Uniform random number generators: a review <s> IMPLEMENTATIONS <s> The twisted GFSR generators proposed in a previous article have a defect in k -distribution for k larger than the order of recurrence. In this follow up article, we introduce and analyze a new TGFSR variant having better k -distribution property. We provide an efficient algorithm to obtain the order of equidistribution, together with a tight upper bound on the order. We discuss a method to search for generators attaining this bound, and we list some of these such generators. The upper bound turns out to be (sometimes far) less than the maximum order of equidistribution for a generator of that period length, but far more than that for a GFSR with a working are of the same size. <s> BIB001 </s> Uniform random number generators: a review <s> IMPLEMENTATIONS <s> Abstract Following some remarks on the quality of pseudorandom number generators commonly used in Monte Carlo calculations in computational physics, we offer a portable Fortran 77 implementation of a high-quality generator called RANLUX (for LUXury RANdom numbers), using the algorithm of Martin Luscher described in an accompanying article. The implementation allows the user to select different quality or luxury levels, where higher quality requires somewhat longer computing time for the generation. There is a convenient way of initialization (appropriate also for massively parallel Monte Carlo computations) as well as two different methods of restarting from a break point. <s> BIB002 </s> Uniform random number generators: a review <s> IMPLEMENTATIONS <s> A new algorithm called Mersenne Twister (MT) is proposed for generating uniform pseudorandom numbers. For a particular choice of parameters, the algorithm provides a super astronomical period of 2 19937 −1 and 623-dimensional equidistribution up to 32-bit accuracy, while using a working area of only 624 words. This is a new variant of the previously proposed generators, TGFSR, modified so as to admit a Mersenne-prime period. The characteristic polynomial has many terms. The distribution up to v bits accuracy for 1 ≤ v ≤ 32 is also shown to be good. An algorithm is also given that checks the primitivity of the characteristic polynomial of MT with computational complexity O(p 2 ) where p is the degree of the polynomial. We implemented this generator in portable C-code. It passed several stringent statistical tests, including diehard. Its speed is comparable to other modern generators. Its merits are due to the efficient algorithms that are unique to polynomial calculations over the two-element field. <s> BIB003
Computer implementations of generators with good theoretical support, well tested, and reasonably efficient, include those of the combined MRGs of L'Ecuyer (1996a), L'Ecuyer and Andres (1997) and L'Ecuyer (1997b), the combined Tausworthe generators of L'Ecuyer (1996b) and L'Ecuyer (1997e), the twisted GFSRs of BIB001 and BIB003 , and perhaps RANLUX in BIB002 