  In this paper, we develop and test a fast numerical algorithm, called MDI-LR,
for efficient implementation of quasi-Monte Carlo lattice rules for computing
$d$-dimensional integrals of a given function. It is based on the idea of
converting and improving the underlying lattice rule into a tensor product rule
by an affine transformation and adopting the multilevel dimension iteration
approach which computes the function evaluations (at the integration points) in
the tensor product multi-summation in cluster and iterates along each
(transformed) coordinate direction so that a lot of computations can be reused.
The proposed algorithm also eliminates the need for storing integration points
and computing function values independently at each point. Extensive numerical
experiments are presented to gauge the performance of the algorithm MDI-LR and
to compare it with standard implementation of quasi-Monte Carlo lattice rules.
It is also showed numerically that the algorithm MDI-LR can achieve a
computational complexity of order $O(N^2d^3)$ or better, where $N$ represents
the number of points in each (transformed) coordinate direction and $d$
standard for the dimension. Thus, the algorithm MDI-LR effectively overcomes
the curse of dimensionality and revitalizes QMC lattice rules for
high-dimensional integration.
