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.[1]
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.
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:[4]
Quintic convergence
Start by setting
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.[5]
^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, ISBN3-540-66572-2
^Arndt, Jörg; Haenel, Christoph (1998). π Unleashed. Springer-Verlag. p. 236. ISBN3-540-66572-2.
^Mak, Ronald (2003). The Java Programmers Guide to Numerical Computation. Pearson Educational. p. 353. ISBN0-13-046041-9.
^Milla, Lorenz (2019), Easy Proof of Three Recursive π-Algorithms, arXiv:1907.04110