Borwein's algorithm

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.