Probabilistic analysis of Simpson's quadrature (Q1342511)

The author studies the probabilistic properties of Simpson's quadrature rule, by assuming that the integrand is equipped with a variant of the \(r\)-fold Wiener measure. In the average settting he shows that the error is minimal when one uses equally spaced nodes. He proves that the composite Simpson's rule using such points is almost optimal among all algorithms iff the regularity degree \(r\) does not exceed the number three. He also derives an a posteriori upper bound on the error of the Simpson's quadrature rule, showing that -- from a probabilistic point of view -- it is significantly better than a bound that is commonly used in practice.