Searching for Large-Order Multiple Recursive Generators

Abstract

Pseudo-random numbers (PRNs) are the basis for almost any statistical simulation and this depends largely on the quality of the pseudo-random number generator (PRNG) used. In this study, we used some results from number theory to propose an efficient method to accelerate the computer search of super-order maximum-period multiple recursive generators (MRGs). We conduct efficient computer searches and identify many efficient and portable MRGs of super-orders, 40751, 50551, and 50873; which respectively have equi-distribution property up to 40751, 50551, and 50873 dimensions, and period lengths of approximately 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">380278.1</sup> , 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">471730.6</sup> , and 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">474729.3</sup> . Using the generalized Mersenne prime algorithm, we extend some known results of some efficient, portable and maximum-period MRGs. In particular, the DX/DL/DS/DT large-order generators are extended to super-order generators. An extensive empirical evaluation shows that these generators behave well when tested with the stringent Small Crush and Crush batteries of the TestU01 package.