research-article

High-order asymptotic expansions of Gaussian quadrature rules with classical and generalized weight functions

Authors:

Daan HuybrechsAuthors Info & Claims

Volume 434, Issue C

https://doi.org/10.1016/j.cam.2023.115317

Published: 15 December 2023 Publication History

Abstract

Gaussian quadrature rules are a classical tool for the numerical approximation of integrals with smooth integrands and positive weight functions. We derive and explicitly list asymptotic expressions for the points and weights of Gaussian quadrature rules for three general classes of positive weight functions: analytic functions on a bounded interval with algebraic singularities at the endpoints, analytic weight functions on the halfline with exponential decay at infinity and an algebraic singularity at the finite endpoint, and analytic functions on the real line with exponential decay in both directions at infinity. The results include the Gaussian rules of classical orthogonal polynomials (Legendre, Jacobi, Laguerre and Hermite) as special cases. Explicit expressions for these cases are included in the appendix. We present experiments indicating the range of the number of points at which these expressions achieve high precision. We provide an algorithm that can compute arbitrarily many terms in these expansions for the classical cases, and many though not all terms for the generalized cases.

Highlights

•

Gaussian quadrature rules can be computed efficiently from asymptotic expansions.

•

Generalized weight functions include all classical orthogonal polynomials as special cases.

•

Extensions of Riemann-Hilbert analyses are validated.

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Cited By

Denich ENovati P(2024)A fast and simple algorithm for the computation of the Lerch transcendentNumerical Algorithms10.1007/s11075-023-01637-396:1(13-32)Online publication date: 1-May-2024
https://dl.acm.org/doi/10.1007/s11075-023-01637-3

Index Terms

High-order asymptotic expansions of Gaussian quadrature rules with classical and generalized weight functions
1. Mathematics of computing
  1. Mathematical analysis
2. Theory of computation
  1. Design and analysis of algorithms

Index terms have been assigned to the content through auto-classification.

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Published In

cover image Journal of Computational and Applied Mathematics

Journal of Computational and Applied Mathematics Volume 434, Issue C

Dec 2023

448 pages

ISSN:0377-0427

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Elsevier B.V.

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Elsevier Science Publishers B. V.

Netherlands

Publication History

Published: 15 December 2023

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Cited By

Denich ENovati P(2024)A fast and simple algorithm for the computation of the Lerch transcendentNumerical Algorithms10.1007/s11075-023-01637-396:1(13-32)Online publication date: 1-May-2024
https://dl.acm.org/doi/10.1007/s11075-023-01637-3

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