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# More Quadratically Converging Algorithms for $\pi$

J. M. Borwein and P. B. Borwein
Mathematics of Computation
Vol. 46, No. 173 (Jan., 1986), pp. 247-253
DOI: 10.2307/2008229
Stable URL: http://www.jstor.org/stable/2008229
Page Count: 7
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## Abstract

We present a quadratically converging algorithm for $\pi$ based on a formula of Legendre's for complete elliptic integrals of modulus $\sin(\pi/12)$ and the arithmetic-geometric mean iteration of Gauss and Legendre. Precise asymptotics are provided which show this algorithm to be (marginally) the most efficient developed to date. As such it provides a natural computational check for the recent large-scale calculations of $\pi$.

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