Design and implementation of efficient and portable Multiple Recursive Generators with few zero coefficients

DX-k, proposed by Deng and Xu [2003], is a special class of Multiple Recursive Generators (MRGs) where all nonzero coefficients of the k-th order recurrence are equal. In particular, a DX-k generator requires only up to four nonzero coefficients in its recurrence equation, hence is very efficient in computation. However, a random number generator with few nonzero coefficients has a drawback that, when the k-dimensional state vector is close to the zero vector, the subsequent numbers generated may stay within a neighborhood of zero for quite many of them before they can break away from this near-zero land, a property apparently not desirable in the sense of randomness. Consequently, two generated sequences using the same DX generator with nearly identical initial state vectors may not depart from each other quickly enough. To avoid the above potential problem, we consider MRGs with very few zero coefficients. To make such generators efficient and portable, we propose selecting the same nonzero value for all coefficients (with at most one exception) in the recurrence equation. With this feature, the proposed generators can be implemented efficiently via a higher-order recurrence of few zero coefficients. Note that the new class of generators is an opposite of the DX generators in terms of the number of nonzero coefficients. Several such generators with the maximum period have been found via computer search and presented in the paper for ready implementation.