Product Integration Rules at Clenshaw-Curtis and Related Points: A Robust Implementation

Abstract

Product integration rules generalizing the Fejér, Clenshaw-Curtis, and Filippi quadrature rules respectively are derived for integrals with trigonometric and hyperbolic weight factors. The Chebyshev moments of the weight functions are found to be given by well-conditioned expressions, in terms of hypergeometric functions 0F1. An a priori error estimator is discussed which is shown both to avoid wasteful invocation of the integration rule and to increase significantly the robustness of the automatic quadrature procedure. Then, specializing to extended Clenshaw-Curtis (ECC) rules, three types of a posteriori error estimates are considered and the existence of a great risk of their failure is demonstrated by large scale validation tests. An empirical error estimator, superseding them for slowly varying integrands, is found to result in a spectacular increase in the output reliability. Finally, enhancements in the control of the interval subdivision strategy aiming at increasing code robustness is discussed. Comparison with the code DQAWO of QUADPACK, with about a hundred thousand solved integrals, is illustrative of the increased robustness and error estimate reliability of our computer code implementation of the ECC rules.