An explicit cubic iteration for \(\pi\). (English) Zbl 0613.65010
Using several relations for elliptic integrals and cubic modular equations, the authors present a simple class of cubically convergent algebraic iterations for \(\pi\). The algorithm is described with several starting values.
Reviewer: N.M.Temme
MSC:
| 65D20 | Computation of special functions and constants, construction of tables |
| 33E05 | Elliptic functions and integrals |
| 11F03 | Modular and automorphic functions |
References:
| [1] | 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. · Zbl 0613.65010 |
| [2] | J. M. Borwein and P. B. Borwein,Cubic and higher order algorithms for {\(\pi\)}, Canad. Math. Bull 27 (1984), 436–443. · doi:10.4153/CMB-1984-067-7 |
| [3] | S. Ramanujan,Modular equations and approximations to {\(\pi\)}, Quart. J. Math. 45 (1914), 350–372. · JFM 45.1249.01 |
| [4] | E. Salamin,Computation of {\(\pi\)} using arithmetic-geometric mean, Math. Comput. 30 (1976), 565–570. · Zbl 0345.10003 |
| [5] | H. Weber,Lehrbuch der Algebra, vol. 3, reprinted Chelsea, 1980. |
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