Fast component-by-component construction of rank-1 lattice rules with a non-prime number of points

Abstract

The component-by-component construction algorithm constructs the generating vector for a rank-1 lattice one component at a time by minimizing the worst-case error in each step. This algorithm can be formulated elegantly as a repeated matrix-vector product, where the matrix-vector product expresses the calculation of the worst-case error in that step. As was shown in an earlier paper, this matrix-vector product can be done in time O ( n log ( n ) ) and with memory O ( n ) when the number of points n is prime. Here we extend this result to general n to obtain a total construction cost of O ( sn log ( n ) ) and memory of O ( n ) for a rank-1 lattice in s dimensions with n points. We thus obtain the same big-Oh result as for n prime. As was the case for n prime, the main calculation cost is significantly reduced by using fast Fourier transforms in the matrix-vector calculation. The number of fast Fourier transforms is dependent on the number of divisors of n and the number of prime factors of n. It is believed that the intrinsic structure present in rank-1 lattices and exploited by this fast construction method will deliver new insights in the applicability of these lattices.