Notes on lattice rules

An elementary introduction to lattices integration lattices and lattice rules is fol lowed 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 estab lished Any s dimensional cubature rule can be associated with an index s s N where is the enhanced degree of the rule andN 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 against Two of these plots are presented They convey a clear idea of the relative cost e ectiveness of various individual rules and sequences of rules The Integration Lattice and the Lattice Rule The rst two sections contain background material Much more detailed descriptions of some of this may be found in Ly and SlJo An s dimensional lattice conventionally denoted by is a set of points R containing no limit points and satisfying