Editing Gauss–Legendre algorithm (section)

== 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}.</math>
# Repeat the following instructions until the difference between <math>a_{n+1}</math> and <math>b_{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}, \\
                      \\
p_{n+1} & = 2p_n, \\
                      \\
t_{n+1} & = t_n - p_n(a_{n+1}-a_{n})^2. \\
        \end{align}
</math>
# {{pi}} is then approximated as: <math display="block">\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):

:<math>3.140\dots</math>
:<math>3.14159264\dots</math>
:<math>3.1415926535897932382\dots</math>
:<math>3.14159265358979323846264338327950288419711\dots</math>
:<math>3.141592653589793238462643383279502884197169399375105820974944592307816406286208998625\dots</math>
The algorithm has [[quadratic convergence]], which essentially means that the number of correct digits doubles with each [[iteration]] of the algorithm.