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Borwein's algorithm

Method for calculating the value of pi From Wikipedia, the free encyclopedia

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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.[1]

Ramanujan–Sato series

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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[2]

Then

Each additional term of the partial sum yields approximately 25 digits.

Class number 4 (1993)

Start by setting[3]

Then

Each additional term of the series yields approximately 50 digits.

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Iterative algorithms

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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]

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See also

References

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