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