Borwein's algorithm

In mathematics, Borwein's algorithm is an algorithm devised by Jonathan and Peter Borwein to calculate the value of 1/π. See also Bailey-Borwein-Plouffe formula.

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.

[edit] 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 π
Bailey-Borwein-Plouffe formula