Chudnovsky algorithm

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

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 in August 2010,[3] 10 trillion digits in October 2011,[4][5] 12.1 trillion digits in December 2013[6] and 22.4 trillion digits of π in November 2016.[7]

The algorithm is based on the negated Heegner number , the j-function , 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:

The terms Mk, Lk, and Xk satisfy the following recurrences and can be computed as such:

The computation of Mk can be further optimized by introducing an additional term Kk as follows:

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.

The time complexity of the algorithm is .[8]

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 range(1, maxK+1):
        M = (K**3 - 16*K) * 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 298242, 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", New Scientist
  6. ^ Yee, Alexander J.; Kondo, Shigeru (28 December 2013). "12.1 Trillion Digits of Pi". www.numberworld.org.
  7. ^ "22.4 Trillion Digits of Pi". www.numberworld.org.
  8. ^ "y-cruncher - Formulas". www.numberworld.org. Retrieved 2018-02-25.