Abstract
A lattice rule is a quadrature rule for integration over ans-dimensional hypercube that employsN abscissas located on a lattice, chosen to conform to certain specifications. In this paper we determine the numberv s(N) of distinctN-points-dimensional lattice rules. We show that, in general,
equality being attained if and only ifM andN are mutually prime. This is used to establish that, whenL has prime factor expansion\(\prod\limits_j {p_j^{t_J } ,} \), then\(v_s (L) = \prod\limits_j {v_s (p_j^{t_J } )'} \) where
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References
G. H. Hardy and E. M. Wright 1954.An Introduction to the Theory of Numbers, Clarendon Press, Oxford, 1954, p. 238 et seq.
J. N. Lyness 1989.An introduction to lattice rules and their generator matrices, to appear in IMA J. Num. Anal., vol. 8.
J. N. Lyness and W. Newman 1989.A classification of lattice rules using the reciprocal lattice generator matrix, Argonne National Laboratory Report ANL-89/20, Argonne, Illinois.
I. H. Sloan 1985.Lattice methods for multiple integration, J. Computational and Applied Math., 12 and 13 (1985), pp. 131–143.
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This work was supported by the Applied Mathematical Sciences subprogram of the Office of Energy Research, U.S. Department of Energy, under contract W-31-109-Eng-38, and by the Norwegian Council for Humanities and Science.
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Lyness, J.N., Sørevik, T. The number of lattice rules. BIT 29, 527–534 (1989). https://doi.org/10.1007/BF02219238
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DOI: https://doi.org/10.1007/BF02219238