RESUMEN

Quadrature rules of the extended Clenshaw-Curtis (ECC) type are derived for integrands with trigonometric or hyperbolic weight functions. The study puts into evidence the existence of a fully analytic solution, in terms of hypergeometric functions ₀F₁. An important implementation problem, namely the possibility of conditional activation of the ECC, is evidence and solved. As compared with the best available similar significantly smaller overhead, while yielding a fully reliable output.

RESUMEN

Product integration rules generalizing the Féjer, Clenshaw-Curtis and Filippi quadrature rules respectively are derived for integrals with trigonometric and hyperbolic weight factors. The study puts in evidence the existence of well-conditioned fully analytic solutions, 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 put into evidence by large scale validation tests. An empirical error estimator, superseding them at slowly varying integrands, is found to result in a spectacular increase in the output reliability. Finally, an enhancements in the control of the interval subdivision strategy aiming at increasing code robustness is discussed. Comparison with the code DQAWO of QUADPACK, extending over a statistics of about hundred thousand solved integrals, is illustrative for the increased robustness and error estimate reliability of our computer code implementation of the ECC rules.