Superconvergence and ultraconvergence of Newton-Cotes rules for supersingular integrals (Q848540)

The main result of the paper consists of an error estimate of general (composite) Newton-Cotes rules for integrals of the form \(=\!\!\!\!\!\!\int{a}{b}f(x)(x-s)^{-(p+1)}dx\), \(s\in (a,b)\), \(p=2,3\dots\), where \(=\!\!\!\!\!\!\int\) is the Hadamard finite-part integral. From this error expansion the authors derive a modified quadrature rule with \(k\)th-order accuracy, though it may be invalid if \(s\) is close to the nodal points. Some numerical examples are shown at the end to confirm the theoretical analysis.