MIXMAX generator explained

Class:	pseudorandom number generator
Data:	Array
Time:	O(n)
Best-Time:	O(n)
Average-Time:	O(n)
Space:	O(n)
Optimal:	Yes

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] It is genuine 64-bit generator. The period of the generator is

10 ^{4389} and the Kolmogorov entropy is 8679 . 2 for the matrix size 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] and a PYTHIA interface. ^[8] It has been recently tested extensively on very wide variety of platforms, as part of the CLHEP/Geant4 release. EU-funded MIXMAX project ^[9]

An analysis by L’Ecuyer, Wambergue and Bourceret,^[10] see also,^[11] showed that MIXMAX 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.

External links

• The open source MIXMAX C/C++ source code on hepforge.org
• William L. Dunn, J. Kenneth Shultis, (2022). Exploring Monte Carlo Methods, 2nd edition, Elsevier Science, .
• K. Anagnostopoulos, (2014). Computational Physics, Lulu.com, .

Notes and References

1. 10.1016/0021-9991(91)90015-D. On the Monte Carlo simulation of physical systems. Journal of Computational Physics. 97. 2. 566. 1991. Savvidy. G.K. Ter-Arutyunyan-Savvidy. N.G. 1991JCoPh..97..566S.
2. K. . Savvidy . 2015 . 1403.5355. The MIXMAX Random Number Generator . . 196 . 161–165 . 10.1016/j.cpc.2015.06.003 . 2015CoPhC.196..161S . 16908633 .
3. K. . Savvidy . G. . Savvidy . 2015 . 1510.06274. Spectrum and Entropy of C-systems MIXMAX Random Number Generator . . 91 . 33–38 . 10.1016/j.chaos.2016.05.003 . 2016CSF....91...33S . 119291387 .
4. Web site: boost. proj-www.boost.org.
5. Web site: CLHEP. proj-clhep.web.cern.ch.
6. Web site: Geant4 . 15 December 2022 . proj-clhep.web.cern.ch.
7. Web site: ROOT - ROOT::Math::MixMaxEngine Class. root.cern.ch. 2016-04-09.
8. Web site: PYTHIA - PYTHIA::Random::MixMaxRndm class. thep.lu.se. 2022-01-01.
9. Web site: Fastest random number generator could cut energy bills. commission.europa.eu/index_en.
10. Web site: Pierre . L’Ecuyer . Paul . Wambergue . Erwan . Bourceret . September 22, 2017 . Spectral Analysis of the MIXMAX Random Number Generators .
11. N. . Martirosyan . K. . Savvidy . G. . Savvidy . Nov 19, 2018 . Spectral Test of the MIXMAX Random Number Generator . . 118 . 242–248 . 10.1016/j.chaos.2018.11.024. 1806.05243. 51687163 .