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

$$a_{n+1} = {\textstyle{1\over 2}}(a_n+b_n)$$ (1)

$$b_{n+1} = \sqrt{a_nb_n}$$ (2)

$$c_{n+1} = {\textstyle{1\over 2}}(a_n-b_n)$$ (3)

$$d_n \equiv {{a_n}^2-{b_n}^2},$$ (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, and $\pi$ is given by

$$\pi = {4[M(1,2^{-1/2})]^2\over 1-\sum_{j=1}^\infty 2^{j+1} d_j}$$ (5)

$$= {4[M(1,2^{-1/2})]^2\over 1-\sum_{j=1}^\infty 2^{j+1} {c_j}^2}.$$ (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

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.