Mathematics in Computer Science

, Volume 5, Issue 4, pp 359–375 | Cite as

Stochastic Arithmetic in Multiprecision

  • Stef Graillat
  • Fabienne Jézéquel
  • Shiyue Wang
  • Yuxiang Zhu
Article

Abstract

Floating-point arithmetic precision is limited in length the IEEE single (respectively double) precision format is 32-bit (respectively 64-bit) long. Extended precision formats can be up to 128-bit long. However some problems require a longer floating-point format, because of round-off errors. Such problems are usually solved in arbitrary precision, but round-off errors still occur and must be controlled. Interval arithmetic has been implemented in arbitrary precision, for instance in the MPFI library. Interval arithmetic provides guaranteed results, but it is not well suited for the validation of huge applications. The CADNA library estimates round-off error propagation using stochastic arithmetic. CADNA has enabled the numerical validation of real-life applications, but it can be used in single precision or in double precision only. In this paper, we present a library called SAM (Stochastic Arithmetic in Multiprecision). It is a multiprecision extension of the classic CADNA library. In SAM (as in CADNA), the arithmetic and relational operators are overloaded in order to be able to deal with stochastic numbers. As a consequence, the use of SAM in a scientific code needs only few modifications. This new library SAM makes it possible to dynamically control the numerical methods used and more particularly to determine the optimal number of iterations in an iterative process. We present some applications of SAM in the numerical validation of chaotic systems modeled by the logistic map.

Keywords

Stochastic arithmetic Multiprecision MPFR MPFI Interval arithmetic 

Mathematics Subject Classification (2010)

Primary 68N01 Secondary 68U20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Li X.S., Demmel J.W., Bailey D.H., Henry G., Hida Y., Iskandar J., Kahan W., Kang S.Y., Kapur A., Martin M.C., Thompson B.J., Tung T., Yoo D.J.: Design, implementation and testing of extended and mixed precision BLAS. ACM Trans. Math. Softw. 28(2), 152–205 (2002)CrossRefGoogle Scholar
  2. 2.
    Fousse, L., Hanrot, G., Lefèvre, V., Pélissier, P., Zimmermann, P.: MPFR: A multiple-precision binary floating-point library with correct rounding. ACM Trans. Math. Softw. 33(2), 13:1–13:15 (2007). (http://www.mpfr.org)Google Scholar
  3. 3.
    Wilkinson J.H.: Rounding errors in algebraic processes. Prentice-Hall Inc., Englewood Cliffs (1963)MATHGoogle Scholar
  4. 4.
    Higham N.J.: Accuracy and stability of numerical algorithms. 2nd edn. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2002)CrossRefMATHGoogle Scholar
  5. 5.
    Einarsson B. et al.: Accuracy and Reliability in Scientific Computing. Software-Environments-Tools. SIAM, Philadelphia (2005)CrossRefGoogle Scholar
  6. 6.
    Chaitin-Chatelin F., Frayssé V.: Lectures on Finite Precision Computations. Society for Industrial and Applied Mathematics, Philadelphia (1996)CrossRefMATHGoogle Scholar
  7. 7.
    Moore R.: Interval analysis. Prentice Hall, Saddle River (1966)MATHGoogle Scholar
  8. 8.
    Alefeld G., Herzberger J.: Introduction to interval analysis. Academic Press, New York (1983)Google Scholar
  9. 9.
    Moore R., Kearfott R., Cloud M.: Introduction to interval analysis. Society for Industrial and Applied Mathematics, Philadelphia (2009)CrossRefMATHGoogle Scholar
  10. 10.
    Chesneaux J.M.: L’arithmétique stochastique et le logiciel CADNA. Habilitation à diriger des recherches Université Pierre et Marie Curie, Paris (1995)Google Scholar
  11. 11.
    Vignes J.: A stochastic arithmetic for reliable scientific computation. Math. Comput. Simul. 35, 233–261 (1993)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Goubault, E., Putot, S., Baufreton, P., Gassino, J.: Static analysis of the accuracy in control systems: Principles and experiments. In: Proceedings of Formal Methods in Industrial Critical Systems, LNCS 4916, Springer, Berlin (2007)Google Scholar
  13. 13.
    Chesneaux, J.M., Graillat, S., Jézéquel, F.: Rounding Errors. In: Encyclopedia of Computer Science and Engineering, vol. 4, pp. 2480–2494. Wiley, New York (2009)Google Scholar
  14. 14.
    Revol, N., Rouillier, F.: MPFI (Multiple Precision Floating-point Interval library) (2009). (Available at http://gforge.inria.fr/projects/mpfi).
  15. 15.
    Bailey D.H.: A Fortran 90-based multiprecision system. ACM Trans. Math. Softw. 21(4), 379–387 (1995)CrossRefMATHGoogle Scholar
  16. 16.
    Brent R.P.: A fortran multiple-precision arithmetic package. ACM Trans. Math. Softw. 4(1), 57–70 (1978)CrossRefGoogle Scholar
  17. 17.
    Priest, D.M.: Algorithms for arbitrary precision floating point arithmetic. In Kornerup, P., Matula, D.W., (eds.) Proceedings of the 10th IEEE Symposium on Computer Arithmetic (Arith-10), Grenoble, France, pp. 132–144. IEEE Computer Society Press, Los Alamitos (1991)Google Scholar
  18. 18.
    Shewchuk J.R.: Adaptive precision floating-point arithmetic and fast robust geometric predicates. Discret. Comput. Geom. 18(3), 305–363 (1997)CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Bailey, D.H.: A Fortran-90 double-double library (2001). (Available at http://crd.lbl.gov/~dhbailey/mpdist/index.html)
  20. 20.
    Hida, Y., Li, X.S., Bailey, D.H.: Algorithms for quad-double precision floating point arithmetic. In: Proceedings of 15th IEEE Symposium on Computer Arithmetic, pp. 155–162. IEEE Computer Society Press, Los Alamitos (2001)Google Scholar
  21. 21.
    IEEE Computer Society, New York: IEEE Standard for Binary Floating-Point Arithmetic, ANSI/IEEE Standard, pp. 754–1985 (1985). (Reprinted in SIGPLAN Notices 22(2), 9–25 (1987))Google Scholar
  22. 22.
    Grandlund, T.: GNU MP: The GNU Multiple Precision Arithmetic Library. (http://gmplib.org)
  23. 23.
    Kulisch U.: Advanced Arithmetic for the Digital Computer. Springer, Wien (2002)CrossRefMATHGoogle Scholar
  24. 24.
    Chesneaux J.M.: Study of the computing accuracy by using probabilistic approach. In: Ullrich, C. (eds) Contribution to Computer Arithmetic and Self-Validating Numerical Methods, pp. 19–30. IMACS, New Brunswick (1990)Google Scholar
  25. 25.
    Chesneaux J.M., Vignes J.: Sur la robustesse de la méthode CESTAC. C. R. Acad. Sci. Paris Sér. I Math. 307, 855–860 (1988)MATHMathSciNetGoogle Scholar
  26. 26.
    Vignes J.: Zéro mathématique et zéro informatique. C. R. Acad. Sci. Paris Sér. I Math 303, 997–1000 (1986)MATHMathSciNetGoogle Scholar
  27. 27.
    Vignes J.: Zéro mathématique et zéro informatique. La Vie des Sciences 4(1), 1–13 (1987)MATHMathSciNetGoogle Scholar
  28. 28.
    Université Pierre et Marie Curie, Paris, F.: CADNA: Control of Accuracy and Debugging for Numerical Applications. (http://www.lip6.fr/cadna)
  29. 29.
    Jézéquel F., Chesneaux J.M.: CADNA: a library for estimating round-off error propagation. Comput. Phys. Commun. 178(12), 933–955 (2008)CrossRefMATHGoogle Scholar
  30. 30.
    Jézéquel F., Chesneaux J.M., Lamotte J.L.: A new version of the CADNA library for estimating round-off error propagation in Fortran programs. Comput. Phys. Commun. 181(11), 1927–1928 (2010)CrossRefGoogle Scholar
  31. 31.
    Lamotte J.L., Chesneaux J.M., Jézéquel F.: CADNA_C: A version of CADNA for use with C or C++ programs. Comput. Phys. Commun. 181(11), 1925–1926 (2010)CrossRefGoogle Scholar
  32. 32.
    Chesneaux J.M., Troff B.: Computational stability study using the CADNA software applied to the Navier-Stokes solver PEGASE. In: Alefeld, G., Frommer, A. (eds) Scientific Computing and Validated Numerics, pp. 84–90. Akademie, Berlin (1996)Google Scholar
  33. 33.
    Alberstein N., Chesneaux J.M., Christiansen S., Wirgin A.: Comparison of four software packages applied to a scattering problem. Math. Comput. Simul. 48, 307–318 (1999)CrossRefGoogle Scholar
  34. 34.
    Jézéquel F., Rico F., Chesneaux J.M., Charikhi M.: Reliable computation of a multiple integral involved in the neutron star theory. Math. Comput. Simul. 71(1), 44–61 (2006)CrossRefMATHGoogle Scholar
  35. 35.
    Scott N., Jézéquel F., Denis C., Chesneaux J.M.: Numerical ’health check’ for scientific codes: the CADNA approach. Comput. Phys. Commun. 176(8), 507–521 (2007)CrossRefGoogle Scholar
  36. 36.
    Scott N., Faro-Maza V., Scott M., Harmer T., Chesneaux J.M., Denis C., Jézéquel F.: E-collisions using e-science. Phys. Part. Nuclei Lett. 5(3), 150–156 (2008)CrossRefGoogle Scholar
  37. 37.
    Rump S.: Reliability in Computing. The Role of Interval Methods in Scientific Computing. Academic Press, Oakville (1988)Google Scholar
  38. 38.
    Muller J.M.: Arithmétique des Ordinateurs. Academic Press, Masson (1989)Google Scholar
  39. 39.
    Chesneaux J.M., Jézéquel F.: Dynamical control of computations using the trapezoidal and Simpson’s rules. J. Univers. Comput. Sci. 4(1), 2–10 (1998)MATHGoogle Scholar
  40. 40.
    Jézéquel F.: Dynamical control of converging sequences computation. Appl. Numer. Math. 50(2), 147–164 (2004)CrossRefMATHMathSciNetGoogle Scholar
  41. 41.
    Jézéquel F., Chesneaux J.M.: Computation of an infinite integral using Romberg’s method. Num. Algo. 36(3), 265–283 (2004)CrossRefMATHGoogle Scholar
  42. 42.
    Jézéquel F.: A dynamical strategy for approximation methods. C. R. Acad. Sci. Paris Mécanique 334, 362–367 (2006)MATHGoogle Scholar
  43. 43.
    Ahmed Z.: Definitely an integral. Am Math. Month. 109(7), 670–671 (2002)CrossRefGoogle Scholar
  44. 44.
    Bailey, D., Li, X.: A comparison of three high-precision quadrature schemes. In: Proceedings of 5th Real Numbers and Computers conference, Lyon, France, pp. 81–95 (2003)Google Scholar
  45. 45.
    Devaney R.L.: An introduction to chaotic dynamical systems. Second edn. Addison-Wesley Studies in Nonlinearity, Addison-Wesley Publishing Company Advanced Book Program, Redwood City (1989)MATHGoogle Scholar
  46. 46.
    Argyris J., Faust G., Haase M.: An exploration of chaos. Texts on Computational Mechanics, vol. VII. North-Holland Publishing Co., Amsterdam (1994)Google Scholar
  47. 47.
    Tucker W.: The Lorenz attractor exists. C. R. Acad. Sci. Paris Sér. I Math. 328(12), 1197–1202 (1999)MATHGoogle Scholar
  48. 48.
    Galias, Z., Tucker, W.: Rigorous study of short periodic orbits for the Lorenz system. In: Proceedings of IEEE International Symposium on Circuits and Systems, pp. 764–767. ISCAS’08, Seattle (2008)Google Scholar
  49. 49.
    Tucker, W.: Fundamentals of chaos. In: Kocarev, L., et al. (eds.) Intelligent computing based on chaos. Studies in Computational Intelligence, vol. 184, pp. 1–23. Springer, Berlin (2009)Google Scholar
  50. 50.
    Pichat, M., Vignes, J.: The numerical study of chaotic systems—future and past. In: 16th IMACS World Congress on Scientific Computation, Applied Mathematics and Simulation. Lausanne, Switzerland (2000)Google Scholar
  51. 51.
    Yao L.S.: Computed chaos or numerical errors. Nonlinear Anal. Model. Contr. 15(1), 109–126 (2010)MATHGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  • Stef Graillat
    • 1
  • Fabienne Jézéquel
    • 1
  • Shiyue Wang
    • 1
  • Yuxiang Zhu
    • 1
  1. 1.Laboratoire d’Informatique de Paris 6UPMC Univ Paris 06, UMR 7606Paris Cedex 05France

Personalised recommendations