Chudnovsky algorithm

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The Chudnovsky algorithm is a fast method for calculating the digits of π, based on Ramanujan’s π formulae. It was published by the Chudnovsky brothers in 1988[1] and was used in the world record calculations of 2.7 trillion digits of π in December 2009,[2] 10 trillion digits in October 2011,[3][4] 22.4 trillion digits in November 2016,[5] 31.4 trillion digits in September 2018–January 2019,[6] and 50 trillion digits on January 29, 2020.[7]

The algorithm is based on the negated Heegner number , the j-function , and on the following rapidly convergent generalized hypergeometric series:[2]

A detailed proof of this formula can be found here:[8]

For a high performance iterative implementation, this can be simplified to

There are 3 big integer terms (the multinomial term Mq, the linear term Lq, and the exponential term Xq) that make up the series and π equals the constant C divided by the sum of the series, as below:

, where:
,
,
,
.

The terms Mq, Lq, and Xq satisfy the following recurrences and can be computed as such:

The computation of Mq can be further optimized by introducing an additional term Kq as follows:

Note that

and

This identity is similar to some of Ramanujan's formulas involving π,[2] and is an example of a Ramanujan–Sato series.

The time complexity of the algorithm is .[9]

See also[edit]

References[edit]

  1. ^ Chudnovsky, David; Chudnovsky, Gregory (1988), Approximation and complex multiplication according to ramanujan, Ramanujan revisited: proceedings of the centenary conference
  2. ^ a b c Baruah, Nayandeep Deka; Berndt, Bruce C.; Chan, Heng Huat (2009), "Ramanujan's series for 1/π: a survey", American Mathematical Monthly, 116 (7): 567–587, doi:10.4169/193009709X458555, JSTOR 40391165, MR 2549375
  3. ^ Yee, Alexander; Kondo, Shigeru (2011), 10 Trillion Digits of Pi: A Case Study of summing Hypergeometric Series to high precision on Multicore Systems, Technical Report, Computer Science Department, University of Illinois, hdl:2142/28348
  4. ^ Aron, Jacob (March 14, 2012), "Constants clash on pi day", New Scientist
  5. ^ "22.4 Trillion Digits of Pi". www.numberworld.org.
  6. ^ "Google Cloud Topples the Pi Record". www.numberworld.org/.
  7. ^ "The Pi Record Returns to the Personal Computer". www.numberworld.org/.
  8. ^ Milla, Lorenz (2018), A detailed proof of the Chudnovsky formula with means of basic complex analysis, arXiv:1809.00533
  9. ^ "y-cruncher - Formulas". www.numberworld.org. Retrieved 2018-02-25. CS1 maint: discouraged parameter (link)