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Borwein's algorithm (others)

Jonathan and Peter Borwein devised various algorithms to calculate the value of &pi. The most prominent and oft-used one is explained under Borwein's algorithm. Other algorithms found by them include the following:

  1. Cubical covergence, 1991:

    • Start out by setting

    • Then iterate

    Then ak converges cubically against 1/π; that is, each iteration approximately triples the number of correct digits.

  2. Quartical covergence, 1984:

    • Start out by setting

    • Then iterate

    Then pk converges quartically against π; that is, each iteration approximately quadruples the number of correct digits. The algorithm is not self-correcting; each iteration must be performed with the desired number of correct digits of π.

  3. Quintical covergence:

    • Start out by setting

    • Then iterate

    Then ak converges quintically against 1/π (that is, each iteration approximately quintuples the number of correct digits), and the following condition holds:

  4. Nonical covergence:

    • Start out by setting

    • Then iterate

    Then ak converges nonically against 1/π; that is, each iteration approximately multiplies the number of correct digits by nine.