Systematic searches for good multiple recursive random number generators

This paper proposes two systematic ways to search for good MRGs, in terms of the lattice structure, in a partially exhaustive manner. One is a backward method and the other is a forward method. Several good MRGs of order 1, 2, and 3, with modulus 2³¹–1, found from these two methods are presented. When computational efficiency is the major concern, another group of MRGs where the approximate factoring technique can be applied are generated. Roughly speaking, the execution time of the k-term MRG is k times that of the one-term PMMCG. By adapting to the approximate factoring technique, there is a reduction of around 40% in execution time, with a trade-off of less satisfactory lattice structure of the RNs produced. These generators should be useful for computer simulation studies with different objectives, and the proposed methods are suitable for finding good MRGs of higher order.