Notes on lattice rules

An 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 ru...

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Bibliographic Details
Published in:Journal of Complexity 2003, Vol.19 (3), p.321-331
Main Author: Lyness, J.N.
Format: Article
Language:English
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Summary:An 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= δ s / 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.
ISSN:0885-064X
1090-2708
DOI:10.1016/S0885-064X(03)00005-0