MATHEMATICS OF COMPUTATION VOLUME 59, NUMBER 200 OCTOBER 1992, PAGES 557-568 . ON COMPUTING THE LATTICE RULE CRITERION R STEPHEN JOE AND IAN H. SLOAN Abstract. Lattice rules are integration rules for approximating integrals of pe- riodic 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 Ns~l terms, where TV is the number of quadrature points. Although various bounds involving R are known, a procedure for cal- culating 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 OiNnx) operations, where nx is the largest 'invariant' of the rule, the use of this asymptotic ex- pansion allows the operation count to be reduced to OiN). A complete error analysis for the asymptotic expansion is given. The results of some calculations of R are also given. 1. Introduction Lattice rules were developed in [15, 16, and 17] for the numerical evaluation of integrals of the form If= f fix)dx, Jus where Us = {x £ Rs : 0 < xk < 1, 1 < k < s} is the half-open unit cube in 5 dimensions, and / is assumed to be 1-periodic in each of its 5 variables. Lattice rules are equal-weight rules of the form (i-i) G/=i¿/(x,), in which the abscissa set {xn, ... , x;v_ i} consists of all the points in Us that also belong to a given 'integration lattice'. A lattice is a discrete set of points in R* such that the sum and difference of every point in the set also belongs to the set; the lattice is an integration lattice if it contains the integer lattice Zs as a sublattice. A lattice rule with N distinct abscissae is said to be of order N. Received by the editor February 7, 1991. 1991 Mathematics Subject Classification. Primary 65D30, 65D32. The continuing financial support of the Australian Research Council is gratefully acknowledged. The second author has also been partly supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University. Cg) 1992 American Mathematical Society 0025-5718/92 $1.00+ $.25 per page 557 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use