Testing random number generators by numerical modeling of an exactly solvable problem

Monte Carlo numerical modeling of a problem with a known exact solution is used to test linear multiplicative generators with modulus M = 2 31 - 1 for their applicability in parallel computing. The deviations of the calculated values of the rotational temperature from the known theoretical values are compared with the possible errors of the Monte Carlo method due to the finiteness of statistical samples. In addition, sample correlation coefficients are used to estimate true correlation coefficients between the values of the rotational temperature obtained on different processors with different multipliers, as well as in the case where additional samples were drawn at the terminal state on each processor in order to increase the size of the total sample. For this purpose, by means of special partial averaging, the random values of the rotational temperature were transformed into approximately normally distributed variables; then, for the variables obtained, true correlation coefficients were estimated by sample correlation coefficients. It was discovered that 204 different multipliers suggested by G.S. Fishman, L. R. Moore exhibit the best performance when used in a parallel implementation of the Monte Carlo method: all deviations are less than the theoretical Monte Carlo errors. Moreover, no correlations between the random variables produced by generators with different multipliers were detected. This suggests that generators with different multipliers produce independent sequences of pseudorandom numbers. However, drawing additional samples on each processor, which is frequently done to increase the size of the total sample, gives rise to correlations. Moreover, in many such cases, the theoretical errors of the Monte Carlo method for multipliers in the bottom part of the list proposed by Fishman and Moore are less than the values of temperature deviations and, therefore, should not be used in this way. © 2009 Allerton Press, Inc.