On computing the lattice rule criterion $R$

Stephen Joe, Ian H. Sloan
1992 Mathematics of Computation  
Lattice rules are integration rules for approximating integrals of periodic functions over the s-dimensional unit cube. One criterion for measuring the 'goodness' of lattice rules is the quantity R . This quantity is defined as a sum which contains about Ns~l terms, where TV is the number of quadrature points. Although various bounds involving R are known, a procedure for calculating R itself does not appear to have been given previously. Here we show how an asymptotic series can be used to
more » ... in an accurate approximation to R . Whereas an efficient direct calculation of R requires OiNnx) operations, where nx is the largest 'invariant' of the rule, the use of this asymptotic expansion allows the operation count to be reduced to OiN). A complete error analysis for the asymptotic expansion is given. The results of some calculations of R are also given.
doi:10.1090/s0025-5718-1992-1134733-2 fatcat:jiosyyodwre7hcglgrklfqjymi