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On the Distribution of Fractional Linear Congruential Pseudorandom Numbers
Yoshinori TAKEI Toshinori YOSHIKAWA Xi ZHANG
IEICE TRANSACTIONS on Information and Systems
Publication Date: 2003/02/01
Print ISSN: 0916-8532
Type of Manuscript: Special Section PAPER (Special Issue on Selected Papers from LA Symposium)
pseudorandom generators, fractional linear transforms, discrepancy, inversive linear congruent generators, leap-frog,
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As pseudorandom number generators for Monte Carlo simulations, inversive linear congruential generators (ICG) have some advantages compared with traditional linear congruential generators. It has been shown that a sequence generated by an ICG has a low discrepancy even if the length of the sequence is far shorter than its period. In this paper, we formulate fractional linear congruential generators (FCG), a generalized concept of the inversive linear congruential generators. It is shown that the sequence generated by an FCG is a geometrical shift of a sequence from an ICG and satisfies the same upper bounds of discrepancy. As an application of the general formulation, we show that under certain condition, "Leap-Frog technique," a way of splitting a random number sequence to parallel sequences, can be applied to the ICG or FCG with no extra cost on discrepancy.