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Non-iterative computation of Gauss-Jacobi quadrature

Abstract: Asymptotic approximations to the zeros of Jacobi polynomials are given, with methods to obtain the coefficients in the expansions. These approximations can be used as standalone methods for the noniterative computation of the nodes of Gauss--Jacobi quadratures of high degree ($n\ge 100$). We also provide asymptotic approximations for functions related to the first-order derivative of Jacobi polynomials which are used for computing the weights of the Gauss--Jacobi quadrature. The performance of the asymptotic approximations is illustrated with numerical examples, and it is shown that nearly double precision relative accuracy is obtained for both the nodes and the weights when $n\ge 100$ and $-1< \alpha, \beta\le 5$. For smaller degrees the approximations are also useful as they provide $10^{-12}$ relative accuracy for the nodes when $n\ge 20$, and just one Newton step would be sufficient to guarantee double precision accuracy in that cases.

Otras publicaciones de la misma revista o congreso con autores/as de la Universidad de Cantabria

 Autoría: Gil A., Segura J., Temme N.,

 Fuente: SIAM J. Sci. Comput., 41(1), A668-A693

Editorial: Society for Industrial and Applied Mathematics

 Año de publicación: 2019

Nº de páginas: 26

Tipo de publicación: Artículo de Revista

 DOI: 10.1137/18M1179006

ISSN: 1064-8275,1095-7197

Proyecto español: MTM2015-67142-P (MINECO/FEDER, UE)