Generalized eigenvalue methods for Gaussian quadrature rules

TL;DR

This paper introduces a new approach to Gaussian quadrature rules by using a bivariate polynomial and eigenvalue problems to determine optimal nodes for measures on the real line.

Contribution

It presents a novel polynomial formulation and determinantal formulas that convert the quadrature node finding problem into a generalized eigenvalue problem.

Findings

Provides explicit polynomial parametrization of quadrature nodes.

Derives symmetric determinantal formulas for the polynomial.

Establishes a method to compute nodes via eigenvalue problem solutions.

Abstract

A quadrature rule of a measure $\mu$ on the real line represents a convex combination of finitely many evaluations at points, called nodes, that agrees with integration against $\mu$ for all polynomials up to some fixed degree. In this paper, we present a bivariate polynomial whose roots parametrize the nodes of minimal quadrature rules for measures on the real line. We give two symmetric determinantal formulas for this polynomial, which translate the problem of finding the nodes to solving a generalized eigenvalue problem.