Chudnovsky algorithm

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The Chudnovsky algorithm is a fast method for calculating the digits of π. It was published by the Chudnovsky brothers in 1989,[1] and was used in the world record calculations of 2.7 trillion digits of π in December 2009,[2] 5 trillion digits of π in August 2010,[3] 10 trillion digits of π in October 2011,[4][5] and 12.1 trillion digits in December 2013.[6]

The algorithm is based on the negated Heegner number d = −163, the j-function j(1+−163/2) = −6403203, and on the following rapidly convergent generalized hypergeometric series:[2]

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 = 42688010005
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

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.

Example: Python Implementation[edit]

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)"

See also[edit]

References[edit]

  1. ^ 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 298242Freely accessible, PMID 16594075 
  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. ^ Geeks slice pi to 5 trillion decimal places, Australian Broadcasting Corporation, August 6, 2010 
  4. ^ 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 
  5. ^ Aron, Jacob (March 14, 2012), "Constants clash on pi day", NewScientist 
  6. ^ Alexander J. Yee; Shigeru Kondo (28 December 2013). "12.1 Trillion Digits of Pi".