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