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 rul es of maximal rank are more effective, as judged by a standard test, t han 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 forme d by taking n(s)-copies, with small values of n (principally n = 2), o f rules of rank 1. There is also empirical evidence that, by the same criterion, these maximal rank rules are preferable, in dimensions grea ter than 2, to the rank 2 rules found by Sloan and Walsh.