, Volume 110, Issue 2, pp 89-100

Error bounds for rank 1 lattice quadrature rules modulo composites

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Abstract

Improved estimates are established regarding the accuracy which can be achieved by a suitable choice of generator in a single-generator lattice quadrature rule (as used in the “method of good lattice points”) in the general case wherem, the number of quadrature points, is not necessarily prime. The result obtained for the general case is asymptotically the same as the best currently-known result for the prime case. However, it is also shown that when these rules are applied to some customary test functions the mean error (over different rules with the same number of points) can be arbitrarily large compared to the corresponding mean value for rules with a comparable but prime value ofm. These mean values are of interest in relation to computerised searches for good generators.