A course in computational algebraic number theory
A course in computational algebraic number theory
When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?
Journal of Complexity
Concrete Math
Constructing Randomly Shifted Lattice Rules in Weighted Sobolev Spaces
SIAM Journal on Numerical Analysis
Component-by-component construction of good lattice rules
Mathematics of Computation
On the convergence rate of the component-by-component construction of good lattice rules
Journal of Complexity
Comparison of Point Sets and Sequences for Quasi-Monte Carlo and for Random Number Generation
SETA '08 Proceedings of the 5th international conference on Sequences and Their Applications
Weighted compound integration rules with higher order convergence for all N
Numerical Algorithms
Variance bounds and existence results for randomly shifted lattice rules
Journal of Computational and Applied Mathematics
Constructions of general polynomial lattice rules based on the weighted star discrepancy
Finite Fields and Their Applications
Constructing adapted lattice rules using problem-dependent criteria
Proceedings of the Winter Simulation Conference
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The component-by-component construction algorithm constructs the generating vector for a rank-1 lattice one component at a time by minimizing the worst-case error in each step. This algorithm can be formulated elegantly as a repeated matrix-vector product, where the matrix-vector product expresses the calculation of the worst-case error in that step. As was shown in an earlier paper, this matrix-vector product can be done in time O(nlog(n)) and with memory O(n) when the number of points n is prime. Here we extend this result to general n to obtain a total construction cost of O(snlog(n)) and memory of O(n) for a rank-1 lattice in s dimensions with n points. We thus obtain the same big-Oh result as for n prime. As was the case for n prime, the main calculation cost is significantly reduced by using fast Fourier transforms in the matrix-vector calculation. The number of fast Fourier transforms is dependent on the number of divisors of n and the number of prime factors of n. It is believed that the intrinsic structure present in rank-1 lattices and exploited by this fast construction method will deliver new insights in the applicability of these lattices.