# MIXMAX generator

Class pseudorandom number generator Array O(n) O(n) O(n) O(n)

The MIXMAX generator is a family of pseudorandom number generators (PRNG) and is based on Anosov C-systems (Anosov diffeomorphism) and Kolmogorov K-systems (Kolmogorov automorphism). It was introduced in a 1986 preprint by G. Savvidy and N. Ter-Arutyunyan-Savvidy and published in 1991.[1]

A fast implementation in C/C++ of the generator was developed by Konstantin Savvidy.[2] The period of the generator is ${\displaystyle 10^{4389}}$ and the Kolmogorov entropy is ${\displaystyle 8679.2}$ for the matrix size ${\displaystyle N=240}$.[3] That generator occupies less than 2 kb, and if a smaller generator state is required, a N = 17 version with less than 200 bytes memory requirement also exists.

The generator works on most 64-bit systems, including 64-bit Linux flavors and Intel Mac. It has also been tested on PPC and ARM architectures. The latest version also runs on 32-bit systems and on Windows. The generator is equally usable with C++ programs,[4] has been chosen as the default generator in CLHEP[5] for use in Geant4[6] and there exists a ROOT interface.[7] It has been recently tested extensively on very wide variety of platforms, as part of the CLHEP/Geant4 release.

An analysis by L’Ecuyer, Wambergue and Bourceret,[8] see also,[9] showed that MIXMAX generators, just as all other Multiple Recursive Generators and linear congruential generators, has a lattice structure when the produced random numbers are considered in n - dimensional space larger than the dimension N of the matrix generator, and only in that high dimensions n > N they lie on a set of parallel hyperplanes and determined the maximum distance between the covering hyperplanes.

## References

1. ^ Savvidy, G.K; Ter-Arutyunyan-Savvidy, N.G (1991). "On the Monte Carlo simulation of physical systems". Journal of Computational Physics. 97 (2): 566. Bibcode:1991JCoPh..97..566S. doi:10.1016/0021-9991(91)90015-D.
2. ^ Savvidy, K. (2015). "The MIXMAX Random Number Generator". Computer Physics Communications. 196: 161–165. arXiv:1403.5355. Bibcode:2015CoPhC.196..161S. doi:10.1016/j.cpc.2015.06.003. S2CID 16908633.
3. ^ Savvidy, K.; Savvidy, G. (2015). "Spectrum and Entropy of C-systems MIXMAX Random Number Generator". Chaos, Solitons and Fractals. 91: 33–38. arXiv:1510.06274. Bibcode:2016CSF....91...33S. doi:10.1016/j.chaos.2016.05.003. S2CID 119291387.
4. ^ "boost". proj-www.boost.org.
5. ^ "CLHEP". proj-clhep.web.cern.ch.
6. ^ "Geant4". proj-clhep.web.cern.ch.
7. ^ "ROOT - ROOT::Math::MixMaxEngine Class". root.cern.ch. Retrieved 2016-04-09.
8. ^ L’Ecuyer, Pierre; Wambergue, Paul; Bourceret, Erwan (September 22, 2017). "Spectral Analysis of the MIXMAX Random Number Generators" (PDF). `{{cite journal}}`: Cite journal requires `|journal=` (help)
9. ^ Martirosyan, N.; Savvidy, K.; Savvidy, G. (Nov 19, 2018). "Spectral Test of the MIXMAX Random Number Generator". Chaos, Solitons and Fractals. 118: 242–248. arXiv:1806.05243. doi:10.1016/j.chaos.2018.11.024. S2CID 51687163.