Reducing the construction cost of the component-by-component construction of good lattice rules

The construction of randomly shifted rank-$1$ lattice rules, where the number of points $n$ is a prime number, has recently been developed by Sloan, Kuo and Joe for integration of functions in weighted Sobolev spaces and was extended by Kuo and Joe and by Dick to composite numbers. To construct $d$-dimensional rules, the shifts were generated randomly and the generating vectors were constructed component-by-component at a cost of $O(n^2d^2)$ operations. Here we consider the situation where $n$ is the product of two distinct prime numbers $p$ and $q$. We still generate the shifts randomly but we modify the algorithm so that the cost of constructing the, now two, generating vectors component-by-component is only $O(n(p+q)d^2)$ operations. This reduction in cost allows, in practice, construction of rules with millions of points. The rules constructed again achieve a worst-case strong tractability error bound, with a rate of convergence $O(p^{-1+\delta}q^{-1/2})$ for $\delta>0$.