Borwein's algorithm

Borwein's algorithm is an algorithm devised by Jonathan and Peter Borwein to calculate the value of 1/π.

It works as follows:

Start out by setting
$a_0 = 6 - 4\sqrt{2}$

$y_0 = \sqrt{2} - 1$
Then iterate
$y_{k+1} = \frac{1-(1-y_k^4)^{1/4}}{1+(1-y_k^4)^{1/4}}$

$a_{k+1} = a_k(1+y_{k+1})^4 - 2^{2k+3} y_{k+1} (1 + y_{k+1} + y_{k+1}^2)$

Then a_k converges quartically against 1/π; that is, each iteration approximately quadruples the number of correct digits.

See also

Borwein's algorithm (others) for an explanation of other algorithms by Jonathan and Peter Borwein to determine the digits of π.
Gauss-Legendre algorithm - another algorithm to calculate pi

See also: Borwein's algorithm, Algorithm, Borwein's algorithm (others), Gauss-Legendre algorithm, Jonathan Borwein, Peter Borwein, Pi