Gauss–Legendre algorithm: Difference between revisions

Revision as of 19:18, 5 April 2019

Revision as of 13:16, 7 April 2019

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== Algorithm ==

#1. 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>

#2. 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, \\

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\end{align}

</math>

#3. {{pi}} is then approximated as:<br />
:<math>\pi \approx \frac{(a_{n+1}+b_{n+1})^2}{4t_{n+1}}.\!</math>

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