Fast and accurate computation of classical Gaussian quadratures

2025-09-20

AI Summary

This paper addresses the efficient and high-precision computation of classical Gaussian quadrature nodes and weights—including Gauss–Jacobi, Gauss–Laguerre, Gauss–Hermite, and their Radau/Lobatto variants. We propose an adaptive hybrid strategy: a globally convergent, fourth-order Newton-type iteration is employed in well-conditioned parameter regions, while highly accurate asymptotic approximations are activated for ill-conditioned or large-parameter regimes. Additionally, we derive and implement a numerically stable algorithm for computing barycentric weights. The method supports arbitrary-precision arithmetic and, for the first time, systematically covers parameter regimes where conventional algorithms fail—particularly in symmetric cases such as Gauss–Gegenbauer and Gauss–Hermite quadratures. Experimental results demonstrate superior numerical stability, computational efficiency, and full-parameter robustness compared to state-of-the-art implementations.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

Abstract

Algorithms for computing classical Gaussian quadrature rules (Gauss-Jacobi, Gauss-Laguerre, and Gauss-Hermite) are presented, based on globally convergent fourth-order iterative methods and asymptotic approximations, which are applied in complementary regions of the parameter space. The combination of these approaches results in methods that surpass previous algorithms in terms of speed, accuracy, and computational range (practically unrestricted). The Gauss-Radau and Gauss-Lobatto variants are also considered, along with the computation of the associated barycentric weights. Arbitrary accuracy algorithms are also provided for the symmetric cases (Gauss-Gegenbauer and Gauss-Hermite).

Problem

• Developing fast algorithms for classical Gaussian quadrature rules
• Combining iterative methods and asymptotic approximations for computation
• Achieving high speed, accuracy, and unrestricted computational range

Innovation

• Globally convergent fourth-order iterative methods
• Asymptotic approximations in complementary parameter regions
• Arbitrary accuracy algorithms for symmetric cases