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MIXMAX generator

A pseudorandom number generator.
Class	pseudorandom number generator
Data structure	Array
Worst-case performance	O(n)
Best-case performance	O(n)
Average performance	O(n)
Worst-case space complexity	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]

The first realization of MIXMAX generator as a Fortran code, as well the first statistical tests were performed by Norayr Akopov. ^[2] The abbreviation MIXMAX for the created generator was first introduced in this article.

A fast implementation in C/C++ of the generator was developed by Konstantin Savvidy.^[3] It is genuine 64-bit generator. The period of the generator is ${\displaystyle 10^{4389}}$ and the Kolmogorov entropy is ${\displaystyle 8679.2}$ for the matrix size ${\displaystyle N=240}$.^[4] 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,^[5] has been chosen as the default generator in CLHEP^[6] for use in Geant4^[7] and there exists a ROOT interface ^[8] and a PYTHIA interface. ^[9] It has been recently tested extensively on very wide variety of platforms, as part of the CLHEP/Geant4 release. EU-funded MIXMAX project ^[10]

An analysis by L’Ecuyer, Wambergue and Bourceret,^[11] see also,^[12] 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.

• 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, ISBN 978-0-12-819739-4.
• K. Anagnostopoulos, (2014). Computational Physics, Lulu.com, ISBN 978-1312464414.