Fast Computations of the Exponential Function

  • Timm Ahrendt
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1563)

Abstract

In this paper we present an algorithm which shows that the exponential function has algebraic complexity O(log2n), i.e., can be evaluated with relative error O(2-n) using O(log2n) infinite-precision additions, subtractions, multiplications and divisions. This solves a question of J. M. Borwein and P. B. Borwein [9].

The best known lower bound for the algebraic complexity of the exponential function is Ω(log n).

The best known upper and lower bounds for the bit complexity of the exponential function are O(μ(n) log n) [10] and Ω(ν(n)) [4], respectively, where μ(n) denotes an upper bound and ν(n) denotes a lower bound for the bit complexity of n-bit integer multiplication.

The presented algorithm has bit complexity O(μ(n) log n).

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References

  1. 1.
    Ahrendt, T. Fast high-precision computation of complex square roots. In Proceedings of the 1996 International Symposium on Symbolic and Algebraic Computation: ISSAC’ 96, Lakshman, Y. N., Ed., ACM, New York, pp. 142–149.Google Scholar
  2. 2.
    Alt, H. Square rooting is as difficult as multiplication. Computing 21 (1979), 221–232.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Alt, H. Comparison of arithmetic functions with respect to Boolean circuit depth. In Proc. 16th Ann. ACM Symposium on Theory of Computing (1984), pp. 466–470.Google Scholar
  4. 4.
    Alt, H. Multiplication is the easiest nontrivial arithmetic function. Theoretical Comput. Sci. 36 (1985), 333–339.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bailey, D. H. A portable high performance multiprecision package. RNR Technical Report RNR-90-022, NAS Applied Research Branch, NASA Ames Research Center, Moffett Filed, CA 94035, May 1993.Google Scholar
  6. 6.
    Blum, L., Cucker, F., Shub, M., AND Smale, S.Complexity and Real Computation. Springer, New York, 1997.MATHGoogle Scholar
  7. 7.
    Borwein, J. M., AND Borwein, P. B. The arithmetic-geometric mean and fast computation of elementary funtions. SIAM Review 26,3 (1984), 351–366.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Borwein, J. M., AND Borwein, P. B.Pi and the AGM. John Wiley and Sons, New York, 1987.MATHGoogle Scholar
  9. 9.
    Borwein, J. M., AND Borwein, P. B. On the complexity of familiar functions and numbers. SIAM Review 30,4 (1988), 589–601.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Brent, R. P. Fast multiple-precision evaluation of elementary functions. J. ACM 23,2 (1976), 242–251.MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Brent, R. P. Multiple-precision zero-finding methods and the complexity of elementary function evaluation. In Analytic Computational Complexity, Traub, J. F., Ed. Academic Press, New York, 1976, pp. 151–176.Google Scholar
  12. 12.
    Kanada, Y. Vectorization of multiple-precision arithmetic program and 201,326,000 decimal digits of π calculation. Supercomputing 88: Volume II, Science and Applications (1988), 117–128. In Pi: A Source Book, Berggren, L., Borwein, J. M., AND Borwein, P. B., Eds. Springer, New York, 1997.Google Scholar
  13. 13.
    Muller, J.-M.Elementary Functions: Algorithms and Implementation. Birkhäuser, Boston, 1997.MATHGoogle Scholar
  14. 14.
    Okabe, Y., Takagi, N., AND Yajima, S. Log-depth circuits for elementary functions using residue number system. Electronics and Communications in Japan, Part 3 74,8 (1991), 31–38.CrossRefGoogle Scholar
  15. 15.
    Reif, J. Logarithmic depth circuits for algebraic functions. In Proc. 24th Ann. IEEE Symposium on Foundations of Computer Science (1983), pp. 138–145.Google Scholar
  16. 16.
    Asaki, T., AND Kanada, Y. Practically fast multiple-precision evaluation of log(x). Journal of Information Processing 4,4 (1982), 247–250.Google Scholar
  17. 17.
    Chonhage, A. Routines for square roots. Program documentation of the TP-routines SQRT, ISQRT, CSQRT, unpublished manuscript, July 1991.Google Scholar
  18. 18.
    Chonhage, A., Grotefeld, A. F. W., AND Vetter, E.Fast Algorithms: a Multitape Turing Machine Implementation. Bibliographisches Institut, Mannheim, 1994.Google Scholar
  19. 19.
    Schonhage, A., AND Strassen, V. Schnelle Multiplikation großer Zahlen. Computing 7 (1971), 281–292.CrossRefMathSciNetGoogle Scholar
  20. 20.
    Wegener, I.Effiziente Algorithmen für grundlegende Funktionen. Teubner, Stuttgart, 1989.MATHGoogle Scholar
  21. 21.
    Wolfram, S.Mathematica: A System for Doing Mathematics by Computer, 2nd ed. Addison-Wesley, Redwood City, California, 1991.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Timm Ahrendt
    • 1
  1. 1.Institut für Informatik IIUniversität BonnBonnGermany

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