def mpc_psi(m, z, prec, rnd=round_fast):
"""
Computation of the polygamma function of arbitrary integer order
m >= 0, for a complex argument z.
"""
if m == 0:
return mpc_psi0(z, prec, rnd)
re, im = z
wp = prec + 20
sign, man, exp, bc = re
if not man:
if re == finf and im == fzero:
return (fzero, fzero)
if re == fnan:
return fnan
# Recurrence
w = to_int(re)
n = int(0.4*wp + 4*m)
s = mpc_zero
if w < n:
for k in xrange(w, n):
t = mpc_pow_int(z, -m-1, wp)
s = mpc_add(s, t, wp)
z = mpc_add_mpf(z, fone, wp)
zm = mpc_pow_int(z, -m, wp)
z2 = mpc_pow_int(z, -2, wp)
# 1/m*(z+N)^m
integral_term = mpc_div_mpf(zm, from_int(m), wp)
s = mpc_add(s, integral_term, wp)
# 1/2*(z+N)^(-(m+1))
s = mpc_add(s, mpc_mul_mpf(mpc_div(zm, z, wp), fhalf, wp), wp)
a = m + 1
b = 2
k = 1
# Important: we want to sum up to the *relative* error,
# not the absolute error, because psi^(m)(z) might be tiny
magn = mpc_abs(s, 10)
magn = magn[2]+magn[3]
eps = mpf_shift(fone, magn-wp+2)
while 1:
zm = mpc_mul(zm, z2, wp)
bern = mpf_bernoulli(2*k, wp)
scal = mpf_mul_int(bern, a, wp)
scal = mpf_div(scal, from_int(b), wp)
term = mpc_mul_mpf(zm, scal, wp)
s = mpc_add(s, term, wp)
szterm = mpc_abs(term, 10)
if k > 2 and mpf_le(szterm, eps):
break
#print k, to_str(szterm, 10), to_str(eps, 10)
a *= (m+2*k)*(m+2*k+1)
b *= (2*k+1)*(2*k+2)
k += 1
# Scale and sign factor
v = mpc_mul_mpf(s, mpf_gamma(from_int(m+1), wp), prec, rnd)
if not (m & 1):
v = mpf_neg(v[0]), mpf_neg(v[1])
return v
def mpc_psi(m, z, prec, rnd=round_fast):
"""
Computation of the polygamma function of arbitrary integer order
m >= 0, for a complex argument z.
"""
if m == 0:
return mpc_psi0(z, prec, rnd)
re, im = z
wp = prec + 20
sign, man, exp, bc = re
if not man:
if re == finf and im == fzero:
return (fzero, fzero)
if re == fnan:
return fnan
# Recurrence
w = to_int(re)
n = int(0.4 * wp + 4 * m)
s = mpc_zero
if w < n:
for k in xrange(w, n):
t = mpc_pow_int(z, -m - 1, wp)
s = mpc_add(s, t, wp)
z = mpc_add_mpf(z, fone, wp)
zm = mpc_pow_int(z, -m, wp)
z2 = mpc_pow_int(z, -2, wp)
# 1/m*(z+N)^m
integral_term = mpc_div_mpf(zm, from_int(m), wp)
s = mpc_add(s, integral_term, wp)
# 1/2*(z+N)^(-(m+1))
s = mpc_add(s, mpc_mul_mpf(mpc_div(zm, z, wp), fhalf, wp), wp)
a = m + 1
b = 2
k = 1
# Important: we want to sum up to the *relative* error,
# not the absolute error, because psi^(m)(z) might be tiny
magn = mpc_abs(s, 10)
magn = magn[2] + magn[3]
eps = mpf_shift(fone, magn - wp + 2)
while 1:
zm = mpc_mul(zm, z2, wp)
bern = mpf_bernoulli(2 * k, wp)
scal = mpf_mul_int(bern, a, wp)
scal = mpf_div(scal, from_int(b), wp)
term = mpc_mul_mpf(zm, scal, wp)
s = mpc_add(s, term, wp)
szterm = mpc_abs(term, 10)
if k > 2 and mpf_le(szterm, eps):
break
#print k, to_str(szterm, 10), to_str(eps, 10)
a *= (m + 2 * k) * (m + 2 * k + 1)
b *= (2 * k + 1) * (2 * k + 2)
k += 1
# Scale and sign factor
v = mpc_mul_mpf(s, mpf_gamma(from_int(m + 1), wp), prec, rnd)
if not (m & 1):
v = mpf_neg(v[0]), mpf_neg(v[1])
return v
def __pow__(s, t):
prec, rounding = prec_rounding
if isinstance(t, int_types):
return make_mpc(mpc_pow_int(s._mpc_, t, prec, rounding))
t = mpc_convert_lhs(t)
if t is NotImplemented:
return t
if isinstance(t, mpf):
return make_mpc(mpc_pow_mpf(s._mpc_, t._mpf_, prec, rounding))
return make_mpc(mpc_pow(s._mpc_, t._mpc_, prec, rounding))
def __pow__(s, t):
prec, rounding = prec_rounding
if isinstance(t, int_types):
return make_mpc(mpc_pow_int(s._mpc_, t, prec, rounding))
t = mpc_convert_lhs(t)
if t is NotImplemented:
return t
if isinstance(t, mpf):
return make_mpc(mpc_pow_mpf(s._mpc_, t._mpf_, prec, rounding))
return make_mpc(mpc_pow(s._mpc_, t._mpc_, prec, rounding))