Gauss–Legendre algorithm: Difference between revisions

Revision as of 19:17, 15 January 2022

Revision as of 00:07, 7 February 2022

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== Algorithm ==

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

1. Initial value setting:

2.# Repeat the following instructions until the difference of <math>a_n</math> and <math>b_n</math> is within the desired accuracy: <math display="block"> \begin{align}

:<math>a_0 = 1\qquad b_0 = \frac{1}{\sqrt{2}}\qquad t_0 = \frac{1}{4}\qquad p_0 = 1.</math>

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

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}, \\

\\

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

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\end{align}

</math>

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

:<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):