## Four-dimensional lattice rules generated by skew-circulant matrices

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- by J. N. Lyness and T. Sørevik;
- Math. Comp.
**73**(2004), 279-295 - DOI: https://doi.org/10.1090/S0025-5718-03-01534-5
- Published electronically: June 3, 2003

## 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.

## References

- Ronald Cools and James N. Lyness,
*Three- and four-dimensional $K$-optimal lattice rules of moderate trigonometric degree*, Math. Comp.**70**(2001), no. 236, 1549–1567. MR**1836918**, DOI 10.1090/S0025-5718-01-01326-6 - Ronald Cools and Ian H. Sloan,
*Minimal cubature formulae of trigonometric degree*, Math. Comp.**65**(1996), no. 216, 1583–1600. MR**1361806**, DOI 10.1090/S0025-5718-96-00767-3 - Philip J. Davis,
*Circulant matrices*, A Wiley-Interscience Publication, John Wiley & Sons, New York-Chichester-Brisbane, 1979. Pure and Applied Mathematics. MR**543191** - J. N. Lyness,
*An introduction to lattice rules and their generator matrices*, IMA J. Numer. Anal.**9**(1989), no. 3, 405–419. MR**1011399**, DOI 10.1093/imanum/9.3.405 - J. N. Lyness and R. Cools,
*Notes on a search for optimal lattice rules*, in Cubature Formulae and Their Applications (M. V. Noskov, ed.), pp. 259–273, Krasnoyarsk STU, 2000. Also available as Argonne National Laboratory Preprint ANL/MCS-P829-0600. - H. Minkowski,
*Gesammelte Abhandlungen*, reprint (originally published in 2 volumes, Leipzig, 1911), Chelsea Publishing Company, 1967. - I. P. Mysovskikh,
*Cubature formulas that are exact for trigonometric polynomials*, Metody Vychisl.**15**(1988), 7–19, 178 (Russian). MR**967440** - M. V. Noskov,
*Cubature formulas for functions that are periodic with respect to some of the variables*, Zh. Vychisl. Mat. i Mat. Fiz.**31**(1991), no. 9, 1414–1419 (Russian); English transl., Comput. Math. Math. Phys.**31**(1991), no. 9, 110–114 (1992). MR**1145212** - J. A. C. Weideman,
*Computing integrals of the complex error function*, Mathematics of Computation 1943–1993: a half-century of computational mathematics (Vancouver, BC, 1993) Proc. Sympos. Appl. Math., vol. 48, Amer. Math. Soc., Providence, RI, 1994, pp. 403–407. MR**1314879**, DOI 10.1090/psapm/048/1314879 - T. Sørevik and J. F. Myklebust,
*GRISK: An Internet based search for K-optimal lattice rules*, in Proceedings of PARA2000,*Lecture Notes in Computer Science*1947, pp. 196–205, Springer Verlag, 2001.

## Bibliographic 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
- Email: lyness@mcs.anl.gov
**T. Sørevik**- Affiliation: Department of Informatics, University of Bergen, N-5020 Bergen, Norway
- Email: tor.sorevik@ii.uib.no
- 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.
- © Copyright 2003 University of Chicago
- Journal: Math. Comp.
**73**(2004), 279-295 - MSC (2000): Primary 65D32; Secondary 42A10
- DOI: https://doi.org/10.1090/S0025-5718-03-01534-5
- MathSciNet review: 2034122