Extensions of Atanassov’s Methods for Halton Sequences

  • Henri Faure
  • Christiane Lemieux
  • Xiaoheng Wang
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 23)


We extend Atanassov’s methods for Halton sequences in two different directions: (1) in the direction of Niederreiter (t, s) − sequences, (2) in the direction of generating matrices for Halton sequences. It is quite remarkable that Atanassov’s method for classical Halton sequences applies almost “word for word” to (t, s) − sequences and gives an upper bound quite comparable to those of Sobol’, Faure, and Niederreiter. But Atanassov also found a way to improve further his bound for classical Halton sequences by means of a clever scrambling producing sequences which he named modified Halton sequences. We generalize his method to nonsingular lower triangular matrices in the last part of this article.


Truncation Operator Lower Triangular Matrice Diophantine Geometry Pairwise Coprime Elementary Interval 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



We wish to thank the referee for his/her detailed comments, which were very helpful to improve the presentation of this manuscript. The second author acknowledges the support of NSERC for this work.


  1. 1.
    E. I. Atanassov, On the discrepancy of the Halton sequences, Math. Balkanica, New Series 18.1–2 (2004), 15–32.Google Scholar
  2. 2.
    E. I. Atanassov and M. Durchova, Generating and testing the modified Halton sequences. In Fifth International Conference on Numerical Methods and Applications, Borovets 2002, Springer-Verlag (Berlin), Lecture Notes in Computer Science 2542 (2003), 91–98.Google Scholar
  3. 3.
    J. Dick and F. Pillichshammer, Digital Nets and Sequences: Discrepancy Theory and Quasi-Monte Carlo Integration, Cambridge University Press, UK (2010).Google Scholar
  4. 4.
    H. Faure, Discrépance de suites associées à un système de numération (en dimension un), Bull. Soc. math. France 109 (1981), 143–182.Google Scholar
  5. 5.
    H. Faure, Discrépance de suites associées à un système de numération (en dimension s), Acta Arith. 61 (1982), 337–351.Google Scholar
  6. 6.
    H. Faure, On the star-discrepancy of generalized Hammersley sequences in two dimensions, Monatsh. Math. 101 (1986), 291–300.Google Scholar
  7. 7.
    H. Faure, Méthodes quasi-Monte Carlo multidimensionnelles, Theoretical Computer Science 123 (1994), 131–137.Google Scholar
  8. 8.
    H. Faure and C. Lemieux, Generalized Halton sequences in 2008: A comparative study, ACM Trans. Model. Comp. Sim. 19 (2009), Article 15.Google Scholar
  9. 9.
    H. Faure and C. Lemieux, Improvements on the star discrepancy of (t, s) − sequences. Submitted for publication, 2011.Google Scholar
  10. 10.
    J. H. Halton, On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals, Numer. Math. 2 (1960), 184–90.Google Scholar
  11. 11.
    R. Hofer and G. Larcher, On the existence and discrepancy of certain digital Niederreiter–Halton sequences, Acta Arith. 141 (2010), 369–394.Google Scholar
  12. 12.
    P. Kritzer, Improved upper bounds on the star discrepancy of (t, m, s)-nets and (t, s)-sequences, J. Complexity 22 (2006), 336–347.Google Scholar
  13. 13.
    C. Lemieux, Monte Carlo and Quasi-Monte Carlo Sampling, Springer Series in Statistics, Springer, New York (2009).Google Scholar
  14. 14.
    H. Niederreiter, Point sets and sequences with small discrepancy, Monatsh. Math. 104 (1987), 273–337.Google Scholar
  15. 15.
    H. Niederreiter and F. Özbudak, Low-discrepancy sequences using duality and global function fields, Acta Arith. 130 (2007), 79–97.Google Scholar
  16. 16.
    H. Niederreiter and C. P. Xing, Quasirandom points and global function fields, Finite Fields and Applications, S. Cohen and H. Niederreiter (Eds), London Math. Soc. Lectures Notes Series 233 (1996), 269–296.Google Scholar
  17. 17.
    S. Tezuka, Polynomial arithmetic analogue of Halton sequences, ACM Trans. Modeling and Computer Simulation 3 (1993), 99–107.Google Scholar
  18. 18.
    S. Tezuka, A generalization of Faure sequences and its efficient implementation, Technical Report RT0105, IBM Research, Tokyo Research Laboratory (1994).Google Scholar
  19. 19.
    X. Wang, C. Lemieux, H. Faure, A note on Atanassov’s discrepancy bound for the Halton sequence, Technical report, University of Waterloo, Canada (2008). Available at

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Henri Faure
    • 1
  • Christiane Lemieux
    • 2
  • Xiaoheng Wang
    • 3
  1. 1.Institut de Mathématiques de LuminyMarseilleFrance
  2. 2.University of WaterlooWaterlooCanada
  3. 3.Harvard UniversityCambridgeUSA

Personalised recommendations