Automatic integration using asymptotically optimal adaptive Simpson quadrature

We present a novel theoretical approach to the analysis of adaptive quadratures and adaptive Simpson quadratures in particular which leads to the construction of a new algorithm for automatic integration. For a given function [Formula: see text] with [Formula: see text] and possible endpoint singularities the algorithm produces an approximation to [Formula: see text] within a given [Formula: see text] asymptotically as [Formula: see text]. Moreover, it is optimal among all adaptive Simpson quadratures, i.e., needs the minimal number [Formula: see text] of function evaluations to obtain an [Formula: see text]-approximation and runs in time proportional to [Formula: see text].