Brent-Salamin Formula

The Brent-Salamin formula, also called the Gauss-Salamin formula or Salamin formula, is a formula that uses the arithmetic-geometric mean to compute pi. It has quadratic convergence. 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.