Borwein's algorithm
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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 ak 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