/usr/lib/python2.7/dist-packages/mpmath/functions/factorials.py is in python-mpmath 0.19-3.
This file is owned by root:root, with mode 0o644.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 | from ..libmp.backend import xrange
from .functions import defun, defun_wrapped
@defun
def gammaprod(ctx, a, b, _infsign=False):
a = [ctx.convert(x) for x in a]
b = [ctx.convert(x) for x in b]
poles_num = []
poles_den = []
regular_num = []
regular_den = []
for x in a: [regular_num, poles_num][ctx.isnpint(x)].append(x)
for x in b: [regular_den, poles_den][ctx.isnpint(x)].append(x)
# One more pole in numerator or denominator gives 0 or inf
if len(poles_num) < len(poles_den): return ctx.zero
if len(poles_num) > len(poles_den):
# Get correct sign of infinity for x+h, h -> 0 from above
# XXX: hack, this should be done properly
if _infsign:
a = [x and x*(1+ctx.eps) or x+ctx.eps for x in poles_num]
b = [x and x*(1+ctx.eps) or x+ctx.eps for x in poles_den]
return ctx.sign(ctx.gammaprod(a+regular_num,b+regular_den)) * ctx.inf
else:
return ctx.inf
# All poles cancel
# lim G(i)/G(j) = (-1)**(i+j) * gamma(1-j) / gamma(1-i)
p = ctx.one
orig = ctx.prec
try:
ctx.prec = orig + 15
while poles_num:
i = poles_num.pop()
j = poles_den.pop()
p *= (-1)**(i+j) * ctx.gamma(1-j) / ctx.gamma(1-i)
for x in regular_num: p *= ctx.gamma(x)
for x in regular_den: p /= ctx.gamma(x)
finally:
ctx.prec = orig
return +p
@defun
def beta(ctx, x, y):
x = ctx.convert(x)
y = ctx.convert(y)
if ctx.isinf(y):
x, y = y, x
if ctx.isinf(x):
if x == ctx.inf and not ctx._im(y):
if y == ctx.ninf:
return ctx.nan
if y > 0:
return ctx.zero
if ctx.isint(y):
return ctx.nan
if y < 0:
return ctx.sign(ctx.gamma(y)) * ctx.inf
return ctx.nan
return ctx.gammaprod([x, y], [x+y])
@defun
def binomial(ctx, n, k):
return ctx.gammaprod([n+1], [k+1, n-k+1])
@defun
def rf(ctx, x, n):
return ctx.gammaprod([x+n], [x])
@defun
def ff(ctx, x, n):
return ctx.gammaprod([x+1], [x-n+1])
@defun_wrapped
def fac2(ctx, x):
if ctx.isinf(x):
if x == ctx.inf:
return x
return ctx.nan
return 2**(x/2)*(ctx.pi/2)**((ctx.cospi(x)-1)/4)*ctx.gamma(x/2+1)
@defun_wrapped
def barnesg(ctx, z):
if ctx.isinf(z):
if z == ctx.inf:
return z
return ctx.nan
if ctx.isnan(z):
return z
if (not ctx._im(z)) and ctx._re(z) <= 0 and ctx.isint(ctx._re(z)):
return z*0
# Account for size (would not be needed if computing log(G))
if abs(z) > 5:
ctx.dps += 2*ctx.log(abs(z),2)
# Reflection formula
if ctx.re(z) < -ctx.dps:
w = 1-z
pi2 = 2*ctx.pi
u = ctx.expjpi(2*w)
v = ctx.j*ctx.pi/12 - ctx.j*ctx.pi*w**2/2 + w*ctx.ln(1-u) - \
ctx.j*ctx.polylog(2, u)/pi2
v = ctx.barnesg(2-z)*ctx.exp(v)/pi2**w
if ctx._is_real_type(z):
v = ctx._re(v)
return v
# Estimate terms for asymptotic expansion
# TODO: fixme, obviously
N = ctx.dps // 2 + 5
G = 1
while abs(z) < N or ctx.re(z) < 1:
G /= ctx.gamma(z)
z += 1
z -= 1
s = ctx.mpf(1)/12
s -= ctx.log(ctx.glaisher)
s += z*ctx.log(2*ctx.pi)/2
s += (z**2/2-ctx.mpf(1)/12)*ctx.log(z)
s -= 3*z**2/4
z2k = z2 = z**2
for k in xrange(1, N+1):
t = ctx.bernoulli(2*k+2) / (4*k*(k+1)*z2k)
if abs(t) < ctx.eps:
#print k, N # check how many terms were needed
break
z2k *= z2
s += t
#if k == N:
# print "warning: series for barnesg failed to converge", ctx.dps
return G*ctx.exp(s)
@defun
def superfac(ctx, z):
return ctx.barnesg(z+2)
@defun_wrapped
def hyperfac(ctx, z):
# XXX: estimate needed extra bits accurately
if z == ctx.inf:
return z
if abs(z) > 5:
extra = 4*int(ctx.log(abs(z),2))
else:
extra = 0
ctx.prec += extra
if not ctx._im(z) and ctx._re(z) < 0 and ctx.isint(ctx._re(z)):
n = int(ctx.re(z))
h = ctx.hyperfac(-n-1)
if ((n+1)//2) & 1:
h = -h
if ctx._is_complex_type(z):
return h + 0j
return h
zp1 = z+1
# Wrong branch cut
#v = ctx.gamma(zp1)**z
#ctx.prec -= extra
#return v / ctx.barnesg(zp1)
v = ctx.exp(z*ctx.loggamma(zp1))
ctx.prec -= extra
return v / ctx.barnesg(zp1)
@defun_wrapped
def loggamma_old(ctx, z):
a = ctx._re(z)
b = ctx._im(z)
if not b and a > 0:
return ctx.ln(ctx.gamma_old(z))
u = ctx.arg(z)
w = ctx.ln(ctx.gamma_old(z))
if b:
gi = -b - u/2 + a*u + b*ctx.ln(abs(z))
n = ctx.floor((gi-ctx._im(w))/(2*ctx.pi)+0.5) * (2*ctx.pi)
return w + n*ctx.j
elif a < 0:
n = int(ctx.floor(a))
w += (n-(n%2))*ctx.pi*ctx.j
return w
'''
@defun
def psi0(ctx, z):
"""Shortcut for psi(0,z) (the digamma function)"""
return ctx.psi(0, z)
@defun
def psi1(ctx, z):
"""Shortcut for psi(1,z) (the trigamma function)"""
return ctx.psi(1, z)
@defun
def psi2(ctx, z):
"""Shortcut for psi(2,z) (the tetragamma function)"""
return ctx.psi(2, z)
@defun
def psi3(ctx, z):
"""Shortcut for psi(3,z) (the pentagamma function)"""
return ctx.psi(3, z)
'''
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