Three- and four-dimensional $K$-optimal lattice rules of moderate trigonometric degree

Authors:
Ronald Cools and James N. Lyness

Journal:
Math. Comp. **70** (2001), 1549-1567

MSC (2000):
Primary 41A55, 41A63, 42A10; Secondary 65D32

DOI:
https://doi.org/10.1090/S0025-5718-01-01326-6

Published electronically:
May 14, 2001

MathSciNet review:
1836918

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A systematic search for optimal lattice rules of specified trigonometric degree $d$ over the hypercube $[0,1)^s$ has been undertaken. The search is restricted to a population $K(s,\delta )$ of lattice rules $Q(\Lambda )$. This includes those where the dual lattice $\Lambda ^\perp$ may be generated by $s$ points $\bf h$ for each of which $|\textbf {h} | = \delta =d+1$. The underlying theory, which suggests that such a restriction might be helpful, is presented. The general character of the search is described, and, for $s=3$, $d \leq 29$ and $s=4$, $d \leq 23$, a list of $K$-optimal rules is given. It is not known whether these are also optimal rules in the general sense; this matter is discussed.

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Additional Information

**Ronald Cools**

Affiliation:
Department of Computer Science, K. U. Leuven, Celestijnenlaan 200A, B-3001 Heverlee, Belgium

MR Author ID:
51325

ORCID:
0000-0002-5567-5836

Email:
Ronald.Cools@cs.kuleuven.ac.be

**James N. Lyness**

Affiliation:
Mathematics and Computer Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439 and School of Mathematics, University of New South Wales, Sydney 2052 Australia

Email:
lyness@mcs.anl.gov

Received by editor(s):
November 29, 1999

Published electronically:
May 14, 2001

Additional Notes:
The second author was supported by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Advanced Scientific Computing Research, U.S. Dept. of Energy, under Contract W-31-109-Eng-38.

Article copyright:
© Copyright 2001
University of Chicago and Katholieke Universiteit Leuven