Chudnovsky algorithm

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, and was used in the world record calculations of 2.7 trillion digits of π in December 2009, 10 trillion digits in October 2011, 22.4 trillion digits in November 2016, 31.4 trillion digits in September 2018–January 2019, and 50 trillion digits on January 29, 2020.

The algorithm is based on the negated Heegner number $d=-163$, the j-function $\scriptstyle j\left({\frac {1+{\sqrt {-163}}}{2}}\right)=-640320^{3}$, and on the following rapidly convergent generalized hypergeometric series:

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

A detailed proof of this formula can be found here:

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

${\frac {(640320)^{3/2}}{12\pi }}={\frac {426880{\sqrt {10005}}}{\pi }}=\sum _{k=0}^{\infty }{\frac {(6k)!(545140134k+13591409)}{(3k)!(k!)^{3}\left(-262537412640768000\right)^{k}}}$

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

$\pi =C\left(\sum _{k=0}^{\infty }{\frac {M_{k}\cdot L_{k}}{X_{k}}}\right)^{-1}$, where:

$C=426880{\sqrt {10005}}$,

$M_{k}={\frac {(6k)!}{(3k)!(k!)^{3}}}$,

$L_{k}=545140134k+13591409$,

$X_{k}=(-262537412640768000)^{k}$.

The terms M_k, L_k, and X_k satisfy the following recurrences and can be computed as such:

${\begin{alignedat}{4}L_{k+1}&=L_{k}+545140134\,\,&&{\textrm {where}}\,\,L_{0}&&=13591409\\[4pt]X_{k+1}&=X_{k}\cdot (-262537412640768000)&&{\textrm {where}}\,\,X_{0}&&=1\\[4pt]M_{k+1}&=M_{k}\cdot \left({\frac {(12k+2)(12k+6)(12k+10)}{(k+1)^{3}}}\right)\,\,&&{\textrm {where}}\,\,M_{0}&&=1\\[4pt]\end{alignedat}}$

The computation of M_k can be further optimized by introducing an additional term K_k as follows:

${\begin{alignedat}{4}K_{k+1}&=K_{k}+12\,\,&&{\textrm {where}}\,\,K_{0}&&=6\\[4pt]M_{k+1}&=M_{k}\cdot \left({\frac {K_{k}^{3}-16K_{k}}{(k+1)^{3}}}\right)\,\,&&{\textrm {where}}\,\,M_{0}&&=1\\[12pt]\end{alignedat}}$

Note that

$e^{\pi {\sqrt {163}}}\approx 640320^{3}+743.99999999999925\dots$ and

$640320^{3}=262537412640768000$

$545140134=163\cdot 127\cdot 19\cdot 11\cdot 7\cdot 3^{2}\cdot 2$

$13591409=13\cdot 1045493$

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

The time complexity of the algorithm is $O(n\log(n)^{3})$.

See also

• Borwein's algorithm
• Numerical approximations of π
• Ramanujan–Sato series