Skip to main content
SpringerLink
Log in

A Constructive Approach to Lattice Rule Canonical Forms

  • Published:
BIT Numerical Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. S. Joe and I. H. Sloan, The direct sum of lattice rules, Department of Mathematics and Statistics Research Report No. 30, 1994, The University of Waikato, Hamilton, New Zealand.

    Google Scholar 

  2. J. N. Lyness, The canonical forms of a lattice rule, in Numerical Integration IV, ISNM 112, H. Bräss and G. Hämmerlin, eds., Birkhäuser, Basel, 1993, pp. 225–240.

    Google Scholar 

  3. J. N. Lyness and S. Joe, Triangular canonical forms for lattice rules of prime-power order, Math. Comp., 65 (1996), pp. 165–178.

    Google Scholar 

  4. J. N. Lyness and P. Keast, Application of the Smith normal form to the structure of lattice rules, SIAM J. Matrix Anal. Appl., 16 (1995), pp. 218–231.

    Google Scholar 

  5. H. Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer. Math. Soc., 84 (1978), pp. 957–1041.

    Google Scholar 

  6. H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods, SIAM, Philadelphia, 1992.

    Google Scholar 

  7. A. Schrijver, Theory of Linear and Integer Programming, Wiley, New York, 1986.

    Google Scholar 

  8. I. H. Sloan and S. Joe, Lattice Methods for Multiple Integration, Clarendon Press, Oxford, 1994.

    Google Scholar 

  9. I. H. Sloan and J. N. Lyness, The representation of lattice quadrature rules as multiple sums, Math. Comp., 52 (1989), pp. 81–94.

    Google Scholar 

Download references

Author information

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
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Joe
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

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

Download citation

  • Issue Date:

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

Search

Navigation