On optimal adaptive quadratures for automatic integration

Abstract

In this paper, the problem of automatic integration is investigated. Quadratures that compute the integral of an r times differentiable function with the assumption that the rth derivative is positive within precision $$\varepsilon >0$$ are constructed. A rigorous analysis of these quadratures is presented. It turns out that the mesh selection procedure proposed in this paper is optimal i.e., it uses the minimal number of function evaluations. It is also shown how to adapt the quadratures to the case where the assumption that the rth derivative has a constant sign is not fulfilled. The theoretical results are illustrated and confirmed with numerical tests.