On computing the lattice rule criterion $R$

Abstract:

Lattice rules are integration rules for approximating integrals of periodic functions over the s-dimensional unit cube. One criterion for measuring the ’goodness’ of lattice rules is the quantity R. This quantity is defined as a sum which contains about ${N^{s - 1}}$ terms, where N is the number of quadrature points. Although various bounds involving R are known, a procedure for calculating R itself does not appear to have been given previously. Here we show how an asymptotic series can be used to obtain an accurate approximation to R. Whereas an efficient direct calculation of R requires $O(N{n_1})$ operations, where ${n_1}$ is the largest ’invariant’ of the rule, the use of this asymptotic expansion allows the operation count to be reduced to $O(N)$. A complete error analysis for the asymptotic expansion is given. The results of some calculations of R are also given.