The Chudnovsky algorithm is a fast method for calculating the digits of π, based on Ramanujan's π formulae. Published by the Chudnovsky brothers in 1988,[1] it was used to calculate π to a billion decimal places.[2]
It was used in the world record calculations of 2.7 trillion digits of π in December 2009,[3] 10 trillion digits in October 2011,[4][5] 22.4 trillion digits in November 2016,[6] 31.4 trillion digits in September 2018–January 2019,[7] 50 trillion digits on January 29, 2020,[8] 62.8 trillion digits on August 14, 2021,[9] 100 trillion digits on March 21, 2022,[10] 105 trillion digits on March 14, 2024,[11] and 202 trillion digits on June 28, 2024.[12]
Algorithm
[edit]The algorithm is based on the negated Heegner number , the j-function , and on the following rapidly convergent generalized hypergeometric series:[13]A detailed proof of this formula can be found here: [14]
This identity is similar to some of Ramanujan's formulas involving π,[13] and is an example of a Ramanujan–Sato series.
The time complexity of the algorithm is .[15]
Optimizations
[edit]The optimization technique used for the world record computations is called binary splitting.[16]
Binary splitting
[edit]A factor of can be taken out of the sum and simplified to
Let , and substitute that into the sum.
can be simplified to , so
from the original definition of , so
This definition of isn't defined for , so compute the first term of the sum and use the new definition of
Let and , so
Let and
can never be computed, so instead compute and as approaches , the approximation will get better.
From the original definition of ,
Recursively computing the functions
[edit]Consider a value such that
Base case for recursion
[edit]Consider
Python code
[edit]import decimal
def binary_split(a, b):
if b == a + 1:
Pab = -(6*a - 5)*(2*a - 1)*(6*a - 1)
Qab = 10939058860032000 * a**3
Rab = Pab * (545140134*a + 13591409)
else:
m = (a + b) // 2
Pam, Qam, Ram = binary_split(a, m)
Pmb, Qmb, Rmb = binary_split(m, b)
Pab = Pam * Pmb
Qab = Qam * Qmb
Rab = Qmb * Ram + Pam * Rmb
return Pab, Qab, Rab
def chudnovsky(n):
"""Chudnovsky algorithm."""
P1n, Q1n, R1n = binary_split(1, n)
return (426880 * decimal.Decimal(10005).sqrt() * Q1n) / (13591409*Q1n + R1n)
print(f"1 = {chudnovsky(2)}") # 3.141592653589793238462643384
decimal.getcontext().prec = 100 # number of digits of decimal precision
for n in range(2,10):
print(f"{n} = {chudnovsky(n)}") # 3.14159265358979323846264338...
Notes
[edit]See also
[edit]References
[edit]- ^ Chudnovsky, David; Chudnovsky, Gregory (1988), Approximation and complex multiplication according to Ramanujan, Ramanujan revisited: proceedings of the centenary conference
- ^ Warsi, Karl; Dangerfield, Jan; Farndon, John; Griffiths, Johny; Jackson, Tom; Patel, Mukul; Pope, Sue; Parker, Matt (2019). The Math Book: Big Ideas Simply Explained. New York: Dorling Kindersley Limited. p. 65. ISBN 978-1-4654-8024-8.
- ^ Baruah, Nayandeep Deka; Berndt, Bruce C.; Chan, Heng Huat (2009-08-01). "Ramanujan's Series for 1/π: A Survey". American Mathematical Monthly. 116 (7): 567–587. doi:10.4169/193009709X458555.
- ^ 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
- ^ Aron, Jacob (March 14, 2012), "Constants clash on pi day", New Scientist
- ^ "22.4 Trillion Digits of Pi". www.numberworld.org.
- ^ "Google Cloud Topples the Pi Record". www.numberworld.org/.
- ^ "The Pi Record Returns to the Personal Computer". www.numberworld.org/.
- ^ "Pi-Challenge - Weltrekordversuch der FH Graubünden - FH Graubünden". www.fhgr.ch. Retrieved 2021-08-17.
- ^ "Calculating 100 trillion digits of pi on Google Cloud". cloud.google.com. Retrieved 2022-06-10.
- ^ Yee, Alexander J. (2024-03-14). "Limping to a new Pi Record of 105 Trillion Digits". NumberWorld.org. Retrieved 2024-03-16.
- ^ Ranous, Jordan (2024-06-28). "StorageReview Lab Breaks Pi Calculation World Record with Over 202 Trillion Digits". StorageReview.com. Retrieved 2024-07-20.
- ^ a b 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
- ^ Milla, Lorenz (2018), A detailed proof of the Chudnovsky formula with means of basic complex analysis, arXiv:1809.00533
- ^ "y-cruncher - Formulas". www.numberworld.org. Retrieved 2018-02-25.
- ^ Rayton, Joshua (Sep 2023), How is π calculated to trillions of digits?, YouTube