Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

   
 
 

 

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

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 $\vert{\bf h} \vert = \delta =d+1$. 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.


References [Enhancements On Off] (What's this?)

  • [BC93] M. Beckers and R. Cools, A relation between cubature formulae of trigonometric degree and lattice rules, International Series of Numerical Mathematics., Vol. 112, Numerical Integration IV (H. Brass and G. Hämmerlin, eds.), Birkhäuser Verlag, Basel, 1993, pp. 13-24. MR 95b:65034
  • [CNR99] R. Cools, E. Novak, and K. Ritter, Smolyak's construction of cubature formulas of arbitrary trigonometric degree, Computing 62, no. 2, (1999), 147-162. MR 2000c:41041
  • [CR97] R. Cools and A. Reztsov, Different quality indexes for lattice rules, J. Complexity 13 (1997), 235-258. MR 98e:65011
  • [CS96] R. Cools and I. H. Sloan, Minimal cubature formulae of trigonometric degree, Math. Comp. 65, no. 216, (1996), 1583-1600. MR 97a:65025
  • [Fro77] K. K. Frolov, On the connection between quadrature formulas and sublattices of the lattice of integral vectors, Dokl. Akad. Nauk SSSR 232 (1977), 40-43, (Russian) Soviet Math. Dokl. 18 (1977), 37-41 (English). MR 55:272
  • [GL87] P. M. Gruber and C. G. Lekkerkerker, Geometry of numbers, North-Holland, Amsterdam, 1987. MR 88j:11034
  • [KR95] B. V. Klyuchnikov and A. V. Reztsov, A relation between cubature formulas and densest lattice packings, East J. Approx. 1, no. 4, (1995), 557-570. MR 97g:41045
  • [LS97] J. N. Lyness and I. H. Sloan, Cubature rules of prescribed merit, SIAM J. Numer. Anal. 34, no. 2, (1997), 586-602. MR 97m:65052
  • [LS93] J. N. Lyness and T. Sørevik, Lattice rules by component scaling, Math. Comp. 61, no. 204, (1993), 799-820. MR 94a:65011
  • [Lyn88] J. N. Lyness, Some comments on quadrature rule construction criteria, International Series of Numerical Mathematics., Vol. 85, Numerical Integration III (G. Hämmerlin and H. Brass, eds.), Birkhäuser Verlag, Basel, 1988, pp. 117-129. MR 91b:65028
  • [Lyn89] J. N. Lyness, An introduction to lattice rules and their generator matrices, IMA J. Numer. Anal. 9 (1989), 405-419. MR 91b:65029
  • [Min67] H. Minkowski, Gesammelte Abhandlungen, Reprint (originally published in 2 volumes, Leipzig, 1911), Chelsea Publishing Company, 1967.
  • [Mys85] I. P. Mysovskikh, Quadrature formulae of the highest trigonometric degree of accuracy, Zh. Vychisl. Mat. i Mat. Fiz. 25 (1985), 1246-1252 (Russian). U.S.S.R. Comput. Maths. Math. Phys. 25 (1985), 180-184 (English). MR 87b:65030
  • [Mys87] I. P. Mysovskikh, Cubature formulas that are exact for trigonometric polynomials, Dokl. Akad. Nauk SSSR 296 (1987), 28-31 (Russian). Soviet Math. Dokl. 36 (1988), 229-232 (English). MR 89b:41038
  • [Mys88] I. P. Mysovskikh, Cubature formulas that are exact for trigonometric polynomials, Metody Vycisl. 15 (1988), 7-19 (Russian). MR 90a:65050
  • [Nos85] M. V. Noskov, Cubature formulae for the approximate integration of periodic functions, Metody Vycisl. 14 (1985), 15-23 (Russian). MR 90f:65038
  • [Nos88a] M. V. Noskov, Cubature formulae for the approximate integration of functions of three variables, Zh. Vychisl. Mat. Mat. Fiz. 28 (1988), 1583-1586 (Russian). U.S.S.R. Comput. Maths. Math. Phys. 28 (1988), 200-202 (English). MR 90j:65042
  • [Nos88b] M. V. Noskov, Formulas for the approximate integration of periodic functions, Metody Vycisl. 15 (1988), 19-22 (Russian). CMP 21:03
  • [Nos91] M. V. Noskov, On the construction of cubature formulae of higher trigonometric degree, Metody Vycisl. 16 (1991), 16-23 (Russian).
  • [NS96] M. V. Noskov and A. R. Semenova, Cubature formulae of high trigonometric accuracy for periodic functions of four variables, Comp. Math. Math. Phys. 36, no. 10, (1996), 1325-1330. MR 97h:65029
  • [Sem96] A. R. Semenova, Computing experiments for construction of cubature formulae of high trigonometric accuracy, Cubature Formulas and Their Applications (Russian) (Ufa) (M. D. Ramazanov, ed.), 1996, pp. 105-115.
  • [SJ94] I. H. Sloan and S. Joe, Lattice methods for multiple integration, Oxford University Press, 1994. MR 98a:65026

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 41A55, 41A63, 42A10, 65D32

Retrieve articles in all journals with MSC (2000): 41A55, 41A63, 42A10, 65D32


Additional Information

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

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

American Mathematical Society