# Gauss–Legendre algorithm

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Gauss–Legendre algorithm

The Gauss–Legendre algorithm is an algorithm to compute the digits of &pi;. It is notable for being rapidly convergent, with only 25 iterations producing 45 million correct digits of &pi;. However, the drawback is that it is memory intensive and it is therefore sometimes not used over Machin-like formulas.

The method is based on the individual work of Carl Friedrich Gauss (1777–1855) and Adrien-Marie Legendre (1752–1833) combined with modern algorithms for multiplication and square roots. It repeatedly replaces two numbers by their arithmetic and geometric mean, in order to approximate their arithmetic-geometric mean.

The version presented below is also known as the Brent–Salamin (or Salamin–Brent) algorithm; it was independently discovered in 1975 by Richard Brent and Eugene Salamin. It was used to compute the first 206,158,430,000 decimal digits of &pi; on September 18 to 20, 1999, and the results were checked with Borwein's algorithm.

1. Initial value setting:

:$a_0 = 1qquad b_0 = frac\left\{1\right\}\left\{sqrt\left\{2qquad t_0 = frac\left\{1\right\}\left\{4\right\}qquad p_0 = 1!$

2. Repeat the following instructions until the difference of $a_n!$ and $b_n!$ is within the desired accuracy:

:$a_\left\{n+1\right\} = frac\left\{a_n + b_n\right\}\left\{2\right\}!$,

:$b_\left\{n+1\right\} = sqrt\left\{a_n b_n\right\}!$,:$t_\left\{n+1\right\} = t_n - p_n\left(a_n - a_\left\{n+1\right\}\right)^2!$,:$p_\left\{n+1\right\} = 2p_n!$.

3. &pi; is approximated with $a_n!$, $b_n!$ and $t_n!$ as:

:$pi approx frac\left\{\left(a_n+b_n\right)^2\right\}\left\{4t_n\right\}!$.

The first three iterations give:

:$3.140...!$:$3.14159264...!$:$3.14159265358979...!$

The algorithm has second-order convergent nature, which essentially means that the number of correct digits doubles with each step of the algorithm.

Mathematical background

Limits of the arithmetic-geometric mean

The arithmetic-geometric mean of two numbers, $a_0!$ and $b_0!$, is found by calculating the limit of the sequences $a_\left\{n+1\right\}=\left\{a_n+b_n over 2\right\}!$, $b_\left\{n+1\right\}=sqrt\left\{a_n b_n\right\}!$, which both converge to the same limit.If $a_0=1!$ and $b_0=cosphi!$ then the limit is $\left\{pi over 2K\left(sinphi\right)\right\}!$ where $K\left(k\right)!$ is the complete elliptic integral of the first kind:$K\left(k\right) = int_0^\left\{frac\left\{pi\right\}\left\{2 frac\left\{d heta\right\}\left\{sqrt\left\{1-k^2 sin^2 heta!$.

If $c_0 = sinphi!$, $c_\left\{i+1\right\} = a_i - a_\left\{i+1\right\}!$. then:$sum_\left\{i=0\right\}^\left\{infty\right\} 2^\left\{i-1\right\} c_i^2 = 1 - \left\{E\left(sinphi\right)over K\left(sinphi\right)\right\}!$where $E\left(k\right)!$ be the complete elliptic integral of the second kind::$E\left(k\right) = int_0^\left\{frac\left\{pi\right\}\left\{2sqrt \left\{1-k^2 sin^2 heta\right\} d heta!$.Gauss knew of both of these results.Citation
last=Brent
first=Richard
publication-date=
date=
year=1975
title=Multiple-precision zero-finding methods and the complexity of elementary function evaluation
periodical=Analytic Computational Complexity
series=
publication-place=New York
place=
editor-last=Traub
editor-first=J F
volume=
issue=
pages=151–176
url=http://wwwmaths.anu.edu.au/~brent/pub/pub028.html
issn=
doi=
oclc=
accessdate=8 September 2007
]

Legendre’s identity

For $phi!$ and $heta!$ such that $phi+ heta=\left\{1 over 2\right\}pi!$ Legendre proved the identity::$K\left(sin phi\right) E\left(sin heta \right) + K\left(sin heta \right) E\left(sin phi\right) - K\left(sin phi\right) K\left(sin heta\right) = \left\{1 over 2\right\}pi!$

Gauss–Legendre method

The values $phi= heta=\left\{piover 4\right\}!$ can be substituted into Legendre’s identity and the approximations to K, E can be found by terms in the sequences for the arithmetic geometric mean with $a_0=1!$ and $b_0=sin\left\{pi over 4\right\}=frac\left\{1\right\}\left\{sqrt\left\{2!$.

ee also

*Borwein's algorithm

References

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