Asymptotic expansions of Gauss-Legendre quadrature rules for integrals with endpoint singularities

Asymptotic expansions of Gauss-Legendre quadrature rules for integrals with endpoint singularities

Let I[f ] = ∫ 1 −1 f(x) dx, where f ∈ C ∞(−1, 1), and let Gn[f ] = ∑n i=1 wnif(xni) be the n-point Gauss–Legendre quadrature approximation to I[f ]. In this paper, we derive an asymptotic expansion as n → ∞ for the error En[f ] = I[f ]−Gn[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 f(x) ∼ ∞ ∑ ...

Variable transformations and Gauss-Legendre quadrature for integrals with endpoint singularities

Gauss–Legendre quadrature formulas have excellent convergence properties when applied to integrals ∫ 1 0 f(x) dx with f ∈ C∞[0, 1]. However, their performance deteriorates when the integrands f(x) are in C∞(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 ...

Asymptotic expansions of Legendre series coefficients for functions with endpoint singularities

Avram Sidi Computer Science Department, Technion – Israel Institute of Technology, Haifa 32000, Israel E-mail: [email protected]; URL: http://www.cs.technion.ac.il/~asidi/ Abstract. Let ∑∞ n=0 en[f ]Pn(x) be the Legendre expansion of a function f (x) on (−1, 1). In this work, we derive an asymptotic expansion as n → ∞ for en[f ], assuming that f ∈ C∞(−1, 1), but may have arbitrary algebra...

Asymptotic expansions of Legendre series coefficients for functions with interior and endpoint singularities

Abstract. Let ∑∞ n=0 en[f ]Pn(x) be the Legendre expansion of a function f(x) on (−1, 1). In an earlier work [A. Sidi, Asymptot. Anal., 65 (2009), pp. 175–190], we derived asymptotic expansions as n → ∞ for en[f ], assuming that f ∈ C∞(−1, 1), but may have arbitrary algebraic-logarithmic singularities at one or both endpoints x = ±1. In the present work, we extend this study to functions f(x) t...

Quadrature Rules for Fuzzy Integrals