Triangular canonical forms for lattice

rules of prime-power order

Authors:
J. N. Lyness and S. Joe

Journal:
Math. Comp. **65** (1996), 165-178

MSC (1991):
Primary 65D30, 65D32

DOI:
https://doi.org/10.1090/S0025-5718-96-00691-6

MathSciNet review:
1325873

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we develop a theory of -cycle representations for -dimensional lattice rules of prime-power order. Of particular interest are canonical forms which, by definition, have a -matrix consisting of the nontrivial invariants. Among these is a family of *triangular* forms, which, besides being canonical, have the defining property that their -matrix is a column permuted version of a unit upper triangular matrix. Triangular forms may be obtained constructively using sequences of elementary transformations based on elementary matrix algebra. Our main result is to define a unique canonical form for prime-power rules. This *ultratriangular* form is a triangular form, is easy to recognize, and may be derived in a straightforward manner.

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Additional Information

**J. N. Lyness**

Affiliation:
Mathematics and Computer Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439

Email:
lyness@mcs.anl.gov

**S. Joe**

Affiliation:
Department of Mathematics and Statistics, The University of Waikato, Private Bag 3105, Hamilton, New Zealand

Email:
stephenj@hoiho.math.waikato.ac.nz

DOI:
https://doi.org/10.1090/S0025-5718-96-00691-6

Received by editor(s):
August 16, 1994

Received by editor(s) in revised form:
February 17, 1995

Additional Notes:
This work was supported in part by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.

Article copyright:
© Copyright 1996
American Mathematical Society