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

The representation of lattice quadrature rules as multiple sums


Authors: Ian H. Sloan and James N. Lyness
Journal: Math. Comp. 52 (1989), 81-94
MSC: Primary 65D32
MathSciNet review: 947468
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Abstract: We provide a classification of lattice rules. Applying elementary group theory, we assign to each s-dimensional lattice rule a rank m and a set of positive integer invariants $ {n_1},{n_2}, \ldots ,{n_s}$. The number $ \nu (Q)$ of abscissas required by the rule is the product $ {n_1}{n_2} \cdots {n_s}$, and the rule may be expressed in a canonical form with m independent summations. Under this classification an N-point number-theoretic rule in the sense of Korobov and Conroy is a rank $ m = 1$ rule having invariants N, 1, 1,..., 1, and the product trapezoidal rule using $ {n^s}$ points is a rank $ m = s$ rule having invariants n, n,..., n. Besides providing a canonical form, we give some of the properties of copy rules and of projections into lower dimensions.


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DOI: http://dx.doi.org/10.1090/S0025-5718-1989-0947468-3
PII: S 0025-5718(1989)0947468-3
Article copyright: © Copyright 1989 American Mathematical Society