Borwein's algorithm
In mathematics, Borwein's algorithm is an algorithm devised by Jonathan and Peter Borwein to calculate the value of 1/π. They devised several other algorithms. They published the book Pi and the AGM – A Study in Analytic Number Theory and Computational Complexity.
Ramanujan–Sato series
These two are examples of a Ramanujan–Sato series. The related Chudnovsky algorithm uses a discriminant with class number 1.
Class number 2 (1989)
Start by setting
Then
Each additional term of the partial sum yields approximately 25 digits.
Class number 4 (1993)
Start by setting
Then
Each additional term of the series yields approximately 50 digits.
Iterative algorithms
Quadratic convergence (1984)
Start by setting
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
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:
Quintic convergence
Start by setting
where 
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.