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Determination of the rank of an integration lattice

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Abstract

The continuing and widespread use of lattice rules for high-dimensional numerical quadrature is driving the development of a rich and detailed theory. Part of this theory is devoted to computer searches for rules, appropriate to particular situations.

In some applications, one is interested in obtaining the (lattice) rank of a lattice rule Q(Λ) directly from the elements of a generator matrix B (possibly in upper triangular lattice form) of the corresponding dual lattice Λ⊥. We treat this problem in detail, demonstrating the connections between this (lattice) rank and the conventional matrix rank deficiency of modulo p versions of B.

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Author notes

  1. J. N. Lyness & S. Joe

    Present address: School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia

Authors and Affiliations

  1. Mathematics and Computer Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL, 60439, USA

    J. N. Lyness

  2. Department of Mathematics, The University of Waikato, Private Bag 3105, Hamilton, New Zealand

    S. Joe

Authors
  1. J. N. Lyness
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  2. S. Joe
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Correspondence to S. Joe.

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AMS subject classification (2000)

65D30

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Lyness, J., Joe, S. Determination of the rank of an integration lattice . Bit Numer Math 48, 79–93 (2008). https://doi.org/10.1007/s10543-008-0161-4

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  • DOI: https://doi.org/10.1007/s10543-008-0161-4

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