Application of the Smith Normal Form to the Structure of Lattice Rules

Abstract

Two independent approaches to the theory of the lattice rule have been exploited at length in the literature. One is based on the generator matrix A of the lattice $\Lambda $ whose elements provide the abscissas of Q. The other, based on the t-cycle form $Q(\Lambda )f$ of Sloan and Lyness [Math. Comput., 52 (1989), pp. 81–94], leads to a canonical form for Q. In this paper, a close connection between these approaches is demonstrated. This connection reflects the close relation between the Kronecker decomposition theorem for Abelian groups and the Smith normal form of an integer matrix. It is shown that the invariants of the canonical form of $Q(\Lambda )f$ coincide with the elements of the Smith normal form of $B = (A^T )^{ - 1} $, the reciprocal lattice generator matrix. This fact may be used to provide a straightforward solution to the previously intransigent problem of identifying and removing a repetition in the general t-cycle form.