Interpolatory quadrature rules for Hadamard finite-part integrals and their superconvergence

Abstract

In this paper, we present a general framework for interpolatory quadrature rules for Hadamard finite-part integrals with a second-order singularity. Gaussian quadrature rules are viewed as a special case and many interesting features can be obtained easily from the framework. We prove theoretically the equivalence of some existing formulas which were obtained in different ways. We show the point-wise superconvergence of these interpolatory quadrature rules, i.e. when the singular point coincides with certain a priori known points, the accuracy is better than what is generally possible. The extension of a popular interpolatory quadrature rule for Cauchy principal value integrals is presented. A new quadrature rule of Gaussian type is proposed for the evaluation of integrals simultaneously involving different types of singularities. Numerical examples confirm our theoretical results