Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

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



Triangular canonical forms for lattice
rules of prime-power order

Authors: J. N. Lyness and S. Joe
Journal: Math. Comp. 65 (1996), 165-178
MSC (1991): Primary 65D30, 65D32
MathSciNet review: 1325873
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we develop a theory of $t$-cycle $D-Z$ representations for $s$-dimensional lattice rules of prime-power order. Of particular interest are canonical forms which, by definition, have a $D$-matrix consisting of the nontrivial invariants. Among these is a family of triangular forms, which, besides being canonical, have the defining property that their $Z$-matrix is a column permuted version of a unit upper triangular matrix. Triangular forms may be obtained constructively using sequences of elementary transformations based on elementary matrix algebra. Our main result is to define a unique canonical form for prime-power rules. This ultratriangular form is a triangular form, is easy to recognize, and may be derived in a straightforward manner.

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

  • [H62] E. Hlawka, Zur angenäherten Berechnung mehrfacher Integrale, Monatsh. Math. 66 (1962), 140--151. MR 26:888.
  • [HW81] L.K. Hua and Y. Wang, Applications of number theory to numerical analysis, Springer-Verlag, Berlin, 1981. MR 83g:10034.
  • [K59] N.M. Korobov, The approximate computation of multiple integrals, Dokl. Akad. Nauk SSSR 124 (1959), 1207--1210. (Russian) MR 21:2848.
  • [L93] J.N. Lyness, The canonical forms of a lattice rule, Numerical Integration IV, ISNM 112 (H. Brass and G. Hämmerlin, eds.), Birkhäuser, Basel, 1993, pp. 225--240. MR 94j:65035.
  • [LK95] J.N. Lyness and P. Keast, Application of the Smith normal form to the structure of lattice rules, SIAM J. Matrix Anal. Appl. 16 (1995), 218--231. CMP 95:06.
  • [LSø93] J.N. Lyness and T. Sørevik, Lattice rules by component scaling, Math. Comp. 61 (1993), 799--820. MR 94a:65011.
  • [N73] H. Niederreiter, Zur quantitativen Theorie der Gleichverteilung, Monatsh. Math. 77 (1973), 55--62. MR 47:4944.
  • [N78] H. Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer. Math. Soc. 84 (1978), 957--1041. MR 80d:65016.
  • [N88] H. Niederreiter, Quasi-Monte Carlo methods for multidimensional numerical integration, Numerical integration III, ISNM 85 (H. Brass and G. Hämmerlin, eds.), Birkhäuser, Basel, 1988, pp. 157--171. MR 91f:65008.
  • [SJ94] I.H. Sloan and S. Joe, Lattice methods for multiple integration, Clarendon Press, Oxford, 1994.
  • [SL89] I.H. Sloan and J.N. Lyness, The representation of lattice quadrature rules as multiple sums, Math. Comp. 52 (1989), 81--94. MR 90a:65053.
  • [SL90] I.H. Sloan and J.N. Lyness, Lattice rules: projection regularity and unique representations, Math. Comp. 54 (1990), 649--660. MR 91a:65062.

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 65D30, 65D32

Retrieve articles in all journals with MSC (1991): 65D30, 65D32

Additional Information

J. N. Lyness
Affiliation: Mathematics and Computer Science Division, Argonne National Laboratory, 9700 South Cass Avenue, Argonne, Illinois 60439

S. Joe
Affiliation: Department of Mathematics and Statistics, The University of Waikato, Private Bag 3105, Hamilton, New Zealand

Received by editor(s): August 16, 1994
Received by editor(s) in revised form: February 17, 1995
Additional Notes: This work was supported in part by the Mathematical, Information, and Computational Sciences Division subprogram of the Office of Computational and Technology Research, U.S. Department of Energy, under Contract W-31-109-Eng-38.
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society