감마함수(gamma function)의 값을 (유효수자 15자리 까지 정밀하게) 계산하는 Python 언어 소스

Lanczos 알고리즘은 Stirlng 공식에 의한 알고리즘 보다 정밀하며, 십진수로 유효숫자 약 15자리 까지는 정확하게 계산해 준다.  단지 exp 함수를 이용하는 부분에서는 exp 함수의 구현에 따라 오차가 더 있을 수 있다.

#!/usr/bin/env python
# -*- encoding:utf-8 -*-

# Filename: testLanczos-01.py
#
#           An approximation for the gamma function by using the Lanczos algorithm
#
# Execute: python testLanczos-01.py
#         or
# Execute: ./testLanczos-01.py
#
# See: http://en.wikipedia.org/wiki/Lanczos_approximation
# See:http://www-history.mcs.st-and.ac.uk/Biographies/Lanczos.html
# See: http://www.thinbasic.com/community/showthread.php?11279-Gamma-function

from cmath import *
 
# Coefficients used by the GNU Scientific Library
g = 7
p = [0.99999999999980993, 676.5203681218851, -1259.1392167224028,
     771.32342877765313, -176.61502916214059, 12.507343278686905,
     -0.13857109526572012, 9.9843695780195716e-6, 1.5056327351493116e-7]
 
def gamma(z):
    z = complex(z)
    # Reflection formula
    if z.real < 0.5:
        return pi / (sin(pi*z)*gamma(1-z))
    else:
        z -= 1
        x = p[0]
        for i in range(1, g+2):
            x += p[i]/(z+i)
        t = z + g + 0.5
        return sqrt(2*pi) * t**(z+0.5) * exp(-t) * x

def factorial(n):
    if n < 2:
        return 1
    else:
        k = 1
        if n % 2 == 0:
            for i in xrange(n/2):
                k *= (i + 1) * (n - i)
        else:
            for i in xrange(n/2):
                k *= (i + 1) * (n - 1 - i)
            k *= n
        return k

def facto(n):
    if n < 2:
        return 1
    else:
        k = 1
        for i in xrange(2, n + 1):
                k *= i
        return k

if __name__ == "__main__":
    print "gamma(10) = 9! = %s asymptotically" % gamma(10)
    # print "gamma(101) = 100! = %16s asymptotically" % gamma(101)
    print "gamma(101) = 100! = %16s asymptotically" % "{0.real:.15f}{0.imag:+.5f}j".format(gamma(101))

    for i in range(11):
        print "%d! = %d" % (i, factorial(i))

    i = 100
    print "factorial(%d) = %d! = %d" % (i, i, factorial(i))
    print "facto(%d) = %d! = %d" % (i, i, facto(i))

"""
Output:
gamma(10) = 9! = (362880+0j) asymptotically
gamma(101) = 100! = 9.33262154439379e+157+0.00000j asymptotically
0! = 1
1! = 1
2! = 2
3! = 6
4! = 24
5! = 120
6! = 720
7! = 5040
8! = 40320
9! = 362880
10! = 3628800
factorial(100) = 100! = 93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000
facto(100) = 100! = 93326215443944152681699238856266700490715968264381621468592963895217599993229915608941463976156518286253697920827223758251185210916864000000000000000000000000
"""