A formula which uses the Arithmetic-Geometric Mean to compute Pi. It has quadratic convergence and
is also called the Gauss-Salamin Formula and Salamin Formula. Let

(1) | |||

(2) | |||

(3) | |||

(4) |

and define the initial conditions to be , . Then iterating and gives the Arithmetic-Geometric Mean, and is given by

(5) | |||

(6) |

King (1924) showed that this formula and the Legendre Relation are equivalent and that either may be derived from the other.

**References**

Borwein, J. M. and Borwein, P. B. *Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity.*
New York: Wiley, pp. 48-51, 1987.

Castellanos, D. ``The Ubiquitous Pi. Part II.'' *Math. Mag.* **61**, 148-163, 1988.

King, L. V. *On the Direct Numerical Calculation of Elliptic Functions and Integrals.* Cambridge, England: Cambridge University Press, 1924.

Lord, N. J. ``Recent Calculations of : The Gauss-Salamin Algorithm.'' *Math. Gaz.* **76**, 231-242, 1992.

Salamin, E. ``Computation of Using Arithmetic-Geometric Mean.'' *Math. Comput.* **30**, 565-570, 1976.

© 1996-9

1999-05-26