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Mathematics of Computation

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Bounds on the lattice rule criterion $ R$

Author: Stephen Joe
Journal: Math. Comp. 61 (1993), 821-831
MSC: Primary 65D30
MathSciNet review: 1195427
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Abstract: Lattice rules are used for the numerical integration of periodic functions over the s-dimensional unit cube. They are normally classified according to their 'rank'; in simple terms, the rank of a lattice rule is the minimum number of sums required to write it down. One criterion for measuring the 'goodness' of a lattice rule is the quantity R which is the quadrature error for a certain test function. Bounds on R exist for rank-1 and rank-2 lattice rules, but not for lattice rules of higher rank. For $ 1 \leq m \leq s$, we shall look at certain rank-m rules and obtain bounds on R for them. These rank-m rules have $ {n^m}r$ quadrature points, where n and r are relatively prime numbers. In order to obtain these bounds, we make use of a result which shows that R may be considered to be the quadrature error obtained when a modified lattice rule with only r quadrature points is applied to a modified test function. Some numerical results are given.

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Article copyright: © Copyright 1993 American Mathematical Society