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m This important special case of Legendre's identity is due to Euler. A clarifying reference is added |
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The method is based on the individual work of [[Carl Friedrich Gauss]] (1777–1855) and [[Adrien-Marie Legendre]] (1752–1833) combined with modern algorithms for multiplication and [[square root]]s. It repeatedly replaces two numbers by their [[arithmetic mean|arithmetic]] and [[geometric mean]], in order to approximate their [[arithmetic-geometric mean]].
The version presented below is also known as the '''Gauss-Euler, Brent–Salamin (or Salamin–Brent) algorithm''';<ref>[[Richard Brent (scientist)|Brent, Richard]] ''Old and New Agorithms for pi'', Letters to the Editor, Notices of the AMS 60(1), p. 7</ref> it was independently discovered in 1975 by [[Richard Brent (scientist)|Richard Brent]] and [[Eugene Salamin (mathematician)|Eugene Salamin]]. It was used to compute the first 206,158,430,000 decimal digits of π on September 18 to 20, 1999, and the results were checked with [[Borwein's algorithm]].
== Algorithm ==
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:<math>K(\sin \varphi) E(\sin \theta ) + K(\sin \theta ) E(\sin \varphi) - K(\sin \varphi) K(\sin \theta) = {1 \over 2}\pi.\!</math><ref name="brent" />
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The values <math>\varphi=\theta={\pi\over 4}\!</math> can be substituted into Legendre’s identity and the approximations to K, E can be found by terms in the sequences for the arithmetic geometric mean with <math>a_0=1\!</math> and <math>b_0=\sin{\pi \over 4}=\frac{1}{\sqrt{2}}\!</math>.<ref>Adlaj,
== See also ==
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