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 a book: Jonathon 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.

Jonathan Borwein and Peter Borwein's version (1993)[edit]

Start out by setting[citation needed]

Then

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

Cubic convergence (1991)[edit]

Start out by setting

Then iterate

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

Another formula for π (1989)[edit]

Start out by setting[citation needed]

Then

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

Quartic algorithm (1985)[edit]

Start out by setting[1]

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.

Quadratic convergence (1984)[edit]

Start out by setting[2] [3]

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.

Quintic convergence[edit]

Start out 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:

[4]

Nonic convergence[edit]

Start out by setting

Then iterate

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

See also[edit]

References[edit]

  1. ^ Mak, Ronald (2003). The Java Programmers Guide to Numerical Computation. Pearson Educational. p. 353. ISBN 0-13-046041-9. 
  2. ^ Arndt, Jörg; Haenel, Christoph (1998). π Unleashed. Springer-Verlag. p. 236. ISBN 3-540-66572-2. 
  3. ^ Template:Pi Unleashed
  4. ^ http://www.cecm.sfu.ca/organics/papers/garvan/paper/html/node12.html