Abstract
Improved estimates are established regarding the accuracy which can be achieved by a suitable choice of generator in a single-generator lattice quadrature rule (as used in the “method of good lattice points”) in the general case wherem, the number of quadrature points, is not necessarily prime. The result obtained for the general case is asymptotically the same as the best currently-known result for the prime case. However, it is also shown that when these rules are applied to some customary test functions the mean error (over different rules with the same number of points) can be arbitrarily large compared to the corresponding mean value for rules with a comparable but prime value ofm. These mean values are of interest in relation to computerised searches for good generators.
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Bourdeau, M., Pitre, A.: Tables of good lattices for four and five dimensions. Numer. Math.47, 39–43 (1985).
Disney, S., Sloan, I. H.: Error bounds for the method of good lattice points. Math. Comp. To appear.
Hua, L. K., Wang, Y.: Applications of Number Theory to Numerical Analysis. Berlin-Heidelber-New York: Springer. 1981.
Kedem, G., Zaremba, S. K.: A table of good lattice points in three dimensions. Number. Math.23, 175–180 (1974).
Maisonneuve, D.: Recherche et utilisation des ‘Bon Treillis’. Programmation et résultats numeriques. In: Applications of Number Theory to Numerical Analysis (S. K. Zaremba, ed.), pp. 121–201. New York: Academic Press. 1972.
Niederreiter, H.: Quasi-Monte Carlo methods and pseudo-random numbers. Bull. Amer. Math. Soc.84, 957–1041 (1978).
Niederreiter, H.: Quasi-Monte Carlo methods for numerical integration. In: Numerical Integration III (H. Brass andG. Hämmerlin eds.), pp. 157–171. Basel: Birkhäuser. 1988.
Niederreiter, H.: Existence of good lattice points in the sense of Hlawka. Mh. Math.86, 203–219 (1978).
Saltykov, A. I.: Tables for computing multiple integrals by the methods of optimal coefficients. Ž. Vyčisl. Mat. i Mat. Fiz.3, 181–186 (1963), (=U.S.S.R. Computational Math. and Math. Phys.3, 235–242 (1963)).
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Disney, S. Error bounds for rank 1 lattice quadrature rules modulo composites. Monatshefte für Mathematik 110, 89–100 (1990). https://doi.org/10.1007/BF01302778
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DOI: https://doi.org/10.1007/BF01302778