Borwein's algorithm

Borwein's algorithm was devised by Jonathan and Peter Borwein to calculate the value of. This and other algorithms can be found in 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 p_k converges quadratically to ; 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 's final result.

Cubic convergence (1991)

Start by setting
Then iterate
Then a_k converges cubically to ; that is, each iteration approximately triples the number of correct digits.

Quartic convergence (1985)

Start by setting
Then iterate
Then a_k converges quartically against ; 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 '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 is the golden ratio. Then iterate
Then a_k converges quintically to, and the following condition holds:

Nonic convergence

Start by setting
Then iterate
Then a_k converges nonically to ; that is, each iteration approximately multiplies the number of correct digits by nine.