Notes on lattice rules

AbstractAn elementary introduction to lattices, integration lattices and lattice rules is followed by a description of the role of the dual lattice in assessing the trigonometric degree of a lattice rule. The connection with the classical lattice-packing problem is established: any s-dimensional cubature rule can be associated with an index ρ=δs/s!N, where δ is the enhanced degree of the rule and N its abscissa count. For lattice rules, this is the packing factor of the associated dual lattice with respect to the unit s-dimensional octahedron.An individual cubature rule may be represented as a point on a plot of ρ against δ. Two of these plots are presented. They convey a clear idea of the relative cost-effectiveness of various individual rules and sequences of rules

Lyness Three and four-dimensional K-optimal lattice rules of moderate trigonometric degree

Abstract. A systematic search for optimal lattice rules of specified trigonometric degree d over the hypercube [0, 1) s has been undertaken. The search is restricted to a population K(s, δ) of lattice rules Q(Λ). This includes those where the dual lattice Λ ⊥ may be generated by s points h for each of which |h | = δ = d + 1. The underlying theory, which suggests that such a restriction might be helpful, is presented. The general character of the search is described, and, for s =3,d ≤ 29 and s =4,d ≤ 21, a list of K-optimal rules is given. It is not known whether these are also optimal rules in the general sense; this matter is discussed. 1