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Mathematics of Computation

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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
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: $ \pi $, arithmetic-geometric mean iteration, high-precision calculation
Article copyright: © Copyright 1986 American Mathematical Society

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