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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



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