Gauss–Legendre algorithm: Difference between revisions

Revision as of 11:14, 30 October 2014

Revision as of 19:42, 25 November 2014

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== Algorithm ==

# Initial value setting:<br /><math>a_0 = 1\qquad b_0 = \frac{1}{\sqrt{2}}\qquad t_0 = \frac{1}{4}\qquad p_0 = 1.\!</math>

# Repeat the following instructions until the difference of <math>a_n\!</math> and <math>b_n\!</math> is within the desired accuracy:<br /><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_n2^{n}.

\end{align}

</math>