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Cubic and Higher Order Algorithms for π

  • J. M. Borwein (a1) and P. B. Borwein (a1)
Abstract

We show that the theory of elliptic integral transformations may be employed to construct iterative approximations for π of order p (p any prime). Details are provided for two, three and seven. The cubic case proves amenable to surprisingly complete analysis.

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References
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1. Borwein, J. M. and Borwein, P. B., A very rapidly convergent product expansion for π, BIT 23 (1983), 538-540.
2. Borwein, J. M. and Borwein, P. B., More Quadratically Converging Algorithms for π, Math. Comput. (to appear).
3. Borwein, J. M. and Borwein, P. B., The arithmetic-geometric mean and fast computation of elementary functions, SIAM Review 26 (1984).
4. Brent, R. P., Fast multiple-precision evaluation of elementary functions, J. Assoc. Comput. Mach. 23 (1976), 242-251.
5. Cayley, A., An Elementary Treatise on Elliptic Functions, Bell and Sons 1895, republished Dover 1961.
6. Cayley, A., A Memoir on the transformation of elliptic functions, Phil. Trans. T. 164 (1874), 397-456.
7. Newman, D. J., Rational approximation versus fast computer Methods, in Lectures on Approximation and Value Distribution, Presses de l'université de Montréal, 1982, 149-174.
8. Ramanujan, S., Modular equations and approximations to π, Quart. J. Math., 44 (1914), 350-372.
9. Salamin, E., Computation of π using arithmetic-geometric mean, Math. Comput. 135 (1976), 565-570.
10. Tamura, Y. and Kanada, Y., Calculation of π to 4,196,293 decimals based on Gauss-Legendre algorithm, preprint.
11. Whitakker, E. T. and Watson, G. N., A Course of Modem Analysis, Cambridge University Press, Ed. 4, 1927.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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