The generation of lattice points for numerical multiple integration (Q1263226)

In a recent paper \textit{I. H. Sloan} and \textit{T. R. Osborn} [J. Comput. Appl. Math. 17, 181-196 (1987; Zbl 0629.65021)] proposed a ``lattice rule'' for numerical approximation of a multiple integral extended over \({\mathbb{R}}^ n\). The corresponding cubature rule is based on an infinite lattice of points, truncated at some suitable radius. In 1981 \textit{N. J. A. Sloane} [IEEE Trans. Inf. Theory IT-27, 327-338 (1981; Zbl 0483.05022)] has found some important lattices. In this paper the author shows, how the generator matrices for the lattices found there may be used to generate the corresponding lattices of points required in the numerical multiple integration. The method proposed for generating the lattice points essentially involves solving sets of Diophantine equations by exhaustive search. The author gives an indication of the computational efficiency of the proposed method.