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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

On Cartesian products of good lattices


Author: S. K. Zaremba
Journal: Math. Comp. 30 (1976), 546-552
MSC: Primary 65D30
MathSciNet review: 0423770
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Abstract: Good lattices yield a powerful method of computing multiple integrals. Asymptotically, a lattice generated by one good lattice point is much more efficient than a Cartesian product of such lattices. However, if the number of dimensions is large, this does not always apply to the case when the number of points remains within reasonable limits. Examples of such products of two or three lattices being more efficient than good lattices generated by single lattice points are systematically presented. Additional symmetries of Cartesian products of lattices offer a further advantage when the integrand has to be symmetrized beforehand.


References [Enhancements On Off] (What's this?)

  • [1] Edmund Hlawka, Zur angenäherten Berechnung mehrfacher Integrale, Monatsh. Math. 66 (1962), 140–151 (German). MR 0143329 (26 #888)
  • [2] Gershon Kedem and S. K. Zaremba, A table of good lattice points in three dimensions, Numer. Math. 23 (1974), 175–180. MR 0373239 (51 #9440)
  • [3] N. M. KOROBOV, Number-Theoretic Methods in Approximate Analysis, Fizmatgiz, Moscow, 1963. (Russian) MR 28 #716.
  • [4] Dominique Maisonneuve, Recherche et utilisation des “bons treillis”. Programmation et résultats numériques, Applications of number theory to numerical analysis (Proc. Sympos., Univ. Montréal, Montreal, Que., 1971) Academic Press, New York, 1972, pp. 121–201 (French, with English summary). MR 0343529 (49 #8270)
  • [5] A. I. Saltykov, Tables for evaluating multiple integrals by the method of optimal coefficients, Ž. Vyčisl. Mat. i Mat. Fiz. 3 (1963), 181–186 (Russian). MR 0150976 (27 #962)
  • [6] S. C. Zaremba, Good lattice points, discrepancy, and numerical integration, Ann. Mat. Pura Appl. (4) 73 (1966), 293–317. MR 0218018 (36 #1107)
  • [7] S. K. Zaremba, La méthode des “bons treillis” pour le calcul des intégrales multiples, Applications of number theory to numerical analysis (Proc. Sympos., Univ. Montreal, Montreal, Que., 1971) Academic Press, New York, 1972, pp. 39–119 (French, with English summary). MR 0343530 (49 #8271)
  • [8] S. K. Zaremba, Good lattice points modulo composite numbers, Monatsh. Math. 78 (1974), 446–460. MR 0371845 (51 #8062)
  • [9] Stanisław K. Zaremba, L’erreur dans le calcul des intégrales doubles par la méthode des bons treillis, Demonstratio Math. 8 (1975), no. 3, 347–364 (French). MR 0381264 (52 #2161)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1976-0423770-X
PII: S 0025-5718(1976)0423770-X
Article copyright: © Copyright 1976 American Mathematical Society