Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

Computation of $ \pi $ using arithmetic-geometric mean


Author: Eugene Salamin
Journal: Math. Comp. 30 (1976), 565-570
MSC: Primary 10A30; Secondary 10A40, 33A25
MathSciNet review: 0404124
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A new formula for $ \pi $ is derived. It is a direct consequence of Gauss' arithmetic-geometric mean, the traditional method for calculating elliptic integrals, and of Legendre's relation for elliptic integrals. The error analysis shows that its rapid convergence doubles the number of significant digits after each step. The new formula is proposed for use in a numerical computation of $ \pi $, but no actual computational results are reported here.


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

  • [1] Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C., 1964. MR 0167642 (29 #4914)
  • [2] R. FINKEL, L. GUIBAS & C. SIMONYI, private communication.
  • [3] K. F. GAUSS, Werke, Bd. 3, Gottingen, 1866, pp. 331-403.
  • [4] Alfred George Greenhill, The applications of elliptic functions, Dover Publications Inc., New York, 1959. MR 0111864 (22 #2724)
  • [5] Harris Hancock, Elliptic integrals, Dover Publications Inc., New York, 1958. MR 0099454 (20 #5893)
  • [6] H. JEFFREYS & B. S. JEFFREYS, Methods of Mathematical Physics, 3rd ed., Cambridge Univ. Press, London, 1962.
  • [7] L. V. KING, On the Direct Numerical Calculation of Elliptic Functions and Integrals, Cambridge Univ. Press, London, 1924.
  • [8] Donald E. Knuth, The art of computer programming. Vol. 2: Seminumerical algorithms, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont, 1969. MR 0286318 (44 #3531)
  • [9] A. M. LEGENDRE, Exercices de calcul intégral. Vol. 1, 1811.
  • [10] A. Schönhage and V. Strassen, Schnelle Multiplikation grosser Zahlen, Computing (Arch. Elektron. Rechnen) 7 (1971), 281–292 (German, with English summary). MR 0292344 (45 #1431)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 10A30, 10A40, 33A25

Retrieve articles in all journals with MSC: 10A30, 10A40, 33A25


Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1976-0404124-9
PII: S 0025-5718(1976)0404124-9
Keywords: $ \pi $, arithmetic-geometric mean, elliptic integral, Landen's transformation, Legendre's relation, fast Fourier transform multiplication
Article copyright: © Copyright 1976 American Mathematical Society