Generalized eigenvalue methods for Gaussian quadrature rules
This offers a new computational approach for constructing Gaussian quadrature rules, benefiting numerical analysis and approximation theory.
The paper introduces a bivariate polynomial whose roots parametrize nodes of minimal quadrature rules, and provides symmetric determinantal formulas that reduce node computation to a generalized eigenvalue problem.
A quadrature rule of a measure $μ$ on the real line represents a convex combination of finitely many evaluations at points, called nodes, that agrees with integration against $μ$ 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.