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

TL;DR

This paper presents asymptotic formulas for calculating nodes and weights of high-degree Gauss-Hermite and Gauss-Laguerre quadratures, enabling efficient and accurate numerical integration for large polynomial degrees.

Contribution

It introduces asymptotic approximations and coefficient computation methods for nodes and weights, improving high-degree Gaussian quadrature calculations.

Findings

Asymptotic methods achieve double precision accuracy for degrees over 100.

Numerical evidence supports the efficiency of the asymptotic approach.

Approximations are suitable for standalone high-degree quadrature computations.

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.