def mpi_gamma(z, prec, type=0):
a, b = z
wp = prec + 20
if type == 1:
return mpi_gamma(mpi_add(z, mpi_one, wp), prec, 0)
# increasing
if mpf_gt(a, gamma_min_b):
if type == 0:
c = mpf_gamma(a, prec, round_floor)
d = mpf_gamma(b, prec, round_ceiling)
elif type == 2:
c = mpf_rgamma(b, prec, round_floor)
d = mpf_rgamma(a, prec, round_ceiling)
elif type == 3:
c = mpf_loggamma(a, prec, round_floor)
d = mpf_loggamma(b, prec, round_ceiling)
# decreasing
elif mpf_gt(a, fzero) and mpf_lt(b, gamma_min_a):
if type == 0:
c = mpf_gamma(b, prec, round_floor)
d = mpf_gamma(a, prec, round_ceiling)
elif type == 2:
c = mpf_rgamma(a, prec, round_floor)
d = mpf_rgamma(b, prec, round_ceiling)
elif type == 3:
c = mpf_loggamma(b, prec, round_floor)
d = mpf_loggamma(a, prec, round_ceiling)
else:
# TODO: reflection formula
znew = mpi_add(z, mpi_one, wp)
if type == 0: return mpi_div(mpi_gamma(znew, prec + 2, 0), z, prec)
if type == 2: return mpi_mul(mpi_gamma(znew, prec + 2, 2), z, prec)
if type == 3:
return mpi_sub(mpi_gamma(znew, prec + 2, 3), mpi_log(z, prec + 2),
prec)
return c, d
def mpi_gamma(z, prec, type=0):
a, b = z
wp = prec+20
if type == 1:
return mpi_gamma(mpi_add(z, mpi_one, wp), prec, 0)
# increasing
if mpf_gt(a, gamma_min_b):
if type == 0:
c = mpf_gamma(a, prec, round_floor)
d = mpf_gamma(b, prec, round_ceiling)
elif type == 2:
c = mpf_rgamma(b, prec, round_floor)
d = mpf_rgamma(a, prec, round_ceiling)
elif type == 3:
c = mpf_loggamma(a, prec, round_floor)
d = mpf_loggamma(b, prec, round_ceiling)
# decreasing
elif mpf_gt(a, fzero) and mpf_lt(b, gamma_min_a):
if type == 0:
c = mpf_gamma(b, prec, round_floor)
d = mpf_gamma(a, prec, round_ceiling)
elif type == 2:
c = mpf_rgamma(a, prec, round_floor)
d = mpf_rgamma(b, prec, round_ceiling)
elif type == 3:
c = mpf_loggamma(b, prec, round_floor)
d = mpf_loggamma(a, prec, round_ceiling)
else:
# TODO: reflection formula
znew = mpi_add(z, mpi_one, wp)
if type == 0: return mpi_div(mpi_gamma(znew, prec+2, 0), z, prec)
if type == 2: return mpi_mul(mpi_gamma(znew, prec+2, 2), z, prec)
if type == 3: return mpi_sub(mpi_gamma(znew, prec+2, 3), mpi_log(z, prec+2), prec)
return c, d
def mpc_zeta(s, prec, rnd=round_fast, alt=0, force=False):
re, im = s
if im == fzero:
return mpf_zeta(re, prec, rnd, alt), fzero
# slow for large s
if (not force) and mpf_gt(mpc_abs(s, 10), from_int(prec)):
raise NotImplementedError
wp = prec + 20
# Near pole
r = mpc_sub(mpc_one, s, wp)
asign, aman, aexp, abc = mpc_abs(r, 10)
pole_dist = -2*(aexp+abc)
if pole_dist > wp:
if alt:
q = mpf_ln2(wp)
y = mpf_mul(q, mpf_euler(wp), wp)
g = mpf_shift(mpf_mul(q, q, wp), -1)
g = mpf_sub(y, g)
z = mpc_mul_mpf(r, mpf_neg(g), wp)
z = mpc_add_mpf(z, q, wp)
return mpc_pos(z, prec, rnd)
else:
q = mpc_neg(mpc_div(mpc_one, r, wp))
q = mpc_add_mpf(q, mpf_euler(wp), wp)
return mpc_pos(q, prec, rnd)
else:
wp += max(0, pole_dist)
# Reflection formula. To be rigorous, we should reflect to the left of
# re = 1/2 (see comments for mpf_zeta), but this leads to unnecessary
# slowdown for interesting values of s
if mpf_lt(re, fzero):
# XXX: could use the separate refl. formula for Dirichlet eta
if alt:
q = mpc_sub(mpc_one, mpc_pow(mpc_two, mpc_sub(mpc_one, s, wp),
wp), wp)
return mpc_mul(mpc_zeta(s, wp), q, prec, rnd)
# XXX: -1 should be done exactly
y = mpc_sub(mpc_one, s, 10*wp)
a = mpc_gamma(y, wp)
b = mpc_zeta(y, wp)
c = mpc_sin_pi(mpc_shift(s, -1), wp)
rsign, rman, rexp, rbc = re
isign, iman, iexp, ibc = im
mag = max(rexp+rbc, iexp+ibc)
wp2 = wp + mag
pi = mpf_pi(wp+wp2)
pi2 = (mpf_shift(pi, 1), fzero)
d = mpc_div_mpf(mpc_pow(pi2, s, wp2), pi, wp2)
return mpc_mul(a,mpc_mul(b,mpc_mul(c,d,wp),wp),prec,rnd)
n = int(wp/2.54 + 5)
n += int(0.9*abs(to_int(im)))
d = borwein_coefficients(n)
ref = to_fixed(re, wp)
imf = to_fixed(im, wp)
tre = MPZ_ZERO
tim = MPZ_ZERO
one = MPZ_ONE << wp
one_2wp = MPZ_ONE << (2*wp)
critical_line = re == fhalf
for k in xrange(n):
log = log_int_fixed(k+1, wp)
# A square root is much cheaper than an exp
if critical_line:
w = one_2wp // sqrt_fixed((k+1) << wp, wp)
else:
w = to_fixed(mpf_exp(from_man_exp(-ref*log, -2*wp), wp), wp)
if k & 1:
w *= (d[n] - d[k])
else:
w *= (d[k] - d[n])
wre, wim = mpf_cos_sin(from_man_exp(-imf * log, -2*wp), wp)
tre += (w * to_fixed(wre, wp)) >> wp
tim += (w * to_fixed(wim, wp)) >> wp
tre //= (-d[n])
tim //= (-d[n])
tre = from_man_exp(tre, -wp, wp)
tim = from_man_exp(tim, -wp, wp)
if alt:
return mpc_pos((tre, tim), prec, rnd)
else:
q = mpc_sub(mpc_one, mpc_pow(mpc_two, r, wp), wp)
return mpc_div((tre, tim), q, prec, rnd)
def mpc_zeta(s, prec, rnd=round_fast, alt=0, force=False):
re, im = s
if im == fzero:
return mpf_zeta(re, prec, rnd, alt), fzero
# slow for large s
if (not force) and mpf_gt(mpc_abs(s, 10), from_int(prec)):
raise NotImplementedError
wp = prec + 20
# Near pole
r = mpc_sub(mpc_one, s, wp)
asign, aman, aexp, abc = mpc_abs(r, 10)
pole_dist = -2 * (aexp + abc)
if pole_dist > wp:
if alt:
q = mpf_ln2(wp)
y = mpf_mul(q, mpf_euler(wp), wp)
g = mpf_shift(mpf_mul(q, q, wp), -1)
g = mpf_sub(y, g)
z = mpc_mul_mpf(r, mpf_neg(g), wp)
z = mpc_add_mpf(z, q, wp)
return mpc_pos(z, prec, rnd)
else:
q = mpc_neg(mpc_div(mpc_one, r, wp))
q = mpc_add_mpf(q, mpf_euler(wp), wp)
return mpc_pos(q, prec, rnd)
else:
wp += max(0, pole_dist)
# Reflection formula. To be rigorous, we should reflect to the left of
# re = 1/2 (see comments for mpf_zeta), but this leads to unnecessary
# slowdown for interesting values of s
if mpf_lt(re, fzero):
# XXX: could use the separate refl. formula for Dirichlet eta
if alt:
q = mpc_sub(mpc_one, mpc_pow(mpc_two, mpc_sub(mpc_one, s, wp), wp),
wp)
return mpc_mul(mpc_zeta(s, wp), q, prec, rnd)
# XXX: -1 should be done exactly
y = mpc_sub(mpc_one, s, 10 * wp)
a = mpc_gamma(y, wp)
b = mpc_zeta(y, wp)
c = mpc_sin_pi(mpc_shift(s, -1), wp)
rsign, rman, rexp, rbc = re
isign, iman, iexp, ibc = im
mag = max(rexp + rbc, iexp + ibc)
wp2 = wp + mag
pi = mpf_pi(wp + wp2)
pi2 = (mpf_shift(pi, 1), fzero)
d = mpc_div_mpf(mpc_pow(pi2, s, wp2), pi, wp2)
return mpc_mul(a, mpc_mul(b, mpc_mul(c, d, wp), wp), prec, rnd)
n = int(wp / 2.54 + 5)
n += int(0.9 * abs(to_int(im)))
d = borwein_coefficients(n)
ref = to_fixed(re, wp)
imf = to_fixed(im, wp)
tre = MPZ_ZERO
tim = MPZ_ZERO
one = MPZ_ONE << wp
one_2wp = MPZ_ONE << (2 * wp)
critical_line = re == fhalf
for k in xrange(n):
log = log_int_fixed(k + 1, wp)
# A square root is much cheaper than an exp
if critical_line:
w = one_2wp // sqrt_fixed((k + 1) << wp, wp)
else:
w = to_fixed(mpf_exp(from_man_exp(-ref * log, -2 * wp), wp), wp)
if k & 1:
w *= (d[n] - d[k])
else:
w *= (d[k] - d[n])
wre, wim = mpf_cos_sin(from_man_exp(-imf * log, -2 * wp), wp)
tre += (w * to_fixed(wre, wp)) >> wp
tim += (w * to_fixed(wim, wp)) >> wp
tre //= (-d[n])
tim //= (-d[n])
tre = from_man_exp(tre, -wp, wp)
tim = from_man_exp(tim, -wp, wp)
if alt:
return mpc_pos((tre, tim), prec, rnd)
else:
q = mpc_sub(mpc_one, mpc_pow(mpc_two, r, wp), wp)
return mpc_div((tre, tim), q, prec, rnd)
def mpci_gamma(z, prec, type=0):
(a1, a2), (b1, b2) = z
# Real case
if b1 == b2 == fzero and (type != 3 or mpf_gt(a1, fzero)):
return mpi_gamma(z, prec, type), mpi_zero
# Estimate precision
wp = prec + 20
if type != 3:
amag = a2[2] + a2[3]
bmag = b2[2] + b2[3]
if a2 != fzero:
mag = max(amag, bmag)
else:
mag = bmag
an = abs(to_int(a2))
bn = abs(to_int(b2))
absn = max(an, bn)
gamma_size = max(0, absn * mag)
wp += bitcount(gamma_size)
# Assume type != 1
if type == 1:
(a1, a2) = mpi_add((a1, a2), mpi_one, wp)
z = (a1, a2), (b1, b2)
type = 0
# Avoid non-monotonic region near the negative real axis
if mpf_lt(a1, gamma_min_b):
if mpi_overlap((b1, b2), (gamma_mono_imag_a, gamma_mono_imag_b)):
# TODO: reflection formula
#if mpf_lt(a2, mpf_shift(fone,-1)):
# znew = mpci_sub((mpi_one,mpi_zero),z,wp)
# ...
# Recurrence:
# gamma(z) = gamma(z+1)/z
znew = mpi_add((a1, a2), mpi_one, wp), (b1, b2)
if type == 0:
return mpci_div(mpci_gamma(znew, prec + 2, 0), z, prec)
if type == 2:
return mpci_mul(mpci_gamma(znew, prec + 2, 2), z, prec)
if type == 3:
return mpci_sub(mpci_gamma(znew, prec + 2, 3),
mpci_log(z, prec + 2), prec)
# Use monotonicity (except for a small region close to the
# origin and near poles)
# upper half-plane
if mpf_ge(b1, fzero):
minre = mpc_loggamma((a1, b2), wp, round_floor)
maxre = mpc_loggamma((a2, b1), wp, round_ceiling)
minim = mpc_loggamma((a1, b1), wp, round_floor)
maxim = mpc_loggamma((a2, b2), wp, round_ceiling)
# lower half-plane
elif mpf_le(b2, fzero):
minre = mpc_loggamma((a1, b1), wp, round_floor)
maxre = mpc_loggamma((a2, b2), wp, round_ceiling)
minim = mpc_loggamma((a2, b1), wp, round_floor)
maxim = mpc_loggamma((a1, b2), wp, round_ceiling)
# crosses real axis
else:
maxre = mpc_loggamma((a2, fzero), wp, round_ceiling)
# stretches more into the lower half-plane
if mpf_gt(mpf_neg(b1), b2):
minre = mpc_loggamma((a1, b1), wp, round_ceiling)
else:
minre = mpc_loggamma((a1, b2), wp, round_ceiling)
minim = mpc_loggamma((a2, b1), wp, round_floor)
maxim = mpc_loggamma((a2, b2), wp, round_floor)
w = (minre[0], maxre[0]), (minim[1], maxim[1])
if type == 3:
return mpi_pos(w[0], prec), mpi_pos(w[1], prec)
if type == 2:
w = mpci_neg(w)
return mpci_exp(w, prec)
def mpi_overlap(x, y):
a, b = x
c, d = y
if mpf_lt(d, a): return False
if mpf_gt(c, b): return False
return True
def mpi_le(s, t):
sa, sb = s
ta, tb = t
if mpf_le(sb, ta): return True
if mpf_gt(sa, tb): return False
return None
def mpi_le(s, t):
sa, sb = s
ta, tb = t
if mpf_le(sb, ta): return True
if mpf_gt(sa, tb): return False
return None
def mpci_gamma(z, prec, type=0):
(a1,a2), (b1,b2) = z
# Real case
if b1 == b2 == fzero and (type != 3 or mpf_gt(a1,fzero)):
return mpi_gamma(z, prec, type), mpi_zero
# Estimate precision
wp = prec+20
if type != 3:
amag = a2[2]+a2[3]
bmag = b2[2]+b2[3]
if a2 != fzero:
mag = max(amag, bmag)
else:
mag = bmag
an = abs(to_int(a2))
bn = abs(to_int(b2))
absn = max(an, bn)
gamma_size = max(0,absn*mag)
wp += bitcount(gamma_size)
# Assume type != 1
if type == 1:
(a1,a2) = mpi_add((a1,a2), mpi_one, wp); z = (a1,a2), (b1,b2)
type = 0
# Avoid non-monotonic region near the negative real axis
if mpf_lt(a1, gamma_min_b):
if mpi_overlap((b1,b2), (gamma_mono_imag_a, gamma_mono_imag_b)):
# TODO: reflection formula
#if mpf_lt(a2, mpf_shift(fone,-1)):
# znew = mpci_sub((mpi_one,mpi_zero),z,wp)
# ...
# Recurrence:
# gamma(z) = gamma(z+1)/z
znew = mpi_add((a1,a2), mpi_one, wp), (b1,b2)
if type == 0: return mpci_div(mpci_gamma(znew, prec+2, 0), z, prec)
if type == 2: return mpci_mul(mpci_gamma(znew, prec+2, 2), z, prec)
if type == 3: return mpci_sub(mpci_gamma(znew, prec+2, 3), mpci_log(z,prec+2), prec)
# Use monotonicity (except for a small region close to the
# origin and near poles)
# upper half-plane
if mpf_ge(b1, fzero):
minre = mpc_loggamma((a1,b2), wp, round_floor)
maxre = mpc_loggamma((a2,b1), wp, round_ceiling)
minim = mpc_loggamma((a1,b1), wp, round_floor)
maxim = mpc_loggamma((a2,b2), wp, round_ceiling)
# lower half-plane
elif mpf_le(b2, fzero):
minre = mpc_loggamma((a1,b1), wp, round_floor)
maxre = mpc_loggamma((a2,b2), wp, round_ceiling)
minim = mpc_loggamma((a2,b1), wp, round_floor)
maxim = mpc_loggamma((a1,b2), wp, round_ceiling)
# crosses real axis
else:
maxre = mpc_loggamma((a2,fzero), wp, round_ceiling)
# stretches more into the lower half-plane
if mpf_gt(mpf_neg(b1), b2):
minre = mpc_loggamma((a1,b1), wp, round_ceiling)
else:
minre = mpc_loggamma((a1,b2), wp, round_ceiling)
minim = mpc_loggamma((a2,b1), wp, round_floor)
maxim = mpc_loggamma((a2,b2), wp, round_floor)
w = (minre[0], maxre[0]), (minim[1], maxim[1])
if type == 3:
return mpi_pos(w[0], prec), mpi_pos(w[1], prec)
if type == 2:
w = mpci_neg(w)
return mpci_exp(w, prec)
def mpi_overlap(x, y):
a, b = x
c, d = y
if mpf_lt(d, a): return False
if mpf_gt(c, b): return False
return True
