Incorporating the external zeros and poles of the integrand into Gauss-type quadrature rules

Abstract

Quadrature formulas are often constructed to be exact on the space of functions that are easily integrated and that are in some sense similar to the integrand. This motivates us to explore how the known properties of the integrand can be used to improve the accuracy of certain quadrature rules. In the present paper, we propose an l-point Gauss-type quadrature rule Gl into which the zeros and poles of the integrand outside the integration interval are incorporated. Formula Gl proves to be exact for certain rational functions which have the same zeros and poles as the integrand. It converges, all its nodes are pairwise distinct and belong to the interior of the integration interval, and all its weights are positive. Theoretical results on the remainder term of Gl suggest that formula Gl is applicable both when the incorporated zeros and poles of the integrand are known exactly, as well as when they are known approximately. To practically and economically estimate the error of Gl, some extensions that inherit the l nodes of Gl are developed. They are analogous to the Gauss-Kronrod, averaged Gauss, and generalized averaged Gauss quadrature rules. Numerical experiments confirm the accuracy of Gl and its extensions.