Three- and four-dimensional \(K\)-optimal lattice rules of moderate trigonometric degree (Q2723528)

In this interesting and well written paper, the authors record their search for lattice rules of optimal degree. Recall that an \(s\)-dimensional lattice rule has the form NEWLINE\[NEWLINEQ[f]= {1\over d_1d_2\dots d_t} \sum^{d_1}_{j_1=1} \sum^{d_2}_{j_2=1} \cdots\sum^{d_t}_{j_t=1} \left(\left\{ {j _1\over d_1}{\mathbf z}_1 +{j_2 \over d_2}{\mathbf z}_2+ \cdots+{j_t \over d_t}{\mathbf z}_t\right\} \right)NEWLINE\]NEWLINE where \(t\geq 1\), each \({\mathbf z}_j\) is an \(s\)-tuple of integers, and each \(d\) is a positive integer. The rule \(Q\) is said to have degree \(d\) if \(Q\) integrates exactly all \(s\)-dimensional trigonometric polynomials of degree \(\leq d\). That is, \(Q\) integrates \(\exp(2\pi i{\mathbf h}\cdot {\mathbf x})\) over \([0,1]^s\) provided \({\mathbf h}=(h_1,h_2, \dots,h_s)\) is an \(s\)-tuple of integers with \(\sum^s_{j=1} |h_j|\leq d\). The number of abscissas in the rule, namely the number of points at which \(f\) must be evaluated in \(Q[f]\), is denoted by \(N(Q)\) and the enhanced degree of \(Q\) is denoted by \(\delta= d+1\). It is obviously of interest to minimize \(N(Q)\) for a given enhanced degree \(\delta\). An optimal rule is one that attains this minimum. The authors review the search for optimal rules, especially optimal rules of lattice type. They note that for \(s=1\) and 2, there is an optimal rule that is a lattice rule, no matter what is \(\delta\). A similar situation is true for \(\delta\leq 4\) and all \(s\). They describe their computer search for optimal lattice rules of a certain type for \(s=3\) and 4. They list new rules for \(\delta\) up to 30 for \(s=3\), and \(\delta\) up to 24 for \(s=4\). The paper is essential reading for anyone interested in cubature rules.