The Chudnovsky algorithm is a fast method for calculating the digits of π. It was published by the Chudnovsky brothers in 1989, and was used in the world record calculations of 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, and 12.1 trillion digits in December 2013.
For a high performance iterative implementation, this can be simplified to
There are 3 big integer terms (the multinomial term Mk, the linear term Lk, and the exponential term Xk) that make up the series and π equals the constant C divided by the sum of the series, as below:
- , where:
C = 426880√
Mk+1 = Mk * (12k+2) * (12k+6) * (12k+10) / (k+1)^3 and M0 = 1 [Mk = (6k)! / ((3k)! * (k!)^3)]
Lk+1 = Lk + 545,140,134 and L0 = 13,591,409 [Lk = 13591409 + 545140134*k]
Xk+1 = Xk * -262,537,412,640,768,000 and X0 = 1 [Xk = (-640320)^3k = (-262537412640768000)^k]
Mk can be optimized further:
Kk+1 = Kk + 12 and K0 = 6
Mk+1 = Mk * (Kk^3 - 16Kk) / (k+1)^3 and M0 = 1
Example: Python Implementation
π can be computed to any precision using the above algorithm in any environment which supports arbitrary-precision arithmetic. As an example, here is a Python implementation:
from decimal import Decimal as Dec, getcontext as gc def PI(maxK=70, prec=1008, disp=1007): # parameter defaults chosen to gain 1000+ digits within a few seconds gc().prec = prec K, M, L, X, S = 6, 1, 13591409, 1, 13591409 for k in xrange(1, maxK+1): M = (K**3 - (K<<4)) * M / k**3 L += 545140134 X *= -262537412640768000 S += Dec(M * L) / X K += 12 pi = 426880 * Dec(10005).sqrt() / S pi = Dec(str(pi)[:disp]) # drop few digits of precision for accuracy print "PI(maxK=%d iterations, gc().prec=%d, disp=%d digits) =\n%s" % (maxK, prec, disp, pi) return pi Pi = PI() print "\nFor greater precision and more digits (takes a few extra seconds) - Try" print "Pi = PI(317,4501,4500)" print "Pi = PI(353,5022,5020)"
- Chudnovsky, David V.; Chudnovsky, Gregory V. (1989), "The Computation of Classical Constants", Proceedings of the National Academy of Sciences of the United States of America, 86 (21): 8178–8182, doi:10.1073/pnas.86.21.8178, ISSN 0027-8424, JSTOR 34831, PMC , PMID 16594075
- 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
- Geeks slice pi to 5 trillion decimal places, Australian Broadcasting Corporation, August 6, 2010
- 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
- Aron, Jacob (March 14, 2012), "Constants clash on pi day", NewScientist
- Alexander J. Yee; Shigeru Kondo (28 December 2013). "12.1 Trillion Digits of Pi".
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