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More quadratically converging algorithms for $\pi$


Authors: J. M. Borwein and P. B. Borwein
Journal: Math. Comp. 46 (1986), 247-253
MSC: Primary 65D20
DOI: https://doi.org/10.1090/S0025-5718-1986-0815846-6
MathSciNet review: 815846
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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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Keywords: <IMG WIDTH="18" HEIGHT="18" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$\pi$">, arithmetic-geometric mean iteration, high-precision calculation
Article copyright: © Copyright 1986 American Mathematical Society