s(\log(1+x))(1+x)^{\beta_s}$   as $ x\to -1+$,    

where $ U_s(y)$ and $ V_s(y)$ are some polynomials in $ y$. Here, $ \alpha_s$ and $ \beta_s$ are, in general, complex and $ \Re\alpha_s,\Re\beta_s>-1$. An important special case is that in which $ U_s(y)$ and $ V_s(y)$ are constant polynomials; for this case, the asymptotic expansion of $ E_n[f]$ assumes the form

$\displaystyle E_n[f]\sim\sum^\infty_{\substack{s=0 \\ \alpha_s\not\in \mathbb{Z... ... \mathbb{Z}^+}}\sum^\infty_{i=1}b_{si}h^{\beta_s+i}\quad\text{as $n\to\infty$},$

where $ h=(n+1/2)^{-2}$, $ \mathbb{Z}^+=\{0,1,2,\ldots\},$ and $ a_{si}$ and $ b_{si}$ are constants independent of $ n$.

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Additional Information

Avram Sidi
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

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