BIT Numerical Mathematics

, Volume 26, Issue 1, pp 123–126 | Cite as

An explicit cubic iteration for π

  • J. M. Borwein
  • P. B. Borwein
Scientific Notes


Using the theory of the cubic modular equation we have discovered a remarkably simple class of cubically convergent algebraic iterations for π.


Computational Mathematic Simple Class Modular Equation Algebraic Iteration 
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.


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    J. M. Borwein and P. B. Borwein,Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, John Wiley, to appear, 1986.Google Scholar
  2. 2.
    J. M. Borwein and P. B. Borwein,Cubic and higher order algorithms for π, Canad. Math. Bull 27 (1984), 436–443.Google Scholar
  3. 3.
    S. Ramanujan,Modular equations and approximations to π, Quart. J. Math. 45 (1914), 350–372.Google Scholar
  4. 4.
    E. Salamin,Computation of π using arithmetic-geometric mean, Math. Comput. 30 (1976), 565–570.Google Scholar
  5. 5.
    H. Weber,Lehrbuch der Algebra, vol. 3, reprinted Chelsea, 1980.Google Scholar

Copyright information

© BIT Foundations 1986

Authors and Affiliations

  • J. M. Borwein
    • 1
  • P. B. Borwein
    • 1
  1. 1.Department of Mathematics, Statistics and Computing ScienceDalhousie UniversityHalifaxCanada

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