Hybrid Gauss-Trapezoidal Quadrature Rules

摘要

A new class of quadrature rules for the integration of both regular and singularfunctions is constructed and analyzed. For each rule the quadrature weights are positive and the class includes rules of arbitrarily high order convergence. The quadratures result from alterations to the trapezoidal rule, in which a small number of nodes and weights as the ends of the integration interval are replaced. The new nodes and weights are determined so that the asymptotic expansion of the resulting rule, provided by a generalization of the Euler-Maclaurin summation formula, has a prescribed number of vanishing terms. The superior performance of the rules is demonstrated with numerical examples and application to several problems is discussed.