On the mean iteration $(a,b)\leftarrow ((a+3b)/4,(\sqrt {ab}+b)/2)$

Authors:
J. M. Borwein and P. B. Borwein

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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

- 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, - 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 https://doi.org/10.1137/1026073
J. M. Borwein & P. B. Borwein, - 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 https://doi.org/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 https://doi.org/10.1145/321941.321944 - B. C. Carlson,
*Algorithms involving arithmetic and geometric means*, Amer. Math. Monthly**78**(1971), 496–505. MR**283246**, DOI https://doi.org/10.2307/2317754 - D. H. Lehmer,
*On the compounding of certain means*, J. Math. Anal. Appl.**36**(1971), 183–200. MR**281696**, DOI https://doi.org/10.1016/0022-247X%2871%2990029-1 - 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 https://doi.org/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 https://doi.org/10.1090/S0025-5718-1976-0404124-9

*Ueber das Arithmetisch-geometrische Mittel aus vier Elementen*, Berl. Monatsber., 1876, pp. 611-621. Werke, Berlin, 1888, pp. 329-338.

*Pi and the AGM—A Study in Analytic Number Theory and Computational Complexity*, Wiley, New York, 1987.

Retrieve articles in *Mathematics of Computation*
with MSC:
30D05,
33A25

Retrieve articles in all journals with MSC: 30D05, 33A25

Additional Information

Article copyright:
© Copyright 1989
American Mathematical Society