Geodesic
Cubic and Higher Order Algorithms for π
Canadian mathematical bulletin, Tome 27 (1984) no. 4, pp. 436-443

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-1984-067-7
Mots-clés : 10A30, 33A25, Pi, Elliptic Integrals, Algorithms, Modular Equations 
Borwein, J. M.; Borwein, P. B. Cubic and Higher Order Algorithms for π. Canadian mathematical bulletin, Tome 27 (1984) no. 4, pp. 436-443. doi: 10.4153/CMB-1984-067-7
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[1] 1. Borwein, J. M. and Borwein, P. B., A very rapidly convergent product expansion for π, BIT 23 (1983), 538-540. Google Scholar | DOI

[2] 2. Borwein, J. M. and Borwein, P. B., More Quadratically Converging Algorithms for π, Math. Comput. (to appear). Google Scholar

[3] 3. Borwein, J. M. and Borwein, P. B., The arithmetic-geometric mean and fast computation of elementary functions, SIAM Review 26 (1984). Google Scholar | DOI

[4] 4. Brent, R. P., Fast multiple-precision evaluation of elementary functions, J. Assoc. Comput. Mach. 23 (1976), 242-251. Google Scholar | DOI

[5] 5. Cayley, A., An Elementary Treatise on Elliptic Functions, Bell and Sons 1895, republished Dover 1961. Google Scholar

[6] 6. Cayley, A., A Memoir on the transformation of elliptic functions, Phil. Trans. T. 164 (1874), 397-456. Google Scholar

[7] 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] 8. Ramanujan, S., Modular equations and approximations to π, Quart. J. Math., 44 (1914), 350-372. Google Scholar

[9] 9. Salamin, E., Computation of π using arithmetic-geometric mean, Math. Comput. 135 (1976), 565-570. Google Scholar

[10] 10. Tamura, Y. and Kanada, Y., Calculation of π to 4,196,293 decimals based on Gauss-Legendre algorithm, preprint. Google Scholar

[11] 11. Whitakker, E. T. and Watson, G. N., A Course of Modem Analysis, Cambridge University Press, Ed. 4, 1927. Google Scholar

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