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A Constructive Approach to Lattice Rule Canonical Forms

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Abstract

The rank and invariants of a general lattice rule are conventionally defined in terms of the group-theoretic properties of the rule. Here we give a constructive definition of the rank and invariants using integer matrices. This underpins a nonabstract algorithm set in matrix algebra for obtaining the Sylow p-decomposition of a lattice rule. This approach is particularly useful when it is not known whether the form in which the lattice rule is specified is canonical or even repetitive. A new set of necessary and sufficient conditions for recognizing a canonical form is given.

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Authors and Affiliations

  1. Mathematics and Computer Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL, 60439, USA. email: lyness@mcs.anl.gov

    J. N. Lyness

  2. Department of Mathematics, The University of Waikato, Private Bag 3105, Hamilton, New Zealand. email: stephenj@math.waikato.ac.nz

    S. Joe

Authors
  1. J. N. Lyness
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  2. S. Joe
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Lyness, J.N., Joe, S. A Constructive Approach to Lattice Rule Canonical Forms. BIT Numerical Mathematics 39, 701–715 (1999). https://doi.org/10.1023/A:1022391207786

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  • DOI: https://doi.org/10.1023/A:1022391207786

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