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.

Start by setting

Then

Each additional term of the series yields approximately 50 digits. This is an example of a Ramanujan–Sato series.

Start by setting

Then iterate

Then a_k converges cubically to 1/π; that is, each iteration approximately triples the number of correct digits.

Start by setting

Then

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

Start by setting

Then iterate

Then a_k 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.

Start by setting

Then iterate

Then p_k converges quadraticly 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.

Start by setting

Then iterate

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

Start by setting

Then iterate

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

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