## Computation of $\pi$ using arithmetic-geometric mean

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- by Eugene Salamin PDF
- Math. Comp.
**30**(1976), 565-570 Request permission

## Abstract:

A new formula for $\pi$ is derived. It is a direct consequence of Gaussâ€™ arithmetic-geometric mean, the traditional method for calculating elliptic integrals, and of Legendreâ€™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

- Milton Abramowitz and Irene A. Stegun,
*Handbook of mathematical functions with formulas, graphs, and mathematical tables*, National Bureau of Standards Applied Mathematics Series, No. 55, U. S. Government Printing Office, Washington, D.C., 1964. For sale by the Superintendent of Documents. MR**0167642**
R. FINKEL, L. GUIBAS & C. SIMONYI, private communication.
K. F. GAUSS, - Alfred George Greenhill,
*The applications of elliptic functions*, Dover Publications, Inc., New York, 1959. MR**0111864** - Harris Hancock,
*Elliptic integrals*, Dover Publications, Inc., New York, 1958. MR**0099454**
H. JEFFREYS & B. S. JEFFREYS, - 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**
A. M. LEGENDRE, - A. SchĂ¶nhage and V. Strassen,
*Schnelle Multiplikation grosser Zahlen*, Computing (Arch. Elektron. Rechnen)**7**(1971), 281â€“292 (German, with English summary). MR**292344**, DOI 10.1007/bf02242355

*Werke*, Bd. 3, Gottingen, 1866, pp. 331-403.

*Methods of Mathematical Physics*, 3rd ed., Cambridge Univ. Press, London, 1962. L. V. KING,

*On the Direct Numerical Calculation of Elliptic Functions and Integrals*, Cambridge Univ. Press, London, 1924.

*Exercices de calcul intĂ©gral.*Vol. 1, 1811.

## Additional Information

- © Copyright 1976 American Mathematical Society
- Journal: Math. Comp.
**30**(1976), 565-570 - MSC: Primary 10A30; Secondary 10A40, 33A25
- DOI: https://doi.org/10.1090/S0025-5718-1976-0404124-9
- MathSciNet review: 0404124