Review of High-Quality Random Number Generators

Abstract

This is a review of pseudorandom number generators (RNG's) of the highest quality, suitable for use in the most demanding Monte Carlo calculations. All the RNG's we recommend here are based on the Kolmogorov-Anosov theory of mixing in classical mechanical systems, which guarantees under certain conditions and in certain asymptotic limits, that points on the trajectories of these systems can be used to produce random number sequences of exceptional quality. We outline this theory of mixing and establish criteria for deciding which RNG's are sufficiently good approximations to the ideal mathematical systems that guarantee highest quality. The well-known RANLUX (at highest luxury level) and its recent variant RANLUX++ are seen to meet our criteria, and some of the proposed versions of MIXMAX can be modified easily to meet the same criteria.1 High-quality random numbersWe are concerned here with pseudorandom number generators (RNG's), in particular those of the highest quality. It turns out to be difficult to find an operational definition of randomness that can be used to measure the quality of a RNG, that is the degree of independence of the numbers in a given sequence, or to prove that they are indeed independent. The situation for traditional RNG's (not based on Kolmogorov-Anasov mixing) is well described by Donald Knuth in [2]. The book contains a wealth of information about random number generation, but nothing about where the randomness comes from, or how to measure the quality (randomness) of a generator. Now with hindsight, it is not surprising that all the widely-used generators described there were later found to have defects (failing tests of randomness and/or giving incorrect results in Monte Carlo (MC) calculations), with the notable exception of RANLUX, which Knuth does mention briefly in the third edition, but without describing the new theoretical basis.

Challenges in Monte Carlo Event Generator Software for High-Luminosity LHC

“…[75,76] for early reviews of this technique in HEP). The distinctive feature of MC methods is their reliance on the generation of random numbers (or, more precisely, of "pseudo-random" [77] numbers). 2 In particular, the starting point of both MC phase space integration and MC unweighted event generation is the calculation of f ( ) for a large sample of events i ∈{ 1 , … , N } , drawn at random from a known probability density function g( ) .…”

“…Finally, work is also ongoing [180] on the efficient exploitation of GPUs in the pseudo-random number generation libraries that are used in all MC generators (see Ref. [77] for a recent review of these components).…”

Co-simulation of linear congruential generator by using Xilinx system generator and MATLAB Simulink

“…The block diagram represented in Figure 1 contains operations like addition, multiplication, addition, comparison and subtraction. To simplify the work process; the circuit is intended using the 'word lengths' lessening method that has recommended in [13][14][15][16][17][18][19][20][21][22][23][24][25]. Then comparator and subtractor blocks can be merged, as shown in Figure 2.…”

Stochastic Properties of Confidence Ellipsoids after Least Squares Adjustment, Derived from GUM Analysis and Monte Carlo Simulations

“…Nowadays, with modern computers and well-established Random Number Generators (RNG), large samples are easy to generate, see [20,21]. According to the variability of input quantities, pseudorandom sequences are computed, which allows us to evaluate and re-run simulations.…”

“…RNG should have high quality, i.e., provides good approximations of the ideal mathematical system, e.g., has long sequences, shows no gaps in data, fulfils distribution requirements, see [21]. As discussed in [20], the RANLUX (random number generator at highest luxury level) and its recent variant RANLUX++, which are used here, can be considered as representative of such high-quality RNGs.…”