What Is the Complexity of Stieltjes Integration?

We study the complexity of approximating the Stieltjes integral R 1 0 f(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() denote the -complexity (the minimal cost of computing an -approximation). We show that r(n) n-min{r,s+1} and that comp() -1/min{r,s+1}. We also present an algorithm that computes an -approximation at nearly-minimal cost.

• Type: Journal
• Year: 2000
• Where: JC
• Authors: Arthur G. Werschulz
• Keywords: Continuous Derivatives, JC 2000, Nth Minimal Error, Stieltjes Integral