A lower triangular Hermite normal form for projection-regular lattice rules
by
Muni V. Reddy
The University of the South Pacific
Coauthors: Stephen Joe (The University of Waikato)

Lattice rules are equal-weight quadrature rules which are used in the approximation of integrals over the s-dimensional unit cube [0, 1]^s. The structure of such rules has been studied using two different approaches. One of them is based on its representation in t-cycle D-Z forms and the other approach is based on the generator matrices of the integration lattice and its dual.

The approach that is based on the generator matrix of the dual lattice has previously made the assumption that its representaion in the so-called Hermite normal form is upper triangular. However, since the unique Z for the special case of projection-regular rules is upper triangular, the corresponding matrix for the dual lattice turns out to be lower triangular. This suggests that the lower triangular Hermite normal form might be an appropriate form to study. We shall consider such representations for projection-regular rules and obtain a unique representation for the generator matrix of the dual lattice for such rules.

Date received: September 27, 2001