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.
Highlights
Given a measure μ on R, a classical problem in numerical analysis is to approximate the integral with respect to μ of a suitably well-behaved function f
One approach is via so called quadrature rules
One classical construction for quadrature rules designed to approximate the integral of continuous functions consists of demanding an exact evaluation of the integral for all polynomials of degree D
Summary
Given a (positive Borel) measure μ on R, a classical problem in numerical analysis is to approximate the integral with respect to μ of a suitably well-behaved function f. We reprove this fact by constructing a polynomial F ∈ R[x, y] with degree d in each of x and y and the property that for every y ∈ R, the d roots of F (x, y) ∈ R[x] d are the other d nodes (among them possibly ∞) of this unique quadrature rule with y as a node. We give symmetric determinantal representations of F , which again translates the problem of finding nodes of a quadrature rule into finding the generalized eigenvalues of a real symmetric matrix, i.e. solving det(x A − B) = 0 where A, B are real symmetric matrices and A is positive semidefinite
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