The Canonical Forms of a Lattice Rule

Lyness, J. N.

doi:10.1007/978-3-0348-6338-4_18

The Canonical Forms of a Lattice Rule

J. N. Lyness⁶

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Part of the book series: ISNM International Series of Numerical Mathematics ((ISNM,volume 112))

Abstract

Much of the elementary theory of lattice rules may be presented as an elegant application of classical results. These include Kronecker group representation theorem and the Hermite and Smith normal forms of integer matrices. The theory of the canonical form is a case in point. In this paper, some of this theory is treated in a constructive rather than abstract manner. A step-by-step approach that parallels the group theory is described, leading to an algorithm to obtain a canonical form of a rule of prime power order. The number of possible distinct canonical forms is derived, and this is used to determine the number of integration lattices having specified invariants.

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References

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Authors and Affiliations

Mathematics and Computer Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, IL, 60439, USA
Dr. J. N. Lyness

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Dr. J. N. Lyness
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Editors and Affiliations

Institut für Angewandte Mathematik, TU Braunschweig, Pockelsstr. 14, D-3300, Braunschweig, Germany
H. Brass
Mathematisches Institut, Ludwig-Maximilian-Universität, Theresienstr. 39, D-8000, München, Germany
G. Hämmerlin

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Lyness, J.N. (1993). The Canonical Forms of a Lattice Rule. In: Brass, H., Hämmerlin, G. (eds) Numerical Integration IV. ISNM International Series of Numerical Mathematics, vol 112. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6338-4_18

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DOI: https://doi.org/10.1007/978-3-0348-6338-4_18
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-6340-7
Online ISBN: 978-3-0348-6338-4
eBook Packages: Springer Book Archive

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