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


and define the initial conditions to be a_0=1, b_0=1/sqrt(2). Then iterating a_n and b_n gives the arithmetic-geometric mean M(a,b), and pi is given by


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

See also

Arithmetic-Geometric Mean, Pi Iterations

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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 pi: The Gauss-Salamin Algorithm." Math. Gaz. 76, 231-242, 1992.Salamin, E. "Computation of pi Using Arithmetic-Geometric Mean." Math. Comput. 30, 565-570, 1976.

Referenced on Wolfram|Alpha

Brent-Salamin Formula

Cite this as:

Weisstein, Eric W. "Brent-Salamin Formula." From MathWorld--A Wolfram Web Resource.

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