A Riemann--Hilbert approach to Jacobi operators and Gaussian quadrature

TL;DR

This paper introduces an efficient $ ext{O}(N)$ numerical method leveraging Riemann--Hilbert problems to compute Jacobi operators and orthogonal polynomials, enhancing Gaussian quadrature computations especially for challenging weights.

Contribution

It develops a novel $ ext{O}(N)$ approach using Riemann--Hilbert techniques to compute Jacobi operators from weights, including those beyond current asymptotic methods.

Findings

Achieves efficient computation of Jacobi operators for large N.

Extends Riemann--Hilbert methods to broader class of weights.

Provides a practical numerical framework for orthogonal polynomial evaluation.

Abstract

The computation of the entries of Jacobi operators associated with orthogonal polynomials has important applications in numerical analysis. From truncating the operator to form a Jacobi matrix, one can apply the Golub--Welsh algorithm to compute the Gaussian quadrature weights and nodes. Furthermore, the entries of the Jacobi operator are the coefficients in the three-term recurrence relationship for the polynomials. This provides an efficient method for evaluating the orthogonal polynomials. Here, we present an $\mathcal O(N)$ method to compute the first $N$ rows of Jacobi operators from the associated weight. The method exploits the Riemann--Hilbert representation of the polynomials by solving a deformed Riemann--Hilbert problem numerically. We further adapt this computational approach to certain entire weights that are beyond the reach of current asymptotic Riemann--Hilbert techniques.