Chudnovsky's algorithm

The Chudnovsky algorithm is a method to find decimal places of the circle number π. It was developed by the Chudnovsky brothers in 1988 and set the world records for calculating 2.7 trillion digits of π in December 2009, 5 trillion digits of π in August 2010, 10 trillion digits of π in October 2011, 12 trillion digits from 2013 as well as for the world record with 31.4 trillion digits from 2019 and the current world record with 50 trillion digits from 2020.

The algorithm is based on the convergence of a generalized hypergeometric series :

${\displaystyle {\frac {1}{\pi }}=12\sum _{k=0}^{\infty }{\frac {(-1)^{k}(6k)!(545140134k+13591409)}{(3k)!(k!)^{3}640320^{3k+3/2}}}.\!}$

A detailed proof of this formula can be found here:

Note:

${\displaystyle {\begin{aligned}e^{\pi {\sqrt {163}}}&=640320^{3}+743{,}9999999999992500725\ldots \\e^{{\frac {\pi }{3}}{\sqrt {163}}}&=640320{,}00000000060486373\ldots \end{aligned}}}$

This is similar to the Ramanujan formula for determining π, and is an example of the Ramanujan-Sato series .

Implementation of the algorithm

${\displaystyle \pi }$can be calculated with any precision by implementing the above-mentioned algorithm in a suitable programming environment. Here is an example using MATLAB :

function P = chud _ pi(d)
% CHUD _ PI Chudnovsky algorithm for pi.
% chud _ pi(d) produces d decimal digits.
k = sym(0);
s = sym(0);
sig = sym(1);
n = ceil(d/14); for j = 1:n
s = s + sig * prod(3*k+1:6*k)/prod(1:k)^3 * ... (13591409+545140134*k) / 640320^(3*k+3/2);
k = k+1;
sig = -sig; end
S= 1/(12*s); P = vpa(S,d);

Individual evidence

1. ↑ David Chudnovsky, Gregory Chudnovsky: Approximation and complex multiplication according to Ramanujan . In: Ramanujan revisited: proceedings of the centenary conference . 1988.
2. ↑ ^{a b c} Nayandeep Deka Baruah, Bruce C. Berndt, Heng Huat Chan: Ramanujan's series for 1 / π: a survey . In: American Mathematical Monthly . tape 116 , no. 7 , 2009, p. 567-587 , doi : 10.4169 / 193009709X458555 ( JSTOR: 40391165 mr: 2549375).
3. ↑ Geeks slice pi to 5 trillion decimal places. Australian Broadcasting Corporation , August 6, 2010; accessed October 31, 2014 . 
4. ↑ Alexander Yee, Shigeru Kondo: 10 Trillion Digits of Pi: A Case Study of summing Hypergeometric Series to high precision on Multicore Systems . 2011 ( hdl: 2142/28348 ).
5. ^ Aron Jacob: Constants clash on pi day. In: New Scientist . March 14, 2012, accessed October 31, 2014 . 
6. ↑ Jens Minor: New world record: Google Cloud calculates the district number Pi to 31.4 trillion digits & makes it freely accessible. In: GoogleWatchBlog. March 14, 2019, accessed on March 14, 2019 (German). 
7. ^ Timothy Mullican: Calculating Pi: My attempt at breaking the Pi World Record. June 26, 2019, accessed May 24, 2020 (American English). 
8. ^ Records set by y-cruncher. Retrieved May 24, 2020 . 
9. ↑ Lorenz Milla: A detailed proof of the Chudnovsky formula with elementary function theory . 2018, arxiv : 1809.00533 .
10. ^ Cleve Moler: Computing π. (PDF) In: mathworks. 2011, accessed March 3, 2018 .