LATTICE INTEGRATION RULES OF MAXIMAL RANK FORMED BY COPYING RANK-1 RULES

For integration of periodic functions over the s-dimensional unit cube, theoretical and empirical evidence suggests that certain lattice rules of maximal rank are more effective, as judged by a standard test, than the widely used rules of rank 1. In a rank 1 rule, all the points are generated by taking multiples of a single rational vector, modulo 1, in the manner suggested by Korobov. The rules in question are formed by taking n(s)-copies, with small values of n (principally n = 2), of rules of rank 1. There is also empirical evidence that, by the same criterion, these maximal rank rules are preferable, in dimensions greater than 2, to the rank 2 rules found by Sloan and Walsh.