Quadrature rules for rational functions

It is shown how recent ideas on rational Gauss-type quadrature rules can be extended to Gauss-Kronrod, Gauss-Turr an, and Cauchy principal value quadrature rules. Numerical examples illustrate the advantages in accuracy thus achievable. 0. Introduction The idea of constructing quadrature rules that are exact for rational functions with prescribed poles, rather than for polynomials, has received some attention in recent years; see, e.g., 9], 10], 11], 2], 4]. These \rational" quadrature rules have proven to be quite eeective if the poles are chosen so as to simulate the poles present in the integrand; see 3] for an application to integrals occurring in solid state physics. The work so far has been exclusively centered around quadrature rules of Gaussian type. Here we construct rational versions of other important quadrature rules, speciically the Gauss-Kronrod and the Gauss-Turr an rule, and Cauchy principal value quadrature rules. It is found that the accuracy is enhanced similarly as has been observed for Gauss-type quadrature rules. 1. Rational Gauss-Kronrod Quadrature Let dd(t) be a positive measure on the real line R and f G g n =1 the nodes of the n-point Gaussian quadrature rule for dd. The Gauss-Kronrod rule is a formula of the type (1:1) Z R g(t)dd(t) = which is exact for all polynomials of degree 3n + 1. Here, in analogy to 2], 4], we are interested in making (1.1) exact on the space of dimension 3n+2, (1:2)