Spectral Analysis of the MIXMAX Random Number Generators

@article{LEcuyer2020SpectralAO,
  title={Spectral Analysis of the MIXMAX Random Number Generators},
  author={Pierre L’Ecuyer and Paul Wambergue and Erwan Bourceret},
  journal={INFORMS J. Comput.},
  year={2020},
  volume={32},
  pages={135-144}
}
We study the lattice structure of random number generators of the MIXMAX family, a class of matrix linear congruential generators that produce a vector of random numbers at each step. These generators were initially proposed and justified as close approximations to certain ergodic dynamical systems having the Kolmogorov K-mixing property, which implies a chaotic (fast-mixing) behavior. But for a K-mixing system, the matrix must have irrational entries, whereas for the MIXMAX it has only integer… 

Spectral test of the MIXMAX random number generators

Method for Generating Pseudorandom Sequence of Permutations Based on Linear Congruential Generator

TLDR
Computer implementation of the algorithm for generating PRS of permutations based on LCG with any type of graph of its states has allowed increasing the speed of the generator compared to the permutation generator using the modern Fisher-Yates algorithm.

Review of High-Quality Random Number Generators

TLDR
This paper outlines the Kolmogorov–Anosov theory of mixing in classical mechanical systems, and establishes criteria for deciding which RNG’s are sufficiently good approximations to the ideal mathematical systems that guarantee highest quality.

MULTIPLE STREAMS WITH RECURRENCE-BASED, COUNTER-BASED, AND SPLITTABLE RANDOM NUMBER GENERATORS

TLDR
The need for multiple independent streams and substreams of random numbers, as well as the advantages (and potential pitfalls) of the increasingly popular counter-based and dynamically splittable generators.

References

SHOWING 1-10 OF 36 REFERENCES

The MIXMAX random number generator

  • K. Savvidy
  • Computer Science, Mathematics
    Comput. Phys. Commun.
  • 2015

An Implementation of the Lattice and Spectral Tests for Multiple Recursive Linear Random Number Generators

TLDR
The implementation of theoretical tests to assess the structural properties of simple or combined linear congruential and multiple recursive random number generators are discussed and a package implementing the so-called spectral and lattice tests for such generators is described.

Spectrum and entropy of C-systems MIXMAX random number generator

Bad Lattice Structures for Vectors of Nonsuccessive Values Produced by Some Linear Recurrences

TLDR
This article considers the case where the t values taken are not successive, but separated by lags that are chosen a priori, and gives lower bounds on the distance between hyperplanes.

On the Lattice Structure of a Special Class of Multiple Recursive Random Number Generators

TLDR
The points produced by certain classes of long-period linear multiple recursive random number generators proposed by L.-Y.

On the lattice structure of the add-with-carry and subtract-with-borrow random number generators

TLDR
It is shown that these sequences are essentially equivalent to linear congruential sequences with very large prime moduli, and how the equivalence can be exploited to implement efficient jumping-ahead facilities for the AWC and SWB sequences.

Uniform random number generation

TLDR
Practical ways of generating uniform variates for several classes of generators, such as linear congruential, multiple recursive, digital multistep, Tausworthe, lagged-Fibonacci, generalized feedback shift register, matrix, linear Congruential over fields of formal series, and combined generators are examined.

Orbits and lattices for linear random number generators with composite moduli

TLDR
A generalized spectral test is introduced and it is shown how to apply the test in large dimensions by a recursive procedure based on the fact that such combinations are subgenerators of other MRGs with composite moduli.

Sparse Serial Tests of Uniformity for Random Number Generators

TLDR
For the classes of alternatives that correspond to linear generators, the most efficient tests turn out to have $k \gg n$ (in contrast to what is usually done or recommended in simulation books) and to use overlapping vectors.