Fast computation of Gauss quadrature nodes and weights on the whole real line

TL;DR

This paper introduces a fast, accurate algorithm for computing Gauss-Hermite and generalized Gauss-Hermite quadrature nodes and weights on the entire real line, achieving near machine precision with optimized complexity.

Contribution

It presents a novel algorithm combining Newton's method with asymptotic formulas and Riemann-Hilbert techniques for efficient quadrature node and weight computation.

Findings

Computes n-point quadrature in O(n) operations with high accuracy.

Achieves complexity as low as O(√n) for large n by exploiting small weights.

Provides explicit asymptotic formulas and Riemann-Hilbert reformulation for initial guesses.

Abstract

A fast and accurate algorithm for the computation of Gauss-Hermite and generalized Gauss-Hermite quadrature nodes and weights is presented. The algorithm is based on Newton's method with carefully selected initial guesses for the nodes and a fast evaluation scheme for the associated orthogonal polynomial. In the Gauss-Hermite case the initial guesses and evaluation scheme rely on explicit asymptotic formulas. For generalized Gauss-Hermite, the initial guesses are furnished by sampling a certain equilibrium measure and the associated polynomial evaluated via a Riemann-Hilbert reformulation. In both cases the $n$-point quadrature rule is computed in $\mathcal{O}(n)$ operations to an accuracy that is close to machine precision. For sufficiently large $n$, some of the quadrature weights have a value less than the smallest positive normalized floating-point number in double precision and we exploit this fact to achieve a complexity as low as $\mathcal{O}(\sqrt{n})$.