High-Order Corrected Trapezoidal Rules for Singular Functions

摘要

A group of quadrature formulae is presented applicable to both non-singularfunctions and functions with end-point singularities, generalizing the classical end-point corrected trapezoidal quadrature rules. We present an algorithm for the construction of very high-order end-point corrected trapezoidal rules, taking advantage of functional information outside the interval of integration. The scheme applies not only to non-singular functions, but also for a wide class of functions with monotonic singularities. Numerical experiments are presented demonstrating the practical usefulness of the new class of quadratures. Tables of quadrature weights are included for singularities of the form s(x) = log(absolute value of x), s(x) = absolute value of x to the lambda power for a variety of values of lambda. (Author).