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

DOI:
https://doi.org/10.1090/S0025-5718-96-00691-6

MathSciNet review:
1325873

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we develop a theory of -cycle representations for -dimensional lattice rules of prime-power order. Of particular interest are canonical forms which, by definition, have a -matrix consisting of the nontrivial invariants. Among these is a family of *triangular* forms, which, besides being canonical, have the defining property that their -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.

**[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**.

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

Email:
lyness@mcs.anl.gov

**S. Joe**

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

Email:
stephenj@hoiho.math.waikato.ac.nz

DOI:
https://doi.org/10.1090/S0025-5718-96-00691-6

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