The ultratriangular form for prime-power lattice rules |
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Institution: | Department of Mathematics, The University of Waikato, Private Bag 3105, Hamilton, New Zealand |
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Abstract: | A recent development in the theory of lattice rules has been the introduction of the unique ultratriangular D-Z form for prime-power rules. It is known that any lattice rule may be decomposed into its Sylow p-components. These components are prime-power rules, each of which has a unique ultratriangular form. By reassembling these ultratriangular forms in a defined way, it is possible to obtain a canonical form for any lattice rule. A special case occurs when the ultratriangular forms of each of the Sylow p-components have a consistent set of column indices. In this case, it is possible to obtain a unique canonical D-Z form. Given the column indices and the invariants for an ultratriangular form, we may obtain a formula for the number of ultratriangular forms, and hence the number of prime-power lattice rules, having these column indices and invariants. |
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