On the mean iteration $(a,b)\leftarrow ((a+3b)/4,(\sqrt {ab}+b)/2)$
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- by J. M. Borwein and P. B. Borwein PDF
- Math. Comp. 53 (1989), 311-326 Request permission
Abstract:
The iterative process \[ {a_{n + 1}} = ({a_n} + 3{b_n})/4,\quad {b_{n + 1}} = (\sqrt {{a_n}{b_n}} + {b_n})/2\] is studied in detail. The limit of this quadratically converging process is explicitly identified, as are the uniformizing parameters. The role of symbolic computation, in discovering these nontrivial identifications, is highlighted.References
- Jonathan Arazy, Tomas Claesson, Svante Janson, and Jaak Peetre, Means and their iterations, Proceedings of the nineteenth Nordic congress of mathematicians (Reykjavík, 1984) Vísindafél. Ísl., XLIV, Icel. Math. Soc., Reykjavík, 1985, pp. 191–212. MR 828035 C. W. Borchardt, Ueber das Arithmetisch-geometrische Mittel aus vier Elementen, Berl. Monatsber., 1876, pp. 611-621. Werke, Berlin, 1888, pp. 329-338.
- J. M. Borwein and P. B. Borwein, The arithmetic-geometric mean and fast computation of elementary functions, SIAM Rev. 26 (1984), no. 3, 351–366. MR 750454, DOI 10.1137/1026073 J. M. Borwein & P. B. Borwein, Pi and the AGM—A Study in Analytic Number Theory and Computational Complexity, Wiley, New York, 1987.
- J. M. Borwein and P. B. Borwein, Unsolved Problems: The Way of All Means, Amer. Math. Monthly 94 (1987), no. 6, 519–522. MR 1541118, DOI 10.2307/2322842
- Richard P. Brent, Fast multiple-precision evaluation of elementary functions, J. Assoc. Comput. Mach. 23 (1976), no. 2, 242–251. MR 395314, DOI 10.1145/321941.321944
- B. C. Carlson, Algorithms involving arithmetic and geometric means, Amer. Math. Monthly 78 (1971), 496–505. MR 283246, DOI 10.2307/2317754
- D. H. Lehmer, On the compounding of certain means, J. Math. Anal. Appl. 36 (1971), 183–200. MR 281696, DOI 10.1016/0022-247X(71)90029-1
- D. J. Newman, A simplified version of the fast algorithms of Brent and Salamin, Math. Comp. 44 (1985), no. 169, 207–210. MR 771042, DOI 10.1090/S0025-5718-1985-0771042-1 J. Peetre, "Generalizing the arithmetic-geometric mean—A hapless computer experiment," preprint.
- Eugene Salamin, Computation of $\pi$ using arithmetic-geometric mean, Math. Comp. 30 (1976), no. 135, 565–570. MR 404124, DOI 10.1090/S0025-5718-1976-0404124-9
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Math. Comp. 53 (1989), 311-326
- MSC: Primary 30D05; Secondary 33A25
- DOI: https://doi.org/10.1090/S0025-5718-1989-0968148-4
- MathSciNet review: 968148