Encyclopedia > Borwein's algorithm

  Article Content

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 ak converges quartically against 1/π; that is, each iteration approximately quadruples the number of correct digits.

See also



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Holtsville, New York

... of 18, 7.5% from 18 to 24, 33.5% from 25 to 44, 23.9% from 45 to 64, and 6.9% who are 65 years of age or older. The median age is 34 years. For every 100 females there ...

 
 
 
This page was created in 22.1 ms