Some properties of rank$2$ lattice rules
Authors:
J. N. Lyness and I. H. Sloan
Journal:
Math. Comp. 53 (1989), 627637
MSC:
Primary 65D32
DOI:
https://doi.org/10.1090/S00255718198909823696
MathSciNet review:
982369
Fulltext PDF Free Access
Abstract  References  Similar Articles  Additional Information
Abstract: A rank2 lattice rule is a quadrature rule for the (unit) sdimensional hypercube, of the form \[ Qf = (1/{n_1}{n_2})\sum \limits _{{j_1} = 1}^{{n_1}} {\sum \limits _{{j_2} = 1}^{{n_2}} {\bar f({j_1}{{\mathbf {z}}_1}/{n_1} + {j_2}{{\mathbf {z}}_2}/{n_2}),} } \] which cannot be reexpressed in an analogous form with a single sum. Here $\bar f$ is a periodic extension of f, and ${{\mathbf {z}}_1}$, ${{\mathbf {z}}_2}$ are integer vectors. In this paper we discuss these rules in detail; in particular, we categorize a special subclass, whose leading one and twodimensional projections contain the maximum feasible number of abscissas. We show that rules of this subclass can be expressed uniquely in a simple tricycle form.

H. Conroy, "Molecular Schrödinger equation, VIII: A new method for the evaluation of multidimensional integrals," J. Chem. Phys., v. 47, 1967, pp. 53075318.
 Seymour Haber, Numerical evaluation of multiple integrals, SIAM Rev. 12 (1970), 481–526. MR 285119, DOI https://doi.org/10.1137/1012102
 Seymour Haber, Parameters for integrating periodic functions of several variables, Math. Comp. 41 (1983), no. 163, 115–129. MR 701628, DOI https://doi.org/10.1090/S0025571819830701628X
 Edmund Hlawka, Zur angenäherten Berechnung mehrfacher Integrale, Monatsh. Math. 66 (1962), 140–151 (German). MR 143329, DOI https://doi.org/10.1007/BF01387711
 Loo Keng Hua and Yuan Wang, Applications of number theory to numerical analysis, SpringerVerlag, BerlinNew York; Kexue Chubanshe (Science Press), Beijing, 1981. Translated from the Chinese. MR 617192
 P. Keast, Optimal parameters for multidimensional integration, SIAM J. Numer. Anal. 10 (1973), 831–838. MR 353636, DOI https://doi.org/10.1137/0710068
 N. M. Korobov, Properties and calculation of optimal coefficients, Soviet Math. Dokl. 1 (1960), 696–700. MR 0120768
 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
 Harald Niederreiter, QuasiMonte Carlo methods and pseudorandom numbers, Bull. Amer. Math. Soc. 84 (1978), no. 6, 957–1041. MR 508447, DOI https://doi.org/10.1090/S000299041978145327
 Ian H. Sloan, Lattice methods for multiple integration, Proceedings of the international conference on computational and applied mathematics (Leuven, 1984), 1985, pp. 131–143. MR 793949, DOI https://doi.org/10.1016/03770427%2885%29900123
 Ian H. Sloan and Philip J. Kachoyan, Lattice methods for multiple integration: theory, error analysis and examples, SIAM J. Numer. Anal. 24 (1987), no. 1, 116–128. MR 874739, DOI https://doi.org/10.1137/0724010
 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 https://doi.org/10.1090/S00255718198909474683
 S. C. Zaremba, Good lattice points, discrepancy, and numerical integration, Ann. Mat. Pura Appl. (4) 73 (1966), 293–317. MR 218018, DOI https://doi.org/10.1007/BF02415091
 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
Retrieve articles in Mathematics of Computation with MSC: 65D32
Retrieve articles in all journals with MSC: 65D32
Additional Information
Article copyright:
© Copyright 1989
American Mathematical Society