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

  <math>a-0 = 6 - 4\sqrt{2}</math>

  <math>y-0 = \sqrt{2} - 1</math>

• Then iterate

  <math>y-{k+1} = \frac{1-(1-y-k^4)^{1/4}}{1+(1-y-k^4)^{1/4}}</math>

  <math>a-{k+1} = a-k(1+y-{k+1})^4 - 2^{2k+3} y-{k+1} (1 + y-{k+1} + y-{k+1}^2)</math>

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.