# Some Bounds on the Figure of Merit of a Lattice Rule

• T. N. Langtry
Conference paper
Part of the Lecture Notes in Statistics book series (LNS, volume 127)

## Abstract

Lattice rules are quasi-Monte Carlo quadrature rules for multiple integrals defined on the s-dimensional cube [0,1)s that generalise the more widely-known method of good lattice points by allowing more than a single generating vector for the set of quadrature points. It is known that any s-dimensional lattice rule may be expressed as a multiple sum in terms of at most s generating vectors. The minimal number of generating vectors required in this form for a given rule is called its rank. A rank 1 rule which has a generator with at least one component that is a unit is called simple. A key problem in the application of lattice rules is the identification of rules of various ranks that perform well with respect to certain standard criteria, one of these being the ‘figure of merit’ ρ. Various strategies have been used in conducting computer searches for rules that perform well with respect to this criterion, and some ‘good’ (with respect to ρ) rules have been found as a result. However, for a variety of reasons, constructive approaches are also of interest. In recent work the author has investigated the connections between the theories of Diophantine approximation and lattice rules. It was shown that bounds on the figure of merit of a rank 1 simple rule with a given generating vector could be expressed in terms of parameters determined by the best simultaneous Diophantine approximations to an (s-1)-dimensional projection of the generating vector. These bounds were interpreted as indicating that generators for good rules might be constructed from rational vectors that are poorly approximated, in a particular sense, by rational vectors with lower denominators. In this paper these results are extended to higher-rank rules.

## References

1. [1]
N. S. Bakhvalov. Approximate computation of multiple integrals. Vestnik Moskov. Univ. Ser. Mat. Meh. Astr. Fix. Him., 4:3–18, 1959.Google Scholar
2. [2]
I. Borosh and H. Niederreiter. Optimal multipliers for pseudo-random number generation by the linear congruential method. BIT, 23:65–74, 1983.
3. [3]
J. W. S. Cassels. An Introduction to Diophantine Approximation. Cambridge University Press, Cambridge, UK, 1957.
4. [4]
J. C. Lagarias. Some new results in simultaneous diophantine approximation. In P. Ribenboim, editor, Proceedings of the Queen’s University Number Theory Conference 1979, volume 54 of Queen’s Papers in Pure and Applied Mathematics, pages 453–474, 1980.Google Scholar
5. [5]
T. N. Langtry. An application of Diophantine approximation to the construction of rank 1 lattice quadrature rules. Math. Comp., 65:1635–1662, 1996.
6. [6]
H. Niederreiter. Random Number Generation and Quasi-Monte Carlo Methods. SIAM (Society for Industrial and Applied Mathematics), Philadelphia, Pennsylvania.Google Scholar
7. [7]
I. H. Sloan and S. Joe. Lattice Methods for Multiple Integration. Oxford University Press, Oxford, 1994.
8. [8]
I. H. Sloan and J. N. Lyness. The representation of lattice quadrature rules as multiple sums. Math. Comp., 52:81–94, 1989.
9. [9]
S. K. Zaremba. Good lattice points, discrepancy and numerical integration. Ann. Mat. Pura Appl., 73:293–317, 1966.