Gauss–Legendre algorithm: Difference between revisions

Revision as of 12:47, 21 August 2024

Revision as of 22:53, 4 September 2024

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== Algorithm ==

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

# Repeat the following instructions until the difference ofbetween <math>a_na_{n+1}</math> and <math>b_nb_{n+1}</math> is within the desired accuracy: <math display="block"> \begin{align}

a_{n+1} & = \frac{a_n + b_n}{2}, \\

\\

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

\\

t_p_{n+1} & = t_n - p_n(a_{n}-a_{n+1})^22p_n, \\

\\

p_{n+1} & = 2p_n.

\\

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

\end{align}

</math>