Brent-Salamin Formula
A formula which uses the Arithmetic-Geometric Mean to compute Pi. It has quadratic convergence andis 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 theArithmetic-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 theother.See also Arithmetic-Geometric Mean, Pi
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 Salamin, E. ``Computation of : The Gauss-Salamin Algorithm.'' Math. Gaz. 76, 231-242, 1992. Using Arithmetic-Geometric Mean.'' Math. Comput. 30, 565-570, 1976.
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