Practical error estimation in numerical integration

摘要

Theoretical error estimates for quadrature rules involve quantities that are usually difficult if not impossible to obtain in practice. Various heuristics to obtain computable error estimates are compared by calculating their performance profiles on the Lyness family of integrands. Two sets of tests are used, corresponding to adaptive and single-rule quadrature. In the single rule case, Gaussian quadrature with error estimate provided by dropping one point from the formula performs best. In the adaptive case, the best heuristic is non-linear extrapolation based on Gaussian quadrature.