The superconvergence of Newton-Cotes rules for the Hadamard finite-part integral on an interval

We study the general (composite) Newton-Cotes rules for the computation of Hadamard finite-part integral with the second-order singularity and focus on their pointwise superconvergence phenomenon, i.e., when the singular point coincides with some a priori known point, the convergence rate is higher than what is globally possible. We show that the superconvergence rate of the (composite) Newton-Cotes rules occurs at the zeros of a special function and prove the existence of the superconvergence points. Several numerical examples are provided to validate the theoretical analysis. © 2007 Springer-Verlag.