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.Jonathan M. Borwein, Peter B. Borwein, Pi and the AGM – A Study in Analytic Number Theory and Computational Complexity, Wiley, New York, 1987. Many of their results are available in: Jorg Arndt, Christoph Haenel, Pi Unleashed, Springer, Berlin, 2001,

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 ; 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 ak converges cubically to ; that is, each iteration approximately triples the number of correct digits.

Quartic convergence (1985)

Start by setting

Then iterate

Then ak 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 ak converges quintically to (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 ; that is, each iteration approximately multiplies the number of correct digits by nine.

Borwein's algorithm

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