Asymptotic expansions of Gauss-Legendre quadrature rules for integrals with endpoint singularities
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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 \begin{align*} f(x)&\sim \sum ^\infty _{s=0}U_s(\log (1-x))(1-x)^{\alpha _s} \quad \text {as $x\to 1-$,} f(x)&\sim \sum ^\infty _{s=0}V_s(\log (1+x))(1+x)^{\beta _s} \quad \text {as $x\to -1+$,}\end{align*} 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 \[ E_n[f] \sim \sum ^\infty _{\substack {s=0\\ \alpha _s \not \in \mathbb {Z}^+}} \sum ^\infty _{i=1} a_{si} h^{\alpha _s+i} + \sum ^\infty _{\substack {s=0\\ \beta _s\not \in \mathbb {Z}^+}} \sum ^\infty _{i=1} b_{si} h^{\beta _s+i} \quad \text {as $n\to \infty $}, \] where $h=(n+1/2)^{-2}$, $\mathbb {Z}^+=\{0,1,2,\ldots \},$ and $a_{si}$ and $b_{si}$ are constants independent of $n$.References
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Additional Information
- Avram Sidi
- Affiliation: Computer Science Department, Technion–Israel Institute of Technology, Haifa 32000, Israel
- Email: asidi@cs.technion.ac.il
- 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.
- © Copyright 2008 American Mathematical Society
- Journal: Math. Comp. 78 (2009), 241-253
- MSC (2000): Primary 40A25, 41A55, 41A60, 65D30
- DOI: https://doi.org/10.1090/S0025-5718-08-02135-2
- MathSciNet review: 2448705
Dedicated: This paper is dedicated to the memory of Professor Philip Rabinowitz