What Is the Complexity of Stieltjes Integration?

Abstract

We study the complexity of approximating the Stieltjes integral ∫10f(x)dg(x) for functions f having r continuous derivatives and functions g whose sth derivative has bounded variation. Let r(n) denote the nth minimal error attainable by approximations using at most n evaluations of f and g, and let comp(ε) denote the ε-complexity (the minimal cost of computing an ε-approximation). We show that r(n)≍n−min{r, s+1} and that comp(ε)≍ε−1/min{r, s+1}. We also present an algorithm that computes an ε-approximation at nearly minimal cost.