More quadratically converging algorithms for
Authors:
J. M. Borwein and P. B. Borwein
Journal:
Math. Comp. 46 (1986), 247253
MSC:
Primary 65D20
MathSciNet review:
815846
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We present a quadratically converging algorithm for based on a formula of Legendre's for complete elliptic integrals of modulus and the arithmeticgeometric mean iteration of Gauss and Legendre. Precise asymptotics are provided which show this algorithm to be (marginally) the most efficient developed to date. As such it provides a natural computational check for the recent largescale calculations of .
 [1]
Petr
Beckmann, A history of 𝜋 (pi), 2nd ed., The Golem
Press, Boulder, Colo., 1971. MR 0449960
(56 #8261)
 [2]
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
 [3]
Richard
P. Brent, Fast multipleprecision evaluation of elementary
functions, J. Assoc. Comput. Mach. 23 (1976),
no. 2, 242–251. MR 0395314
(52 #16111)
 [4]
K. F. Gauss, Werke, Bd. 3, Noordhoff, Göttingen, 1866, pp. 361403.
 [5]
L. V. King, On The Direct Numerical Calculation of Elliptic Functions and Integrals, Cambridge Univ. Press, New York, 1924.
 [6]
A. M. Legendre, Exercises de Calcul Integral, Vol. 1, Dunod, Paris, 1811.
 [7]
Donald
J. Newman, Rational approximation versus fast computer
methods, Lectures on approximation and value distribution,
Sém. Math. Sup., vol. 79, Presses Univ. Montréal,
Montreal, Que., 1982, pp. 149–174. MR 654686
(83e:41021)
 [8]
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
 [9]
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
 [10]
Y. Tamura & Y. Kanada, Calculation of to 4,194,293 Decimals Based on GaussLegendre Algorithm. (Preprint.)
 [11]
E.
T. Whittaker and G.
N. Watson, A course of modern analysis, Cambridge Mathematical
Library, Cambridge University Press, Cambridge, 1996. An introduction to
the general theory of infinite processes and of analytic functions; with an
account of the principal transcendental functions; Reprint of the fourth
(1927) edition. MR 1424469
(97k:01072)
 [12]
J. W. Wrench, Jr., "The evolution of extended decimal approximations to ," The Mathematics Teacher, v. 53, 1960, pp. 644650.
 [1]
 P. Beckman, A History of Pi, 4th ed., Golem Press, Boulder, Colorado, 1977. MR 0449960 (56:8261)
 [2]
 J. M. Borwein & P. B. Borwein, "The arithmeticgeometric mean and fast computation of elementary functions," SIAM Rev., v. 26, 1984, pp. 351366. MR 750454 (86d:65029)
 [3]
 R. P. Brent, "Fast multipleprecision evaluation of elementary functions," J. Assoc. Comput. Mach., v. 23, 1976, pp. 242251. MR 0395314 (52:16111)
 [4]
 K. F. Gauss, Werke, Bd. 3, Noordhoff, Göttingen, 1866, pp. 361403.
 [5]
 L. V. King, On The Direct Numerical Calculation of Elliptic Functions and Integrals, Cambridge Univ. Press, New York, 1924.
 [6]
 A. M. Legendre, Exercises de Calcul Integral, Vol. 1, Dunod, Paris, 1811.
 [7]
 D. J. Newman, "Rational approximation versus fast computer methods," Lectures on Approximation and Value Distribution, Presses de l'Université de Montréal, 1982, pp. 149174. MR 654686 (83e:41021)
 [8]
 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)
 [9]
 E. Salamin, "Computation of using arithmeticgeometric mean," Math. Comp., v. 30, 1976, pp. 565570. MR 0404124 (53:7928)
 [10]
 Y. Tamura & Y. Kanada, Calculation of to 4,194,293 Decimals Based on GaussLegendre Algorithm. (Preprint.)
 [11]
 E. T. Whittaker & G. N. Watson, A Course of Modern Analysis, 4th. ed., Cambridge Univ. Press, New York, 1927. MR 1424469 (97k:01072)
 [12]
 J. W. Wrench, Jr., "The evolution of extended decimal approximations to ," The Mathematics Teacher, v. 53, 1960, pp. 644650.
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65D20
Retrieve articles in all journals
with MSC:
65D20
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198608158466
PII:
S 00255718(1986)08158466
Keywords:
,
arithmeticgeometric mean iteration,
highprecision calculation
Article copyright:
© Copyright 1986 American Mathematical Society
