Gauss–Legendre algorithm: Difference between revisions

Revision as of 02:18, 16 May 2014

Revision as of 09:40, 22 May 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>

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}, \\

1. Initial value setting:

:<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:

<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>

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

3. π is then approximated as:

<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):