gamma functions (Python recipe) by nirinA raseliarison

ActiveState Code (http://code.activestate.com/recipes/576393/)

''' gamma function and its natural logarithm'''

##/* @(#)er_lgamma.c 5.1 93/09/24 */
##/*
## * ====================================================
## * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
## *
## * Developed at SunPro, a Sun Microsystems, Inc. business.
## * Permission to use, copy, modify, and distribute this
## * software is freely granted, provided that this notice 
## * is preserved.
## * ====================================================
## *
## */
##original code from: http://www.sourceware.org/cgi-bin/cvsweb.cgi/~checkout~/src/newlib/libm/mathfp/er_lgamma.c?rev=1.6&content-type=text/plain&cvsroot=src
##/* Method:
## *   1. Argument Reduction for 0 < x <= 8
## * 	Since gamma(1+s)=s*gamma(s), for x in [0,8], we may 
## * 	reduce x to a number in [1.5,2.5] by
## * 		lgamma(1+s) = log(s) + lgamma(s)
## *	for example,
## *		lgamma(7.3) = log(6.3) + lgamma(6.3)
## *			    = log(6.3*5.3) + lgamma(5.3)
## *			    = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
## *   2. Polynomial approximation of lgamma around its
## *	minimun ymin=1.461632144968362245 to maintain monotonicity.
## *	On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
## *		Let z = x-ymin;
## *		lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
## *	where
## *		poly(z) is a 14 degree polynomial.
## *   2. Rational approximation in the primary interval [2,3]
## *	We use the following approximation:
## *		s = x-2.0;
## *		lgamma(x) = 0.5*s + s*P(s)/Q(s)
## *	with accuracy
## *		|P/Q - (lgamma(x)-0.5s)| < 2**-61.71
## *	Our algorithms are based on the following observation
## *
## *                             zeta(2)-1    2    zeta(3)-1    3
## * lgamma(2+s) = s*(1-Euler) + --------- * s  -  --------- * s  + ...
## *                                 2                 3
## *
## *	where Euler = 0.5771... is the Euler constant, which is very
## *	close to 0.5.
## *
## *   3. For x>=8, we have
## *	lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
## *	(better formula:
## *	   lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
## *	Let z = 1/x, then we approximation
## *		f(z) = lgamma(x) - (x-0.5)(log(x)-1)
## *	by
## *	  			    3       5             11
## *		w = w0 + w1*z + w2*z  + w3*z  + ... + w6*z
## *	where 
## *		|w - f(z)| < 2**-58.74
## *		
## *   4. For negative x, since (G is gamma function)
## *		-x*G(-x)*G(x) = pi/sin(pi*x),
## * 	we have
## * 		G(x) = pi/(sin(pi*x)*(-x)*G(-x))
## *	since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
## *	Hence, for x<0, signgam = sign(sin(pi*x)) and 
## *		lgamma(x) = log(|Gamma(x)|)
## *			  = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
## *	Note: one should avoid compute pi*(-x) directly in the 
## *	      computation of sin(pi*(-x)).
## *		
## *   5. Special Cases
## *		lgamma(2+s) ~ s*(1-Euler) for tiny s
## *		lgamma(1)=lgamma(2)=0
## *		lgamma(x) ~ -log(x) for tiny x
## *		lgamma(0) = lgamma(inf) = inf
## *	 	lgamma(-integer) = +-inf
## *	
## */
##

from math import *

two52=  4.50359962737049600000e+15
half=  5.00000000000000000000e-01
one =  1.00000000000000000000e+00
pi  =  3.14159265358979311600e+00
a0  =  7.72156649015328655494e-02
a1  =  3.22467033424113591611e-01
a2  =  6.73523010531292681824e-02
a3  =  2.05808084325167332806e-02
a4  =  7.38555086081402883957e-03
a5  =  2.89051383673415629091e-03
a6  =  1.19270763183362067845e-03
a7  =  5.10069792153511336608e-04
a8  =  2.20862790713908385557e-04
a9  =  1.08011567247583939954e-04
a10 =  2.52144565451257326939e-05
a11 =  4.48640949618915160150e-05
tc  =  1.46163214496836224576e+00
tf  = -1.21486290535849611461e-01
##/* tt = -(tail of tf) */
tt  = -3.63867699703950536541e-18
t0  =  4.83836122723810047042e-01
t1  = -1.47587722994593911752e-01
t2  =  6.46249402391333854778e-02
t3  = -3.27885410759859649565e-02
t4  =  1.79706750811820387126e-02
t5  = -1.03142241298341437450e-02
t6  =  6.10053870246291332635e-03
t7  = -3.68452016781138256760e-03
t8  =  2.25964780900612472250e-03
t9  = -1.40346469989232843813e-03
t10 =  8.81081882437654011382e-04
t11 = -5.38595305356740546715e-04
t12 =  3.15632070903625950361e-04
t13 = -3.12754168375120860518e-04
t14 =  3.35529192635519073543e-04
u0  = -7.72156649015328655494e-02
u1  =  6.32827064025093366517e-01
u2  =  1.45492250137234768737e+00
u3  =  9.77717527963372745603e-01
u4  =  2.28963728064692451092e-01
u5  =  1.33810918536787660377e-02
v1  =  2.45597793713041134822e+00
v2  =  2.12848976379893395361e+00
v3  =  7.69285150456672783825e-01
v4  =  1.04222645593369134254e-01
v5  =  3.21709242282423911810e-03
s0  = -7.72156649015328655494e-02
s1  =  2.14982415960608852501e-01
s2  =  3.25778796408930981787e-01
s3  =  1.46350472652464452805e-01
s4  =  2.66422703033638609560e-02
s5  =  1.84028451407337715652e-03
s6  =  3.19475326584100867617e-05
r1  =  1.39200533467621045958e+00
r2  =  7.21935547567138069525e-01
r3  =  1.71933865632803078993e-01
r4  =  1.86459191715652901344e-02
r5  =  7.77942496381893596434e-04
r6  =  7.32668430744625636189e-06
w0  =  4.18938533204672725052e-01
w1  =  8.33333333333329678849e-02
w2  = -2.77777777728775536470e-03
w3  =  7.93650558643019558500e-04
w4  = -5.95187557450339963135e-04
w5  =  8.36339918996282139126e-04
w6  = -1.63092934096575273989e-03
zero=  0.00000000000000000000e+00

##inf = float('inf')
##nan = float('nan')
inf = float(9e999)

def sin_pi(x):
    x = float(x)
    e,ix = frexp(x)
    if(abs(x)<0.25):
        return -sin(pi*x)
    y = -x  ##/* x is assume negative */

## * argument reduction, make sure inexact flag not raised if input
## * is an integer
    z = floor(y)
    if(z!=y):
        y *= 0.5
        y = 2.0*(y - floor(y))  ##/* y = |x| mod 2.0 */
        n = int(y*4.0)
    else:
        if(abs(ix)>=53):
            y = zero
            n = 0  ##/* y must be even */
        else:
            if(abs(ix)<52):
                z = y+two52 ##/* exact */
            e,n=frexp(z)
            n &= 1
            y  = n
            n<<= 2

    if n == 0:
        y = sin(pi*y)
    elif (n == 1 or n == 2):
        y = cos(pi*(0.5-y))
    elif (n == 3 or n == 4):
        y = sin(pi*(one-y))
    elif (n == 5 or n == 6):
        y = -cos(pi*(y-1.5))
    else:
        y = sin(pi*(y-2.0))

    z = cos(pi*(z+1.0));
    return -y*z

def Lgamma(x):
    '''return natural logarithm of gamma function of x
    raise ValueError if x is negative integer'''
    x = float(x)

    ##/* purge off +-inf, NaN, +-0, and negative arguments */
    if ((x == inf) or (x == -inf)):
        return inf
##    if (x is nan):
##        return nan
    
    e,ix = frexp(x)
    nadj = 0
    signgamp = 1

    if ((e==0.0) and (ix==0)):
        return inf

    if (ix>1020):
        return inf
    
    if ((e != 0.0) and (ix<-71)):
        if (x<0):
            return -log(-x)
        else:
            return -log(x)

    if (e<0):
        if (ix>52):
            return inf ##one/zero
        t = sin_pi(x)
        if (t==zero):
            ##return inf
            raise ValueError('gamma not defined for negative integer')
        nadj = log(pi/fabs(t*x))
        if (t<zero):
            signgamp = -1
        x = -x

    ##/* purge off 1 and 2 */
    if ((x==2.0) or (x==1.0)):
        r = 0.0
        
    ##/* for x < 2.0 */
    elif (ix<2):
        if(x<=0.9): ##/* lgamma(x) = lgamma(x+1)-log(x) */
            r = -log(x)
            if (x>=0.7316):
                y = one-x
                z = y*y
                p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))))
                p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))))
                p  = y*p1+p2
                r  += (p-0.5*y)
            elif (x>=0.23164):
                y= x-(tc-one)
                z = y*y
                w = z*y
                p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))) ##	/* parallel comp */
                p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)))
                p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)))
                p  = z*p1-(tt-w*(p2+y*p3))
                r += (tf + p)
            else:
                y = x
                p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))))
                p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))))
                r += (-0.5*y + p1/p2)
        else:
            r = zero
            if(x>=1.7316):
                y=2.0-x ##/* [1.7316,2] */
                z = y*y
                p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))))
                p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))))
                p  = y*p1+p2
                r  += (p-0.5*y)
            elif(x>=1.23164):
                y=x-tc ##/* [1.23,1.73] */
                z = y*y
                w = z*y
                p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))) ##	/* parallel comp */
                p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)))
                p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)))
                p  = z*p1-(tt-w*(p2+y*p3))
                r += (tf + p)
            else:
                y=x-one
                p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))))
                p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))))
                r += (-0.5*y + p1/p2)
    ##/* x < 8.0 */
    elif(ix<4):
        i = int(x)
        t = zero
        y = x-i
        p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))))
        q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))))
        r = half*y+p/q
        z = one ##/* lgamma(1+s) = log(s) + lgamma(s) */
        while (i>2):
            i -=1
            z *= (y+i)
        r += log(z)

    ##/* 8.0 <= x < 2**58 */
    elif (ix<58):
        t = log(x)
        z = one/x
        y = z*z
        w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))))
        r = (x-half)*(t-one)+w

    ##/* 2**58 <= x <= inf */
    else:
        r =  x*(log(x)-one)

    if (e<0):
        r = nadj - r
    return signgamp*r

def Gamma(x):
    '''return gamma function of x
    raise ValueError  if x is negative integer'''
    x = float(x)
    if x == 0.0:
        return inf
    s = 1.0
    if (x<0):
        s = cos(pi*floor(x))
    return s*exp(Lgamma(x))