On the mean iteration
Authors:
J. M. Borwein and P. B. Borwein
Journal:
Math. Comp. 53 (1989), 311326
MSC:
Primary 30D05; Secondary 33A25
MathSciNet review:
968148
Fulltext PDF Free Access
Abstract 
References 
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Additional Information
Abstract: The iterative process 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.
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 [1]
 J. Arazy, J. Claesson, S. Janson & J. Peetre, Means and Their Iterations, Proc. 19th Nordic Congr. Math. (J. R. Stefánson, ed.), Reykjavík, 1985. MR 828035 (87f:01012)
 [2]
 C. W. Borchardt, Ueber das Arithmetischgeometrische Mittel aus vier Elementen, Berl. Monatsber., 1876, pp. 611621. Werke, Berlin, 1888, pp. 329338.
 [3]
 J. M. Borwein & P. B. Borwein, "The arithmeticgeometric mean and fast computation of the elementary functions," SIAM Rev., v. 26, 1984, pp. 351365. MR 750454 (86d:65029)
 [4]
 J. M. Borwein & P. B. Borwein, Pi and the AGMA Study in Analytic Number Theory and Computational Complexity, Wiley, New York, 1987.
 [5]
 J. M. Borwein & P. B. Borwein, "The way of all means," Amer. Math. Monthly, v. 94, 1987, pp. 519522. MR 1541118
 [6]
 R. P. Brent, "Fast multipleprecision evaluation of elementary functions," J. Assoc. Comput. Mach., v. 23, 1976, pp. 242251. MR 0395314 (52:16111)
 [7]
 B. C. Carlson, "Algorithms involving arithmetic and geometric means," Amer. Math. Monthly, v. 98, 1971, pp. 496505. MR 0283246 (44:479)
 [8]
 D. H. Lehmer, "On the compounding of certain means," J. Math. Anal. Appl., v. 36, 1971, pp. 183200. MR 0281696 (43:7411)
 [9]
 D. J. Newman, "A simplified version of the fast algorithms of Brent and Salamin," Math. Comp., v. 44, 1985, pp. 207210. MR 771042 (86e:65030)
 [10]
 J. Peetre, "Generalizing the arithmeticgeometric meanA hapless computer experiment," preprint.
 [11]
 E. Salamin, "Computation of using arithmeticgeometric mean," Math. Comp., v. 30, 1976, pp. 565570. MR 0404124 (53:7928)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198909681484
PII:
S 00255718(1989)09681484
Article copyright:
© Copyright 1989
American Mathematical Society
