The Arithmetic-Geometric Mean and Fast Computation of Elementary Functions

  • J. M. Borwein
  • P. B. Borwein


We produce a self contained account of the relationship between the Gaussian arithmeticgeometric mean iteration and the fast computation of elementary functions. A particularly pleasant algorithm for π is one of the by-products.


Elementary Function Elliptic Function Root Extraction Elliptic Integral Exponential Convergence 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    N. H. Abel, Oeuvres complètes, Grondahl and Son, Christiana, 1881.Google Scholar
  2. [2]
    P. Beckman, A History of Pi, Golem Press, Ed. 4, 1977.Google Scholar
  3. [3]
    F. Bowman, Introduction to Elliptic Functions, English Univ. Press, London, 1953.MATHGoogle Scholar
  4. [4]
    R. P. Brent, Multiple-precision zero-finding and the complexity of elementary function evaluation, in Analytic Computational Complexity, J. F. Traub, ed., Academic Press, New York, 1975, pp. 151–176.Google Scholar
  5. [5]
    R. P. Brent, Fast multiple-precision evaluation of elementary functions, J. Assoc. Comput. Mach., 23 (1976), pp. 242–251.CrossRefMATHMathSciNetGoogle Scholar
  6. [6]
    B. C. Carlson, Algorithms involving arithmetic and geometric means, Amer. Math. Monthly, 78(1971), pp. 496–505.CrossRefMATHMathSciNetGoogle Scholar
  7. [7]
    A. Cayley, An Elementary Treatise on Elliptic Functions, Bell and Sons, 1895, republished by Dover, New York, 1961.Google Scholar
  8. [8]
    A. Eagle, The Elliptic Functions as They Should Be, Galloway and Porter, Cambridge, 1958.MATHGoogle Scholar
  9. [9]
    C. H. Edwards, Jr. The Historical Development of the Calculus, Springer-Verlag, New York, 1979.CrossRefMATHGoogle Scholar
  10. [10]
    K. F. Gauss, Werke, Bd 3, Gottingen, 1866, 361–403.Google Scholar
  11. [11]
    L. V. King, On the direct numerical calculation of elliptic functions and integrals, Cambridge Univ. Press, Cambridge, 1924.Google Scholar
  12. [12]
    D. Knuth, The Art of Computer Programming. Vol. 2: Seminumerical Algorithms, Addison-Wesley, Reading, Ma, 1969.Google Scholar
  13. [13]
    H. T. Kung and J. F. Traub, All algebraic functions can be computed fast, J. Assoc. Comput. Mach., 25(1978), 245–260.CrossRefMATHMathSciNetGoogle Scholar
  14. [14]
    A. M. Legendre, Exercices de calcul integral, Vol. 1, Paris, 1811.Google Scholar
  15. [15]
    D. J. Neuman,Rational approximation versus fast computer methods, in Lectures on Approximation and Value Distribution, Presses de l’Université de Montréal, Montreal, 1982, pp. 149–174.Google Scholar
  16. [16]
    E. Salamin, Computation of using arithmetic-geometric mean, Math. Comput. 135(1976), pp. 565–570.MathSciNetGoogle Scholar
  17. [17]
    D. Shanks and J. W. Wrench Jr. Calculation of π to 100,000 decimals, Math. Comput., 16 (1962), pp. 76–79.MATHMathSciNetGoogle Scholar
  18. [18]
    Y. Tamura and Y. Kanada, Calculation of π to 4,196,293 decimals based on Gauss-Legendre algorithm, preprint.Google Scholar
  19. [19]
    J. Todd, Basic Numerical Mathematics, Vol. 1, Academic Press, New York, 1979.CrossRefMATHGoogle Scholar
  20. [20]
    G. N. Watson, The marquis and the land agent, Math. Gazette, 17 (1933), pp. 5–17.CrossRefMATHGoogle Scholar
  21. [21]
    J. W. Wrench, Jr. The evolution of extended decimal approximations to π. The Mathematics Teacher, 53(1960), pp. 644–650.Google Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • J. M. Borwein
    • 1
  • P. B. Borwein
    • 1
  1. 1.Department of MathematicsDathousie UniversityHalifaxCanada

Personalised recommendations