Lattice Methods for Multiple Integration: Theory, Error Analysis and Examples

Abstract

This paper is concerned with numerical integration over the unit cube in s dimensions of functions that have reasonably smooth periodic extensions in each dimension. The concept of a lattice rule is introduced, and error bounds are developed in terms of the dual lattice. Examples of lattice rules discussed in the paper are the rectangle rule, the number-theoretic "good-lattice" methods of Korobov and others, the body-centred cubic rule, and a generalization of the latter to a family $\{ W_{nr} \} $. The $W_{nr} $ rules appear to have interesting extrapolation properties, both for fixed r and for $r = n$.