## Brent-Salamin Formula

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.