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 S k ′ ( τ ) . 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.