Borwein's Algorithm

• Start out by setting

  $a\_0\; =\; 6\; -\; 4\backslash sqrt\{2\}$

  $y\_0\; =\; \backslash sqrt\{2\}\; -\; 1$

• Then iterate

  $y\_\{k+1\}\; =\; \backslash frac\{1-(1-y\_k^4)^\{1/4\}\}\{1+(1-y\_k^4)^\{1/4\}\}$

  $a\_\{k+1\}\; =\; a\_k(1+y\_\{k+1\})^4\; -\; 2^\{2k+3\}\; y\_\{k+1\}\; (1\; +\; y\_\{k+1\}\; +\; y\_\{k+1\}^2)$

See also

• Borwein's algorithm (others) for an explanation of other algorithms by Jonathan and Peter Borwein to determine the digits of π.

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