Note on error bounds for numerical integration

Abstract:

A length functional was defined in a previous publication which associates a length scale r to every function analytic on a given interval. The ratio of r to the mean panel size for numerical integration on that interval was defined as the sketchability. Stronger error bounds are given here for interpolative integration schemes in terms of the sketchability. Since these are not least upper bounds we also present lower limits for the least upper bounds obtained from actual numerical integrations of specific sketchability. We find the gap between the bounds and their lower limits to be small enough to be ignored for most applications.