Triangular canonical forms for lattice rules of prime-power order

Lyness, J.; Joe, S.

doi:10.1090/S0025-5718-96-00691-6

Triangular canonical forms for lattice rules of prime-power order
HTML articles powered by AMS MathViewer

by J. N. Lyness and S. Joe PDF

Math. Comp. 65 (1996), 165-178 Request permission

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

Edmund Hlawka, Zur angenäherten Berechnung mehrfacher Integrale, Monatsh. Math. 66 (1962), 140–151 (German). MR 143329, DOI 10.1007/BF01387711
Loo Keng Hua and Yuan Wang, Applications of number theory to numerical analysis, Springer-Verlag, Berlin-New York; Kexue Chubanshe (Science Press), Beijing, 1981. Translated from the Chinese. MR 617192, DOI 10.1007/978-3-642-67829-5
N. M. Korobov, Approximate evaluation of repeated integrals, Dokl. Akad. Nauk SSSR 124 (1959), 1207–1210 (Russian). MR 0104086
J. N. Lyness, The canonical forms of a lattice rule, Numerical integration, IV (Oberwolfach, 1992) Internat. Ser. Numer. Math., vol. 112, Birkhäuser, Basel, 1993, pp. 225–240. MR 1248407
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, .
J. N. Lyness and T. Sørevik, Lattice rules by component scaling, Math. Comp. 61 (1993), no. 204, 799–820. MR 1185247, DOI 10.1090/S0025-5718-1993-1185247-6
H. Niederreiter, Zur quantitativen Theorie der Gleichverteilung, Monatsh. Math. 77 (1973), 55–62 (German). MR 316397, DOI 10.1007/BF01300529
Harald Niederreiter, Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer. Math. Soc. 84 (1978), no. 6, 957–1041. MR 508447, DOI 10.1090/S0002-9904-1978-14532-7
Harald Niederreiter, Quasi-Monte Carlo methods for multidimensional numerical integration, Numerical integration, III (Oberwolfach, 1987) Internat. Schriftenreihe Numer. Math., vol. 85, Birkhäuser, Basel, 1988, pp. 157–171. MR 1021532, DOI 10.1007/978-3-0348-6398-8_{1}5
I.H. Sloan and S. Joe, Lattice methods for multiple integration, Clarendon Press, Oxford, 1994.
Ian H. Sloan and James N. Lyness, The representation of lattice quadrature rules as multiple sums, Math. Comp. 52 (1989), no. 185, 81–94. MR 947468, DOI 10.1090/S0025-5718-1989-0947468-3
I. H. Sloan and J. N. Lyness, Lattice rules: projection regularity and unique representations, Math. Comp. 54 (1990), no. 190, 649–660. MR 1011443, DOI 10.1090/S0025-5718-1990-1011443-1