Quadratic convergence (1984)
Start by setting[4]

Then iterate

Then pk converges quadratically to π; that is, each iteration approximately doubles the number of correct digits. The algorithm is not self-correcting; each iteration must be performed with the desired number of correct digits for π's final result.
Cubic convergence (1991)
Start by setting

Then iterate

Then ak converges cubically to 1/π; that is, each iteration approximately triples the number of correct digits.
Quartic convergence (1985)
Start by setting[5]

Then iterate

Then ak converges quartically against 1/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 for π's final result.
One iteration of this algorithm is equivalent to two iterations of the Gauss–Legendre algorithm.
A proof of these algorithms can be found here:[6]
Quintic convergence
Start by setting

where
is the golden ratio. Then iterate

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

Nonic convergence
Start by setting

Then iterate

Then ak converges nonically to 1/π; that is, each iteration approximately multiplies the number of correct digits by nine.[7]