Gauss–Legendre algorithm: Difference between revisions

Revision as of 18:22, 10 September 2009

Revision as of 18:26, 10 September 2009

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2. Repeat the following instructions until the difference of <math>a_n\!</math> and <math>b_n\!</math> is within the desired accuracy:

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

:<math> b_{n+1} & = \sqrt{a_n b_n}\!</math>, \\

:<math> t_{n+1} & = t_n - p_n(a_n - a_{n+1})^2, \\!</math>

p_{n+1} & = 2p_n.

\end{align}

</math>

3. &pi; is then approximated as:

:<math>b_{n+1} = \sqrt{a_n b_n}\!</math>,

:<math>t_{n+1} = t_n - p_n(a_n - a_{n+1})^2,\!</math>

:<math>p_{n+1} = 2p_n.\!</math>

3. &pi; is approximated with <math>a_n\!</math>, <math>b_n\!</math> and <math>t_n\!</math> as:

:<math>\pi \approx \frac{(a_n+b_n)^2}{4t_n}.\!</math>