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:
- 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.
- Start out by setting
- 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.
- Start out by setting
- 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:
- Start out by setting
- 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.
- Start out by setting