Gauss-type Quadrature Rules for Rational Functions

When integrating functions that have poles outside the interval of integration, but are regular otherwise, it is suggested that the quadrature rule in question ought to integrate exactly not only polynomials (if any), but also suitable rational functions. The latter are to be chosen so as to match the most important poles of the integrand. We describe two methods for generating such quadrature rules numerically and report on computational experience with them. Introduction Traditionally, Gauss quadrature rules are designed to integrate exactly polynomials of maximum possible degree. This is meaningful for integrand functions that are “polynomial-like”. For integrands having poles (outside the interval of integration) it would be more natural to include also rational functions among the functions to be exactly integrated. In this paper we consider n-point quadrature rules that exactly integrate m rational functions (with prescribed location and multiplicity of the poles) as well as polynomials of degree 2n−m−1, where 0 ≤ m ≤ 2n. The limit case m = 2n, in which only rational functions are being integrated exactly, is a rational counterpart of the classical Gauss formula; the latter corresponds to the other limit case m = 0. In §1 we characterize these new quadrature rules in terms of classical (polynomial) Gauss formulae with modified weight functions. We also identify special choices of poles that are of interest in applications. The computation of the quadrature rules is discussed in §2, and numerical examples are given in §3. ∗Work supported in part by the National Science Foundation under grant DMS–9023403. 1. Gauss quadrature for rational functions Let dλ be a measure on the real line having finite moments of all orders. Let ζμ ∈ C, μ = 1, 2, . . . ,M , be distinct real or complex numbers such that ζμ 6= 0 and 1 + ζμt 6= 0 for t ∈ supp(dλ), μ = 1, 2, . . . ,M . (1.1) For given integers m, n with 1 ≤ m ≤ 2n, we wish to find an n-point quadrature rule that integrates exactly (against the measure dλ) polynomials of degree 2n−m− 1 as well as the m rational functions (1 + ζμt) , μ = 1, 2, . . . ,M, s = 1, 2, . . . , sμ, (1.2) where sμ ≥ 1 and M