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 
Similar Articles 
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.
 [1]
Jonathan
Arazy, Tomas
Claesson, Svante
Janson, and Jaak
Peetre, Means and their iterations, (Reykjavík, 1984)
Vísindafél. Ísl., XLIV, Icel. Math. Soc.,
Reykjavík, 1985, pp. 191–212. 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 and P.
B. Borwein, The arithmeticgeometric mean and fast computation of
elementary functions, SIAM Rev. 26 (1984),
no. 3, 351–366. MR 750454
(86d:65029), http://dx.doi.org/10.1137/1026073
 [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 and P.
B. Borwein, Unsolved Problems: The Way of All Means, Amer.
Math. Monthly 94 (1987), no. 6, 519–522. MR
1541118, http://dx.doi.org/10.2307/2322842
 [6]
Richard
P. Brent, Fast multipleprecision evaluation of elementary
functions, J. Assoc. Comput. Mach. 23 (1976),
no. 2, 242–251. MR 0395314
(52 #16111)
 [7]
B.
C. Carlson, Algorithms involving arithmetic and geometric
means, Amer. Math. Monthly 78 (1971), 496–505.
MR
0283246 (44 #479)
 [8]
D.
H. Lehmer, On the compounding of certain means, J. Math. Anal.
Appl. 36 (1971), 183–200. MR 0281696
(43 #7411)
 [9]
D.
J. Newman, A simplified version of the fast
algorithms of Brent and Salamin, Math.
Comp. 44 (1985), no. 169, 207–210. MR 771042
(86e:65030), http://dx.doi.org/10.1090/S00255718198507710421
 [10]
J. Peetre, "Generalizing the arithmeticgeometric meanA hapless computer experiment," preprint.
 [11]
Eugene
Salamin, Computation of 𝜋 using
arithmeticgeometric mean, Math. Comp.
30 (1976), no. 135, 565–570. MR 0404124
(53 #7928), http://dx.doi.org/10.1090/S00255718197604041249
 [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)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
30D05,
33A25
Retrieve articles in all journals
with MSC:
30D05,
33A25
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198909681484
PII:
S 00255718(1989)09681484
Article copyright:
© Copyright 1989 American Mathematical Society
