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Notes on integration and integer sublattices

Authors: J. N. Lyness, T. Sørevik and P. Keast
Journal: Math. Comp. 56 (1991), 243-255
MSC: Primary 65D32
MathSciNet review: 1052101
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Abstract: A lattice rule is a quadrature rule over an s-dimensional hypercube, using N abscissas located on an integration lattice. In this paper we study sublattices and superlattices of integration lattices and of integer lattices. We exploit the properties of generator matrices of a lattice to provide an easy and elegant description of the relation between a lattice and a sublattice of given order. We also obtain necessary and sufficient criteria for existence of sublattices and information about the number of these.

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  • [1] S. Haber, Numerical evaluation of multiple integrals, SIAM Rev. 12 (1970), 481-526. MR 0285119 (44:2342)
  • [2] N. M. Korobov, The approximate computation of multiple integrals, Dokl. Akad. Nauk SSSR 124 (1959), 1207-1210. (Russian) MR 0104086 (21:2848)
  • [3] J. N. Lyness, An introduction to lattice rules and their generator matrices, IMA J. Numer. Anal. 9 (1989), 405-419. MR 1011399 (91b:65029)
  • [4] J. N. Lyness and W. Newman, A classification of lattice rules using the reciprocal lattice generator matrix, Argonne National Laboratory Report ANL-89/20, Argonne, Illinois, 1989.
  • [5] J. N. Lyness and T. Sørevik, The number of lattice rules, BIT 29 (1989), 527-534. MR 1009654 (91a:65059)
  • [6] M. Newman, Integral matrices, Academic Press, New York, 1972. MR 0340283 (49:5038)
  • [7] H. Niederreiter, Quasi-Monte Carlo methods for multidimensional numerical integration, Numerical Integration III (G. Hämmerlin and H. Brass, eds.), ISNM, vol. 85, Birkhäuser Verlag, 1988, pp. 157-171. MR 1021532 (91f:65008)
  • [8] A. Schrijver, Theory of linear and integer programming, Wiley, 1986. MR 874114 (88m:90090)
  • [9] I. H. Sloan and P. J. Kachoyan, Lattice methods for multiple integration: theory, error analysis and examples, SIAM J. Numer. Anal. 24 (1987), 116-128. MR 874739 (88e:65023)

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Article copyright: © Copyright 1991 American Mathematical Society

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