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 f C⁴ with f^ (4) 0 and possible endpoint singularities the algorithm produces an approximation to ₐᵇf (x) \, dx within a given asymptotically as 0. Moreover, it is optimal among all adaptive Simpson quadratures, i. e. , needs the minimal number n (f, ) of function evaluations to obtain an -approximation and runs in time proportional to n (f, ).