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Mathematics of Computation

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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
MathSciNet review: 1325873
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Abstract: In this paper we develop a theory of $t$-cycle $D-Z$ representations for $s$-dimensional lattice rules of prime-power order. Of particular interest are canonical forms which, by definition, have a $D$-matrix consisting of the nontrivial invariants. Among these is a family of triangular forms, which, besides being canonical, have the defining property that their $Z$-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

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

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