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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}}$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})$ is the largest 'invariant' of the rule, the use of this asymptotic expansion allows the operation count to be reduced to $ O(N)$R are also given.

Lattice rules are used for the numerical integration of periodic functions over the s-dimensional unit cube. They are normally classified according to their 'rank'; in simple terms, the rank of a lattice rule is the minimum number of sums required to write it down. One criterion for measuring the 'goodness' of a lattice rule is the quantity R which is the quadrature error for a certain test function. Bounds on R exist for rank-1 and rank-2 lattice rules, but not for lattice rules of higher rank. For $ 1 \leq m \leq s$m rules and obtain bounds on R for them. These rank-m rules have $ {n^m}r$n and r are relatively prime numbers. In order to obtain these bounds, we make use of a result which shows that R may be considered to be the quadrature error obtained when a modified lattice rule with only r quadrature points is applied to a modified test function. Some numerical results are given.