Automatic integration using asymptotically optimal adaptive Simpson quadrature

摘要

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 ε asymptotically as ε → 0. Moreover, it is optimal among all adaptive Simpson quadratures, i.e., needs the minimal number n(f, ε) of function evaluations to obtain an ε-approximation and runs in time proportional to n(f, ε).