Bounds on the lattice rule criterion

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 , we shall look at certain rank-*m* rules and obtain bounds on *R* for them. These rank-*m* rules have 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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DOI:
https://doi.org/10.1090/S0025-5718-1993-1195427-1

Article copyright:
© Copyright 1993
American Mathematical Society