Asymptotic approximations to the nodes and weights of Gauss-Hermite and Gauss-Laguerre quadratures

2017/09/27 by Amparo Gil, Javier Segura, Gil, A. +3 · 1 citation
Mathematics · Physics and Astronomy · #Classical Analysis and ODEs (math.CA) #Electromagnetic Scattering and Analysis #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.1709.09656

openalex publication_date 2017/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Asymptotic approximations to the zeros of Hermite and Laguerre polynomials are given, together with methods for obtaining the coefficients in the expansions. These approximations can be used as a standalone method of computation of Gaussian quadratures for high enough degrees, with Gaussian weights computed from asymptotic approximations for the orthogonal polynomials. We provide numerical evidence showing that for degrees greater than 100 the asymptotic methods are enough for a double precision accuracy computation (15-16 digits) of the nodes and weights of the Gauss--Hermite and Gauss--Laguerre quadratures.

Citations

• Calculation of Gauss quadrature rules

Cited by

• Fast and accurate computation of classical Gaussian quadratures

Related

• Fast, reliable and unrestricted iterative computation of Gauss--Hermite and Gauss--Laguerre quadratures
• Non-iterative computation of Gauss-Jacobi quadrature
• Fast computation of Gauss quadrature nodes and weights on the whole real line
• Fast and accurate computation of classical Gaussian quadratures
• Asymptotic expansions of Jacobi polynomials and of the nodes and weights of Gauss-Jacobi quadrature for large degree and parameters in terms of elementary functions