Gauss-Legendre algorithm

The Gauss-Legendre algorithm is an algorithm to compute the digits of π.

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 Salamin-Brent algorithm; it was independently discovered in 1976 by Eugene Salamin and Richard Brent. It was used to compute the first 206,158,430,000 decimal digits of π on September 18 to 20, 1999, and the results were checked with Borwein's algorithm.

1. Initial value setting;

a = 1\qquad b = \frac{1}{\sqrt{2}}\qquad t = \frac{1}{4}\qquad p = 1

2. Repeat the following instructions until the difference of a and b is within the desired accuracy:

x = \frac{a + b}{2}
y = \sqrt{ab}
t = t − p(a − x)2
a = x
b = y
p = 2p

3. π is approximated with a, b and t as:

\pi \approx \frac{(a+b)^2}{4t}

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

See also

See also: Gauss-Legendre algorithm, π, 1752, 1777, 1833, 1855, 1976, Adrien-Marie Legendre, Algorithm, Arithmetic-geometric mean