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]

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[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.

Iterative algorithms

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]

See also

References

  1. 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, ISBN 3-540-66572-2
  2. Bailey, David H (2023-04-01). "Peter Borwein: A Visionary Mathematician". Notices of the American Mathematical Society. 70 (04): 610–613. doi:10.1090/noti2675. ISSN 0002-9920.
  3. Borwein, J.M.; Borwein, P.B. (1993). "Class number three Ramanujan type series for 1/π". Journal of Computational and Applied Mathematics. 46 (1–2): 281–290. doi:10.1016/0377-0427(93)90302-R.
  4. Arndt, Jörg; Haenel, Christoph (1998). π Unleashed. Springer-Verlag. p. 236. ISBN 3-540-66572-2.
  5. Mak, Ronald (2003). The Java Programmers Guide to Numerical Computation. Pearson Educational. p. 353. ISBN 0-13-046041-9.
  6. Milla, Lorenz (2019), Easy Proof of Three Recursive π-Algorithms, arXiv:1907.04110
  7. Henrik Vestermark (4 November 2016). "Practical implementation of π Algorithms" (PDF). Retrieved 29 November 2020.
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