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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

a_(n+1)=1/2(a_n+b_n)
(1)
b_(n+1)=sqrt(a_nb_n)
(2)
c_(n+1)=1/2(a_n-b_n)
(3)
d_n=a_n^2-b_n^2,
(4)

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

pi=(4[M(1,2^(-1/2))]^2)/(1-sum_(j=1)^(infty)2^(j+1)d_j)
(5)
=(4[M(1,2^(-1/2))]^2)/(1-sum_(j=1)^(infty)2^(j+1)c_j^2).
(6)

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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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 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. https://mathworld.wolfram.com/Brent-SalaminFormula.html

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