Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

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

Abstract | References | Similar Articles | Additional Information

Abstract: The iterative process

$\displaystyle {a_{n + 1}} = ({a_n} + 3{b_n})/4,\quad {b_{n + 1}} = (\sqrt {{a_n}{b_n}} + {b_n})/2$

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 [Enhancements On Off] (What's this?)

  • [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 Arithmetisch-geometrische Mittel aus vier Elementen, Berl. Monatsber., 1876, pp. 611-621. Werke, Berlin, 1888, pp. 329-338.
  • [3] J. M. Borwein & P. B. Borwein, "The arithmetic-geometric mean and fast computation of the elementary functions," SIAM Rev., v. 26, 1984, pp. 351-365. MR 750454 (86d:65029)
  • [4] J. M. Borwein & P. B. Borwein, Pi and the AGM--A 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. 519-522. MR 1541118
  • [6] R. P. Brent, "Fast multiple-precision evaluation of elementary functions," J. Assoc. Comput. Mach., v. 23, 1976, pp. 242-251. MR 0395314 (52:16111)
  • [7] B. C. Carlson, "Algorithms involving arithmetic and geometric means," Amer. Math. Monthly, v. 98, 1971, pp. 496-505. MR 0283246 (44:479)
  • [8] D. H. Lehmer, "On the compounding of certain means," J. Math. Anal. Appl., v. 36, 1971, pp. 183-200. 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. 207-210. MR 771042 (86e:65030)
  • [10] J. Peetre, "Generalizing the arithmetic-geometric mean--A hapless computer experiment," preprint.
  • [11] E. Salamin, "Computation of $ \pi $ using arithmetic-geometric mean," Math. Comp., v. 30, 1976, pp. 565-570. 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: https://doi.org/10.1090/S0025-5718-1989-0968148-4
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society