Asymptotic Approximations to the Nodes and Weights of Gauss–Hermite and Gauss–Laguerre Quadratures. Issue 3 (11th January 2018)

• Record Type: Journal Article
• Authors: Gil, Amparo; Segura, Javier; Temme, Nico M.
• Is Part Of: Studies in applied mathematics. Volume 140:Issue 3(2018)
• Page Start: 298
• Page End: 332
• Publication Date: 2018-01-11
• Subjects: Mathematics -- Periodicals; Mathématiques -- Périodiques; Mathématique; Mathématique appliquée; Ressource Internet (Descripteur de forme); Périodique électronique (Descripteur de forme); 519
• DOI: 10.1111/sapm.12201
• Languages: English
• ISSNs: 0022-2526

Abstract

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 stand‐alone 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.