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Four-dimensional lattice rules generated by skew-circulant matrices

Authors: J. N. Lyness and T. Sørevik
Journal: Math. Comp. 73 (2004), 279-295
MSC (2000): Primary 65D32; Secondary 42A10
Published electronically: June 3, 2003
MathSciNet review: 2034122
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Abstract: We introduce the class of skew-circulant lattice rules. These are $s$-dimensional lattice rules that may be generated by the rows of an $s \times s$ skew-circulant matrix. (This is a minor variant of the familiar circulant matrix.) We present briefly some of the underlying theory of these matrices and rules. We are particularly interested in finding rules of specified trigonometric degree $d$. We describe some of the results of computer-based searches for optimal four-dimensional skew-circulant rules. Besides determining optimal rules for $\delta = d+1 \leq 47,$ we have constructed an infinite sequence of rules ${\hat Q}(4,\delta )$ that has a limit rho index of $27/34 \approx 0.79$. This index is an efficiency measure, which cannot exceed 1, and is inversely proportional to the abscissa count.

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Additional Information

J. N. Lyness
Affiliation: Mathematics and Computer Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439-4844, and School of Mathematics, The University of New South Wales, Sydney 2052, Australia

T. Sørevik
Affiliation: Department of Informatics, University of Bergen, N-5020 Bergen, Norway

Keywords: Multidimensional cubature, optimal lattice rules, skew-circulant matrices, $K$-optimal rules, and optimal trigonometric rules.
Received by editor(s): October 5, 2001
Received by editor(s) in revised form: May 23, 2002
Published electronically: June 3, 2003
Additional Notes: This work was supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Advanced Scientific Computing Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.
Article copyright: © Copyright 2003 University of Chicago