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Uniform random number generation. (English) Zbl 0843.65004

This article provides a survey of pseudorandom number generators (i.e., deterministic recursive formulas from \(\{1, 2,\dots,m\}\) to \(\{1, 2,\dots,m\}\) for a given large prime number \(m\)) that attempt to produce a sequence (here, \(k/m\)) which approximates a sequence of truly independent random variables uniformly distributed over the interval \([0, 1]\). Aside from necessary definitions and descriptions of most commonly used generators, the center of attention is paid to discussing criteria for a good generator such as: statistical properties (testing the generator versus known laws of probability), long period (in demand for massive simulations in physics and chemistry), speed and low memory requirement (to make the generator practical for currently available computers).
A word of caution is given to an inexperienced user about a host of packages known to offer notoriously bad generators. The author tested 9 generators versus 10 standard tests (including the poker test, the runs-up test, the birthday spacing test) and recommends the multiple recursive generator (MRG): \(x_n= (a_1 x_{n- 1}+\cdots+ a_k x_{n- k})\text{ mod }m\) with \(m= 2^{31}- 1\), \(k= 5\), \(a_1= 107374182\), \(a_5= 104480\), \(a_2= a_3= a_4= 0\). An extensive bibliography makes this survey a good source for both novice and a specialist. It would be of interest to learn about good generators with long period \(m\) of order \(\sim 10^p\), \(p\geq 100\).

MSC:

65C10 Random number generation in numerical analysis
11K45 Pseudo-random numbers; Monte Carlo methods
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