The superconvergence of the Newton–Cotes rule for Cauchy principal value integrals

We consider the general (composite) Newton–Cotes method for the computation of Cauchy principal value integrals and focus on its pointwise superconvergence phenomenon, which means that the rate of convergence of the Newton–Cotes quadrature rule is higher than what is globally possible when the singular point coincides with some a priori known point. The necessary and sufficient conditions satisfied by the superconvergence point are given. Moreover, the superconvergence estimate is obtained and the properties of the superconvergence points are investigated. Finally, some numerical examples are provided to validate the theoretical results.