Probabilistic Analysis of Simpson′s Quadrature

AbstractWe study probabilistic properties of Simpson′s quadrature, assuming that the class of integrands is equipped with a variant of the r-fold Wiener measure. In the average case setting, we show that the error of Simpson′s quadrature is minimal (modulo a multiplicative constant) when equally spaced points are used. Furthermore, composite Simpson′s quadrature with equally spaced points is almost optimal among all algorithms iff the regularity degree r does not exceed 3. We are also interested in computing a posteriori bounds on the error of Simpson′s quadrature. The error bounds as well as the approximation to ∫10 ƒ(x) dx are computed based on a (fixed) finite number of function values. We derive a new a posteriori error bound for Simpson′s quadrature and show that, from a probabilistic point of view, it is significantly better than a bound that is commonly used in practice