Borwein's algorithm

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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[edit]

These two are examples of a Ramanujan–Sato series. The related Chudnovsky algorithm uses a discriminant with class number 1.

Class number 2 (1989)[edit]

Start by setting[citation needed]

Then

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

Class number 4 (1993)[edit]

Start by setting[citation needed]

Then

Each additional term of the series yields approximately 50 digits.

Iterative algorithms[edit]

Quadratic convergence (1984)[edit]

Start by setting[2]

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)[edit]

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)[edit]

Start by setting[3]

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:[4]

Quintic convergence[edit]

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[edit]

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]

See also[edit]

References[edit]

  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. ^ Arndt, Jörg; Haenel, Christoph (1998). π Unleashed. Springer-Verlag. p. 236. ISBN 3-540-66572-2.
  3. ^ Mak, Ronald (2003). The Java Programmers Guide to Numerical Computation. Pearson Educational. p. 353. ISBN 0-13-046041-9.
  4. ^ Milla, Lorenz (2019), Easy Proof of Three Recursive π-Algorithms, arXiv:1907.04110
  5. ^ Henrik Vestermark (4 November 2016). "Practical implementation of π Algorithms" (PDF). Retrieved 29 November 2020.

External links[edit]