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Computation of $\pi$ Using Arithmetic-Geometric Mean

Eugene Salamin
Mathematics of Computation
Vol. 30, No. 135 (Jul., 1976), pp. 565-570
DOI: 10.2307/2005327
Stable URL: http://www.jstor.org/stable/2005327
Page Count: 6
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Computation of $\pi$ Using Arithmetic-Geometric Mean
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Abstract

A new formula for $\pi$ is derived. It is a direct consequence of Gauss' arithmetic-geometric mean, the traditional method for calculating elliptic integrals, and of Legendre's relation for elliptic integrals. The error analysis shows that its rapid convergence doubles the number of significant digits after each step. The new formula is proposed for use in a numerical computation of $\pi$, but no actual computational results are reported here.

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