Determination of the rank of an integration lattice

Abstract

The continuing and widespread use of lattice rules for high-dimensional numerical quadrature is driving the development of a rich and detailed theory. Part of this theory is devoted to computer searches for rules, appropriate to particular situations.In some applications, one is interested in obtaining the (lattice) rank of a lattice rule Q(Λ) directly from the elements of a generator matrix B (possibly in upper triangular lattice form) of the corresponding dual lattice Λ ⊥ . We treat this problem in detail, demonstrating the connections between this (lattice) rank and the conventional matrix rank deficiency of modulo p versions of B. (2000): 65D30. AMS subject classification