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Mathematics of Computation

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An application of Diophantine approximation
to the construction of rank-1 lattice
quadrature rules

Author: T. N. Langtry
Journal: Math. Comp. 65 (1996), 1635-1662
MSC (1991): Primary 65D30; Secondary 65D32, 11J25, 11J70
Supplement: Additional information related to this article.
MathSciNet review: 1351203
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Abstract: Lattice quadrature rules were introduced by Frolov (1977), Sloan (1985) and Sloan and Kachoyan (1987). They are quasi-Monte Carlo rules for the approximation of integrals over the unit cube in ${\mathbb {R}}^{s}$ and are generalizations of `number-theoretic' rules introduced by Korobov (1959) and Hlawka (1962)---themselves generalizations, in a sense, of rectangle rules for approximating one-dimensional integrals, and trapezoidal rules for periodic integrands. Error bounds for rank-1 rules are known for a variety of classes of integrands. For periodic integrands with unit period in each variable, these bounds are conveniently characterized by the figure of merit $\rho $, which was originally introduced in the context of number-theoretic rules. The problem of finding good rules of order $N$ (that is, having $N$ nodes) then becomes that of finding rules with large values of $\rho $. This paper presents a new approach, based on the theory of simultaneous Diophantine approximation, which uses a generalized continued fraction algorithm to construct rank-1 rules of high order.

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Additional Information

T. N. Langtry
Affiliation: School of Mathematical Sciences, University of Technology, Sydney, PO Box 123, Broadway, NSW, 2007, Australia

Keywords: Numerical quadrature, numerical cubature, multiple integration, lattice rules, continued fractions, Diophantine approximation
Received by editor(s): February 22, 1995
Received by editor(s) in revised form: July 26, 1995
Additional Notes: This work was carried out as part of a doctoral program under the supervision of Prof. I. H. Sloan and Dr. S. A. R. Disney of the University of New South Wales. The author expresses his appreciation of their guidance and support. The comments of an anonymous referee also helped to improve the paper.
Article copyright: © Copyright 1996 American Mathematical Society

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