Report on the numerical experiments of Haselgrove’s method applied to the numerical solution of PDEs

We are interested in the efficiency of the so called Haselgrove’s method (cf. [Math. Comp. 15 (76) (1961) 323; Math. Comp. 39 (160) (1982) 549]) for the evaluation of the multiple integral I=∫01∫01⋯∫01f(x1,x2,…,xp)dx1dx2⋯dxp.This is a kind of Monte Carlo method but different from it in two points:(1)procedure of taking arithmetic mean is improved,(2)employment the Weyl sequences, as random numbers, with a suitable set of irrational numbers.The precision of the usual Monte Carlo method is O(N−1/2), where N denotes the sample size, while it is O(N−r),r≥1 for Haselgrove’s method.