Non-iterative computation of Gauss-Jacobi quadrature

TL;DR

This paper introduces asymptotic methods for direct, non-iterative computation of Gauss-Jacobi quadrature nodes and weights, achieving high accuracy for high-degree polynomials without iterative algorithms.

Contribution

It provides novel asymptotic formulas for Jacobi polynomial zeros and related functions, enabling efficient high-precision quadrature node and weight computation without iteration.

Findings

Achieves nearly double precision accuracy for nodes and weights when n ≥ 100.

Provides 10^{-12} relative accuracy for nodes at n ≥ 20.

Enables high-accuracy quadrature computations with minimal iterative steps.

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 non-iterative 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 both for 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.