Parallel Streams of Nonlinear Congruential Pseudorandom Numbers

AbstractThis paper deals with the general nonlinear congruential method for generating uniform pseudorandom numbers, in which permutation polynomials over finite prime fields play an important role. It is known that these pseudorandom numbers exhibit an attractive equidistribution and statistical independence behavior. In the context of parallelized simulation methods, a large number of parallel streams of pseudorandom numbers with strong mutual statistical independence properties are required. In the present paper, such properties of parallelized nonlinear congruential generators are studied based on the discrepancy of certain point sets. Upper and lower bounds for the discrepancy both over the full period and over (sufficiently large) parts of the period are established. The method of proof rests on the classical Weil bound for exponential sums

Fractional jumps: complete characterisation and an explicit infinite family

In this paper we provide a complete characterisation of transitive fractional jumps by showing that they can only arise from transitive projective automorphisms. Furthermore, we prove that such construction is feasible for arbitrarily large dimension by exhibiting an infinite class of projectively primitive polynomials whose companion matrix can be used to define a full orbit sequence over an affine space

Distribution of Random Streams for Simulation Practitioners

International audienceThere is an increasing interest in the distribution of parallel random number streamsin the high-performance computing community particularly, with the manycore shift. Even ifwe have at our disposal statistically sound random number generators according to the latestand thorough testing libraries, their parallelization can still be a delicate problem. Indeed, aset of recent publications shows it still has to be mastered by the scientific community. Withthe arrival of multi-core and manycore processor architectures on the scientist desktop, modelerswho are non-specialists in parallelizing stochastic simulations need help and advice in distributingrigorously their experimental plans and replications according to the state of the art in pseudo-random numbers parallelization techniques. In this paper, we discuss the different partitioningtechniques currently in use to provide independent streams with their corresponding software. Inaddition to the classical approaches in use to parallelize stochastic simulations on regular processors,this paper also presents recent advances in pseudo-random number generation for general-purposegraphical processing units. The state of the art given in this paper is written for simulationpractitioners