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Cubic and Higher Order Algorithms for π

Published online by Cambridge University Press:  20 November 2018

J. M. Borwein  and
P. B. Borwein
J. M. Borwein
Affiliation:
Department of Mathematics., Statistics and Computing Science, Dalhousie University, Halifax, N.S., B3H 4H8
P. B. Borwein
Affiliation:
Department of Mathematics., Statistics and Computing Science, Dalhousie University, Halifax, N.S., B3H 4H8
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Abstract

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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.

Keywords

10A3033A25PiElliptic IntegralsAlgorithmsModular Equations
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 27 , Issue 4 , 01 December 1984 , pp. 436 - 443
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Borwein, J. M. and Borwein, P. B., A very rapidly convergent product expansion for π, BIT 23 (1983), 538-540.CrossRefGoogle Scholar
2. Borwein, J. M. and Borwein, P. B., More Quadratically Converging Algorithms for π, Math. Comput. (to appear).Google Scholar
3. Borwein, J. M. and Borwein, P. B., The arithmetic-geometric mean and fast computation of elementary functions, SIAM Review 26 (1984).CrossRefGoogle Scholar
4. Brent, R. P., Fast multiple-precision evaluation of elementary functions, J. Assoc. Comput. Mach. 23 (1976), 242-251.CrossRefGoogle Scholar
5. Cayley, A., An Elementary Treatise on Elliptic Functions, Bell and Sons 1895, republished Dover 1961.Google Scholar
6. Cayley, A., A Memoir on the transformation of elliptic functions, Phil. Trans. T. 164 (1874), 397-456.Google Scholar
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.Google Scholar
8. Ramanujan, S., Modular equations and approximations to π, Quart. J. Math., 44 (1914), 350-372.Google Scholar
9. Salamin, E., Computation of π using arithmetic-geometric mean, Math. Comput. 135 (1976), 565-570.Google Scholar
10. Tamura, Y. and Kanada, Y., Calculation of π to 4,196,293 decimals based on Gauss-Legendre algorithm, preprint.Google Scholar
11. Whitakker, E. T. and Watson, G. N., A Course of Modem Analysis, Cambridge University Press, Ed. 4, 1927.Google Scholar