Superconvergence and ultraconvergence of Newton-Cotes rules for supersingular integrals

In this article, the general (composite) Newton-Cotes rules for evaluating Hadamard finite-part integrals with third-order singularity (which is also called “supersingular integrals”) are investigated and the emphasis is placed on their pointwise superconvergence and ultraconvergence. The main error of the general Newton-Cotes rules is derived, which is shown to be determined by a certain function . Based on the error expansion, the corresponding modified quadrature rules are also proposed. At last, some numerical experiments are carried out to validate the theoretical analysis.