Spectral Analysis of the MIXMAX Random Number Generators

References

• Afflerbach L, Grothe H (1988) The lattice structure of pseudo-random vectors generated by matrix generators. J. Comput. Appl. Math. 23(1):127–131.Crossref, Google Scholar
• Akopov NZ, Savvidy GK, Savvidy NGTA (1991) Matrix generators for pseudorandom numbers. J. Comput. Phys. 97(2):573–579.Crossref, Google Scholar
• Conway JH, Sloane NJA (1999) Sphere Packings, Lattices and Groups. Grundlehren der Mathematischen Wissenschaften, vol. 290, 3rd ed. (Springer-Verlag, New York).Crossref, Google Scholar
• Couture R, L’Ecuyer P (1994) On the lattice structure of certain linear congruential sequences related to AWC/SWB generators. Math. Comput. 62(206):798–808.Crossref, Google Scholar
• Couture R, L’Ecuyer P (1996) Orbits and lattices for linear random number generators with composite moduli. Math. Comput. 65(213):189–201.Crossref, Google Scholar
• Coveyou RR, MacPherson RD (1967) Fourier analysis of uniform random number generators. J. ACM 14(1):100–119.Crossref, Google Scholar
• Ferrenberg AM, Landau DP, Wong YJ (1992) Monte Carlo simulations: Hidden errors from “good” random number generators. Phys. Rev. Lett. 69(23):3382–3384.Crossref, Google Scholar
• Fincke U, Pohst M (1985) Improved methods for calculating vectors of short length in a lattice, including a complexity analysis. Math. Comput. 44(170):463–471.Crossref, Google Scholar
• Grothe H (1988) Matrixgeneratoren zur erzeugung gleichverteilter pseudozufallsvektoren. Dissertation, Technische Hochschule, Darmstadt, Germany.Google Scholar
• James F (1994) RANLUX: A Fortran implementation of the high-quality pseudorandom number generator of Lüscher. Comput. Phys. Commun. 79(1):111–114.Crossref, Google Scholar
• Knuth DE (1998) The Art of Computer Programming: Seminumerical Algorithms, vol. 2, 3rd ed. (Addison-Wesley, Reading, MA).Google Scholar
• L’Ecuyer P (1990) Random numbers for simulation. Commun. ACM 33(10):85–97.Crossref, Google Scholar
• L’Ecuyer P (1994) Uniform random number generation. Ann. Oper. Res. 53(1):77–120.Crossref, Google Scholar
• L’Ecuyer P (1997) Bad lattice structures for vectors of non-successive values produced by some linear recurrences. INFORMS J. Comput. 9(1):57–60.Link, Google Scholar
• L’Ecuyer P (1999) Tables of linear congruential generators of different sizes and good lattice structure. Math. Comput. 68(225):249–260. Errata accessed October 10, 2018, http://www.iro.umontreal.ca/∼lecuyer/myftp/papers/latrules99Errata.pdf.Crossref, Google Scholar
• L’Ecuyer P (2006) Uniform random number generation. Henderson SG, Nelson BL, eds. Simulation, Handbooks in Operations Research and Management Science (Elsevier, Amsterdam), 55–81.Crossref, Google Scholar
• L’Ecuyer P (2012) Random number generation. Gentle JE, Haerdle W, Mori Y, eds. Handbook of Computational Statistics, 2nd ed. (Springer-Verlag, Berlin), 35–71.Crossref, Google Scholar
• L’Ecuyer P (2017) History of uniform random number generation. Proc. 2017 Winter Simulation Conf. (IEEE Press, Piscataway, NJ), 202–230.Crossref, Google Scholar
• L’Ecuyer P, Couture R (1997) An implementation of the lattice and spectral tests for multiple recursive linear random number generators. INFORMS J. Comput. 9(2):206–217.Link, Google Scholar
• L’Ecuyer P, Simard R (2001) On the performance of birthday spacings tests for certain families of random number generators. Math. Comput. Simul. 55(1–3):131–137.Crossref, Google Scholar
• L’Ecuyer P, Simard R (2007) TestU01: A C library for empirical testing of random number generators. ACM Trans. Math. Software 33(4):Article No. 22.Crossref, Google Scholar
• L’Ecuyer P, Simard R (2014) On the lattice structure of a special class of multiple recursive random number generators. INFORMS J. Comput. 26(2):449–460.Link, Google Scholar
• L’Ecuyer P, Touzin R (2004) On the Deng-Lin random number generators and related methods. Statist. Comput. 14(1):5–9.Crossref, Google Scholar
• L’Ecuyer P, Simard R, Wegenkittl S (2002) Sparse serial tests of uniformity for random number generators. SIAM J. Sci. Comput. 24(2):652–668.Crossref, Google Scholar
• L’Ecuyer P, Munger D, Oreshkin B, Simard R (2017) Random numbers for parallel computers: Requirements and methods, with emphasis on GPUs. Math. Comput. Simul. 135:3–17.Crossref, Google Scholar
• Lüscher M (1994) A portable high-quality random number generator for lattice field theory simulations. Comput. Phys. Commun. 79(1):100–110.Crossref, Google Scholar
• Marsaglia G (1968) Random numbers fall mainly in the planes. Proc. Natl. Acad. Sci. USA 60:25–28.Crossref, Google Scholar
• Marsaglia G, Zaman A (1991) A new class of random number generators. Ann. Appl. Probab. 1(3):462–480.Crossref, Google Scholar
• Niederreiter H (1986) A pseudorandom vector generator based on finite field arithmetic. Math. Jpn. 31:759–774.Google Scholar
• Savvidy GK, Ter-Arutyuntan-Savvidy NG (1991) On the Monte Carlo simulation of physical systems. J. Comput. Phys. 97(2):566–572.Crossref, Google Scholar
• Savvidy KG (2015) The MIXMAX random number generator. Comput. Phys. Commun. 196:161–165.Crossref, Google Scholar
• Savvidy KG (2017) MIXMAX manual. Accessed October 10, 2018, https://www.hepforge.org/archive/mixmax/MANUAL.pdf.Google Scholar
• Savvidy KG, Savvidy GK (2016) Spectrum and entropy of C-systems MIXMAX random number generator. Chaos Solitons Fractals 91:33–38.Crossref, Google Scholar
• Tahmi E (1982) Contribution aux générateurs de valeurs aléatoires. Dissertation, Université des Sciences et Technologies Houari Boumédienne, Algeria.Google Scholar
• Tezuka S, L’Ecuyer P (1992) An analysis of add-with-carry and subtract-with-borrow generators. Proc. 1992 Winter Simulation Conf. (IEEE Press, Piscataway, NJ), 443–447.Crossref, Google Scholar
• Tezuka S, L’Ecuyer P, Couture R (1993) On the add-with-carry and subtract-with-borrow random number generators. ACM Trans. Model. Comput. Simul. 3(4):315–331.Crossref, Google Scholar