Gauss–Legendre algorithm: Difference between revisions

Revision as of 00:20, 29 July 2013

Revision as of 12:10, 9 August 2013

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:<math> \begin{align} a_{n+1} & = \frac{a_n + b_n}{2}, \\

b_{n+1} & = \sqrt{a_n b_n}, \\

t_{n+1} & = t_n -+ p_n(a_n - a_{n+1})^2, \\

p_{n+1} & = 2p_n.

\end{align}

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3. π is then approximated as:

:<math>\pi \approx \frac{(a_n^2+b_n)^2}{4t_n1-t_{n+1}}.\!</math>

The first three iterations give (approximations given up to and including the first incorrect digit):