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Spectral test of the MIXMAX random number generators. (English) Zbl 1442.65003
Summary: An important statistical test on the pseudo-random number generators is called the spectral test. The test is aimed at answering the question of distribution of the generated pseudo-random vectors in dimensions \(d\) that are larger than the genuine dimension of a generator \(N\). In particular, the default MIXMAX generators have various dimensions: \(N=8,17,240\) and higher. Therefore the spectral test is important to perform in dimensions \(d>8\) for \(N=8\) generator, \(d>17\) for \(N=17\) and \(d>240\) for \(N=240\) generator. These tests have been performed by L’Ecuyer and collaborators. When \(d>N\) the vectors of the generated numbers fall into the parallel hyperplanes and the distances between them can be larger than the genuine “resolution” of the MIXMAX generators, which is \(l=2^{-61}\). The aim of this article is to further study the spectral properties of the MIXMAX generators, to investigate the dependence of the spectral properties of the MIXMAX generators as a function of their internal parameters and in particular their dependence on the parameter \(m\). We found that the best spectral properties are realized when \(m\) is between \(2^{24}\) and \(2^{36}\), a range which is inclusive of the value of the \(N=17\) generator. We also provide the alternative parameters for the generators, \(N=8\) and \(N=240\) with \(m\) in this optimized range.
MSC:
65C10 Random number generation in numerical analysis
11K45 Pseudo-random numbers; Monte Carlo methods
Software:
MIXMAX; TestU01
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