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Lattice Methods for Multiple Integration: Theory, Error Analysis and Examples

Ian H. Sloan and Philip J. Kachoyan
SIAM Journal on Numerical Analysis
Vol. 24, No. 1 (Feb., 1987), pp. 116-128
Stable URL: http://www.jstor.org/stable/2157389
Page Count: 13
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Lattice Methods for Multiple Integration: Theory, Error Analysis and Examples
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Abstract

This paper is concerned with numerical integration over the unit cube in s dimensions of functions that have reasonably smooth periodic extensions in each dimension. The concept of a lattice rule is introduced, and error bounds are developed in terms of the dual lattice. Examples of lattice rules discussed in the paper are the rectangle rule, the number-theoretic "good-lattice" methods of Korobov and others, the body-centred cubic rule, and a generalization of the latter to a family {Wnr}. The Wnr rules appear to have interesting extrapolation properties, both for fixed r and for r = n.

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