86 research outputs found
A generalization of short-period Tausworthe generators and its application to Markov chain quasi-Monte Carlo
A one-dimensional sequence is said to be
completely uniformly distributed (CUD) if overlapping -blocks , , are uniformly distributed
for every dimension . This concept naturally arises in Markov chain
quasi-Monte Carlo (QMC). However, the definition of CUD sequences is not
constructive, and thus there remains the problem of how to implement the Markov
chain QMC algorithm in practice. Harase (2021) focused on the -value, which
is a measure of uniformity widely used in the study of QMC, and implemented
short-period Tausworthe generators (i.e., linear feedback shift register
generators) over the two-element field that approximate CUD
sequences by running for the entire period. In this paper, we generalize a
search algorithm over to that over arbitrary finite fields
with elements and conduct a search for Tausworthe generators
over with -values zero (i.e., optimal) for dimension
and small for , especially in the case where , and . We
provide a parameter table of Tausworthe generators over , and
report a comparison between our new generators over and existing
generators over in numerical examples using Markov chain QMC
Improved long-period generators based on linear recurrences modulo 2
Fast uniform random number generators with extremely long periods have been defined and implemented based on linear recurrences modulo 2. The twisted GFSR and the Mersenne twister are famous recent examples. Besides the period length, the statistical quality of these generators is usually assessed via their equidistribution properties. The huge-period generators proposed so far are not quite optimal in this respect. In this article, we propose new generators of that form with better equidistribution and "bit-mixing" properties for equivalent period length and speed. The state of our new generators evolves in a more chaotic way than for the Mersenne twister. We illustrate how this can reduce the impact of persistent dependencies among successive output values, which can be observed in certain parts of the period of gigantic generators such as the Mersenne twister
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