def mpi_pow(s, t, prec):
ta, tb = t
if ta == tb and ta not in (finf, fninf):
if ta == from_int(to_int(ta)):
return mpi_pow_int(s, to_int(ta), prec)
if ta == fhalf:
return mpi_sqrt(s, prec)
u = mpi_log(s, prec + 20)
v = mpi_mul(u, t, prec + 20)
return mpi_exp(v, prec)
def mpi_pow(s, t, prec):
ta, tb = t
if ta == tb and ta not in (finf, fninf):
if ta == from_int(to_int(ta)):
return mpi_pow_int(s, to_int(ta), prec)
if ta == fhalf:
return mpi_sqrt(s, prec)
u = mpi_log(s, prec + 20)
v = mpi_mul(u, t, prec + 20)
return mpi_exp(v, prec)
def mpf_psi0(x, prec, rnd=round_fast):
"""
Computation of the digamma function (psi function of order 0)
of a real argument.
"""
sign, man, exp, bc = x
wp = prec + 10
if not man:
if x == finf:
return x
if x == fninf or x == fnan:
return fnan
if x == fzero or (exp >= 0 and sign):
raise ValueError("polygamma pole")
# Reflection formula
if sign and exp + bc > 3:
c, s = mpf_cos_sin_pi(x, wp)
q = mpf_mul(mpf_div(c, s, wp), mpf_pi(wp), wp)
p = mpf_psi0(mpf_sub(fone, x, wp), wp)
return mpf_sub(p, q, prec, rnd)
# The logarithmic term is accurate enough
if (not sign) and bc + exp > wp:
return mpf_log(mpf_sub(x, fone, wp), prec, rnd)
# Initial recurrence to obtain a large enough x
m = to_int(x)
n = int(0.11 * wp) + 2
s = MP_ZERO
x = to_fixed(x, wp)
one = MP_ONE << wp
if m < n:
for k in xrange(m, n):
s -= (one << wp) // x
x += one
x -= one
# Logarithmic term
s += to_fixed(mpf_log(from_man_exp(x, -wp, wp), wp), wp)
# Endpoint term in Euler-Maclaurin expansion
s += (one << wp) // (2 * x)
# Euler-Maclaurin remainder sum
x2 = (x * x) >> wp
t = one
prev = 0
k = 1
while 1:
t = (t * x2) >> wp
bsign, bman, bexp, bbc = mpf_bernoulli(2 * k, wp)
offset = bexp + 2 * wp
if offset >= 0:
term = (bman << offset) // (t * (2 * k))
else:
term = (bman >> (-offset)) // (t * (2 * k))
if k & 1:
s -= term
else:
s += term
if k > 2 and term >= prev:
break
prev = term
k += 1
return from_man_exp(s, -wp, wp, rnd)
def mpc_zeta(s, prec, rnd):
re, im = s
wp = prec + 20
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 = MP_ZERO
tim = MP_ZERO
one = MP_ONE << wp
one_2wp = MP_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 = cos_sin(from_man_exp(-imf * log_int_fixed(k+1, wp), -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)
q = mpc_sub(mpc_one, mpc_pow(mpc_two, mpc_sub(mpc_one, s, wp), wp), wp)
return mpc_div((tre, tim), q, prec, rnd)
def mpc_zeta(s, prec, rnd=round_fast, alt=0):
re, im = s
if im == fzero:
return mpf_zeta(re, prec, rnd, alt), fzero
wp = prec + 20
# 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 = MP_ZERO
tim = MP_ZERO
one = MP_ONE << wp
one_2wp = MP_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 = cos_sin(from_man_exp(-imf * log_int_fixed(k+1, wp), -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, mpc_sub(mpc_one, s, wp), wp), wp)
return mpc_div((tre, tim), q, prec, rnd)
def mpc_zeta(s, prec, rnd):
re, im = s
wp = prec + 20
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 = MP_ZERO
tim = MP_ZERO
one = MP_ONE << wp
one_2wp = MP_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 = cos_sin(
from_man_exp(-imf * log_int_fixed(k + 1, wp), -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)
q = mpc_sub(mpc_one, mpc_pow(mpc_two, mpc_sub(mpc_one, s, wp), wp), wp)
return mpc_div((tre, tim), q, prec, rnd)
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 mpf_psi0(x, prec, rnd=round_fast):
"""
Computation of the digamma function (psi function of order 0)
of a real argument.
"""
sign, man, exp, bc = x
wp = prec + 10
if not man:
if x == finf: return x
if x == fninf or x == fnan: return fnan
if x == fzero or (exp >= 0 and sign):
raise ValueError("polygamma pole")
# Reflection formula
if sign and exp + bc > 3:
c, s = mpf_cos_sin_pi(x, wp)
q = mpf_mul(mpf_div(c, s, wp), mpf_pi(wp), wp)
p = mpf_psi0(mpf_sub(fone, x, wp), wp)
return mpf_sub(p, q, prec, rnd)
# The logarithmic term is accurate enough
if (not sign) and bc + exp > wp:
return mpf_log(mpf_sub(x, fone, wp), prec, rnd)
# Initial recurrence to obtain a large enough x
m = to_int(x)
n = int(0.11 * wp) + 2
s = MP_ZERO
x = to_fixed(x, wp)
one = MP_ONE << wp
if m < n:
for k in xrange(m, n):
s -= (one << wp) // x
x += one
x -= one
# Logarithmic term
s += to_fixed(mpf_log(from_man_exp(x, -wp, wp), wp), wp)
# Endpoint term in Euler-Maclaurin expansion
s += (one << wp) // (2 * x)
# Euler-Maclaurin remainder sum
x2 = (x * x) >> wp
t = one
prev = 0
k = 1
while 1:
t = (t * x2) >> wp
bsign, bman, bexp, bbc = mpf_bernoulli(2 * k, wp)
offset = (bexp + 2 * wp)
if offset >= 0: term = (bman << offset) // (t * (2 * k))
else: term = (bman >> (-offset)) // (t * (2 * k))
if k & 1: s -= term
else: s += term
if k > 2 and term >= prev:
break
prev = term
k += 1
return from_man_exp(s, -wp, wp, rnd)
def mpc_nthroot_fixed(a, b, n, prec):
# a, b signed integers at fixed precision prec
start = 50
a1 = int(rshift(a, prec - n * start))
b1 = int(rshift(b, prec - n * start))
try:
r = (a1 + 1j * b1)**(1.0 / n)
re = r.real
im = r.imag
# XXX: workaround bug in gmpy
if abs(re) < 0.1: re = 0
if abs(im) < 0.1: im = 0
re = MP_BASE(re)
im = MP_BASE(im)
except OverflowError:
a1 = from_int(a1, start)
b1 = from_int(b1, start)
fn = from_int(n)
nth = mpf_rdiv_int(1, fn, start)
re, im = mpc_pow((a1, b1), (nth, fzero), start)
re = to_int(re)
im = to_int(im)
extra = 10
prevp = start
extra1 = n
for p in giant_steps(start, prec + extra):
# this is slow for large n, unlike int_pow_fixed
re2, im2 = complex_int_pow(re, im, n - 1)
re2 = rshift(re2, (n - 1) * prevp - p - extra1)
im2 = rshift(im2, (n - 1) * prevp - p - extra1)
r4 = (re2 * re2 + im2 * im2) >> (p + extra1)
ap = rshift(a, prec - p)
bp = rshift(b, prec - p)
rec = (ap * re2 + bp * im2) >> p
imc = (-ap * im2 + bp * re2) >> p
reb = (rec << p) // r4
imb = (imc << p) // r4
re = (reb + (n - 1) * lshift(re, p - prevp)) // n
im = (imb + (n - 1) * lshift(im, p - prevp)) // n
prevp = p
return re, im
def mpc_nthroot_fixed(a, b, n, prec):
# a, b signed integers at fixed precision prec
start = 50
a1 = int(rshift(a, prec - n*start))
b1 = int(rshift(b, prec - n*start))
try:
r = (a1 + 1j * b1)**(1.0/n)
re = r.real
im = r.imag
# XXX: workaround bug in gmpy
if abs(re) < 0.1: re = 0
if abs(im) < 0.1: im = 0
re = MP_BASE(re)
im = MP_BASE(im)
except OverflowError:
a1 = from_int(a1, start)
b1 = from_int(b1, start)
fn = from_int(n)
nth = mpf_rdiv_int(1, fn, start)
re, im = mpc_pow((a1, b1), (nth, fzero), start)
re = to_int(re)
im = to_int(im)
extra = 10
prevp = start
extra1 = n
for p in giant_steps(start, prec+extra):
# this is slow for large n, unlike int_pow_fixed
re2, im2 = complex_int_pow(re, im, n-1)
re2 = rshift(re2, (n-1)*prevp - p - extra1)
im2 = rshift(im2, (n-1)*prevp - p - extra1)
r4 = (re2*re2 + im2*im2) >> (p + extra1)
ap = rshift(a, prec - p)
bp = rshift(b, prec - p)
rec = (ap * re2 + bp * im2) >> p
imc = (-ap * im2 + bp * re2) >> p
reb = (rec << p) // r4
imb = (imc << p) // r4
re = (reb + (n-1)*lshift(re, p-prevp))//n
im = (imb + (n-1)*lshift(im, p-prevp))//n
prevp = p
return re, im
def mpc_psi0(z, prec, rnd=round_fast):
"""
Computation of the digamma function (psi function of order 0)
of a complex argument.
"""
re, im = z
# Fall back to the real case
if im == fzero:
return (mpf_psi0(re, prec, rnd), fzero)
wp = prec + 20
sign, man, exp, bc = re
# Reflection formula
if sign and exp + bc > 3:
c = mpc_cos_pi(z, wp)
s = mpc_sin_pi(z, wp)
q = mpc_mul(mpc_div(c, s, wp), (mpf_pi(wp), fzero), wp)
p = mpc_psi0(mpc_sub(mpc_one, z, wp), wp)
return mpc_sub(p, q, prec, rnd)
# Just the logarithmic term
if (not sign) and bc + exp > wp:
return mpc_log(mpc_sub(z, mpc_one, wp), prec, rnd)
# Initial recurrence to obtain a large enough z
w = to_int(re)
n = int(0.11 * wp) + 2
s = mpc_zero
if w < n:
for k in xrange(w, n):
s = mpc_sub(s, mpc_div(mpc_one, z, wp), wp)
z = mpc_add_mpf(z, fone, wp)
z = mpc_sub(z, mpc_one, wp)
# Logarithmic and endpoint term
s = mpc_add(s, mpc_log(z, wp), wp)
s = mpc_add(s, mpc_div(mpc_half, z, wp), wp)
# Euler-Maclaurin remainder sum
z2 = mpc_mul(z, z, wp)
t = mpc_one
prev = mpc_zero
k = 1
eps = mpf_shift(fone, -wp + 2)
while 1:
t = mpc_mul(t, z2, wp)
bern = mpf_bernoulli(2 * k, wp)
term = mpc_div((bern, fzero), mpc_mul_int(t, 2 * k, wp), wp)
s = mpc_sub(s, term, wp)
szterm = mpc_abs(term, 10)
if k > 2 and mpf_le(szterm, eps):
break
prev = term
k += 1
return s
def mpf_zeta(s, prec, rnd=round_fast):
sign, man, exp, bc = s
if not man:
if s == fzero:
return mpf_neg(fhalf)
if s == finf:
return fone
return fnan
wp = prec + 20
# First term vanishes?
if (not sign) and (exp + bc > (math.log(wp,2) + 2)):
if rnd in (round_up, round_ceiling):
return mpf_add(fone, mpf_shift(fone,-wp-10), prec, rnd)
return fone
elif exp >= 0:
return mpf_zeta_int(to_int(s), prec, rnd)
# Less than 0.5?
if sign or (exp+bc) < 0:
# XXX: -1 should be done exactly
y = mpf_sub(fone, s, 10*wp)
a = mpf_gamma(y, wp)
b = mpf_zeta(y, wp)
c = mpf_sin_pi(mpf_shift(s, -1), wp)
wp2 = wp + (exp+bc)
pi = mpf_pi(wp+wp2)
d = mpf_div(mpf_pow(mpf_shift(pi, 1), s, wp2), pi, wp2)
return mpf_mul(a,mpf_mul(b,mpf_mul(c,d,wp),wp),prec,rnd)
t = MP_ZERO
#wp += 16 - (prec & 15)
# Use Borwein's algorithm
n = int(wp/2.54 + 5)
d = borwein_coefficients(n)
t = MP_ZERO
sf = to_fixed(s, wp)
for k in xrange(n):
u = from_man_exp(-sf*log_int_fixed(k+1, wp), -2*wp, wp)
esign, eman, eexp, ebc = mpf_exp(u, wp)
offset = eexp + wp
if offset >= 0:
w = ((d[k] - d[n]) * eman) << offset
else:
w = ((d[k] - d[n]) * eman) >> (-offset)
if k & 1:
t -= w
else:
t += w
t = t // (-d[n])
t = from_man_exp(t, -wp, wp)
q = mpf_sub(fone, mpf_pow(ftwo, mpf_sub(fone, s, wp), wp), wp)
return mpf_div(t, q, prec, rnd)
def mpf_zeta(s, prec, rnd=round_fast):
sign, man, exp, bc = s
if not man:
if s == fzero:
return mpf_neg(fhalf)
if s == finf:
return fone
return fnan
wp = prec + 20
# First term vanishes?
if (not sign) and (exp + bc > (math.log(wp, 2) + 2)):
if rnd in (round_up, round_ceiling):
return mpf_add(fone, mpf_shift(fone, -wp - 10), prec, rnd)
return fone
elif exp >= 0:
return mpf_zeta_int(to_int(s), prec, rnd)
# Less than 0.5?
if sign or (exp + bc) < 0:
# XXX: -1 should be done exactly
y = mpf_sub(fone, s, 10 * wp)
a = mpf_gamma(y, wp)
b = mpf_zeta(y, wp)
c = mpf_sin_pi(mpf_shift(s, -1), wp)
wp2 = wp + (exp + bc)
pi = mpf_pi(wp + wp2)
d = mpf_div(mpf_pow(mpf_shift(pi, 1), s, wp2), pi, wp2)
return mpf_mul(a, mpf_mul(b, mpf_mul(c, d, wp), wp), prec, rnd)
t = MP_ZERO
#wp += 16 - (prec & 15)
# Use Borwein's algorithm
n = int(wp / 2.54 + 5)
d = borwein_coefficients(n)
t = MP_ZERO
sf = to_fixed(s, wp)
for k in xrange(n):
u = from_man_exp(-sf * log_int_fixed(k + 1, wp), -2 * wp, wp)
esign, eman, eexp, ebc = mpf_exp(u, wp)
offset = eexp + wp
if offset >= 0:
w = ((d[k] - d[n]) * eman) << offset
else:
w = ((d[k] - d[n]) * eman) >> (-offset)
if k & 1:
t -= w
else:
t += w
t = t // (-d[n])
t = from_man_exp(t, -wp, wp)
q = mpf_sub(fone, mpf_pow(ftwo, mpf_sub(fone, s, wp), wp), wp)
return mpf_div(t, q, prec, rnd)
def mpc_psi0(z, prec, rnd=round_fast):
"""
Computation of the digamma function (psi function of order 0)
of a complex argument.
"""
re, im = z
# Fall back to the real case
if im == fzero:
return (mpf_psi0(re, prec, rnd), fzero)
wp = prec + 20
sign, man, exp, bc = re
# Reflection formula
if sign and exp+bc > 3:
c = mpc_cos_pi(z, wp)
s = mpc_sin_pi(z, wp)
q = mpc_mul_mpf(mpc_div(c, s, wp), mpf_pi(wp), wp)
p = mpc_psi0(mpc_sub(mpc_one, z, wp), wp)
return mpc_sub(p, q, prec, rnd)
# Just the logarithmic term
if (not sign) and bc + exp > wp:
return mpc_log(mpc_sub(z, mpc_one, wp), prec, rnd)
# Initial recurrence to obtain a large enough z
w = to_int(re)
n = int(0.11*wp) + 2
s = mpc_zero
if w < n:
for k in xrange(w, n):
s = mpc_sub(s, mpc_reciprocal(z, wp), wp)
z = mpc_add_mpf(z, fone, wp)
z = mpc_sub(z, mpc_one, wp)
# Logarithmic and endpoint term
s = mpc_add(s, mpc_log(z, wp), wp)
s = mpc_add(s, mpc_div(mpc_half, z, wp), wp)
# Euler-Maclaurin remainder sum
z2 = mpc_square(z, wp)
t = mpc_one
prev = mpc_zero
k = 1
eps = mpf_shift(fone, -wp+2)
while 1:
t = mpc_mul(t, z2, wp)
bern = mpf_bernoulli(2*k, wp)
term = mpc_mpf_div(bern, mpc_mul_int(t, 2*k, wp), wp)
s = mpc_sub(s, term, wp)
szterm = mpc_abs(term, 10)
if k > 2 and mpf_le(szterm, eps):
break
prev = term
k += 1
return s
def mpf_ei(x, prec, rnd=round_fast, e1=False):
if e1:
x = mpf_neg(x)
sign, man, exp, bc = x
if e1 and not sign:
if x == fzero:
return finf
raise ComplexResult("E1(x) for x < 0")
if man:
xabs = 0, man, exp, bc
xmag = exp+bc
wp = prec + 20
can_use_asymp = xmag > wp
if not can_use_asymp:
if exp >= 0:
xabsint = man << exp
else:
xabsint = man >> (-exp)
can_use_asymp = xabsint > int(wp*0.693) + 10
if can_use_asymp:
if xmag > wp:
v = fone
else:
v = from_man_exp(ei_asymptotic(to_fixed(x, wp), wp), -wp)
v = mpf_mul(v, mpf_exp(x, wp), wp)
v = mpf_div(v, x, prec, rnd)
else:
wp += 2*int(to_int(xabs))
u = to_fixed(x, wp)
v = ei_taylor(u, wp) + euler_fixed(wp)
t1 = from_man_exp(v,-wp)
t2 = mpf_log(xabs,wp)
v = mpf_add(t1, t2, prec, rnd)
else:
if x == fzero: v = fninf
elif x == finf: v = finf
elif x == fninf: v = fzero
else: v = fnan
if e1:
v = mpf_neg(v)
return v
def mpf_ei(x, prec, rnd=round_fast, e1=False):
if e1:
x = mpf_neg(x)
sign, man, exp, bc = x
if e1 and not sign:
if x == fzero:
return finf
raise ComplexResult("E1(x) for x < 0")
if man:
xabs = 0, man, exp, bc
xmag = exp + bc
wp = prec + 20
can_use_asymp = xmag > wp
if not can_use_asymp:
if exp >= 0:
xabsint = man << exp
else:
xabsint = man >> (-exp)
can_use_asymp = xabsint > int(wp * 0.693) + 10
if can_use_asymp:
if xmag > wp:
v = fone
else:
v = from_man_exp(ei_asymptotic(to_fixed(x, wp), wp), -wp)
v = mpf_mul(v, mpf_exp(x, wp), wp)
v = mpf_div(v, x, prec, rnd)
else:
wp += 2 * int(to_int(xabs))
u = to_fixed(x, wp)
v = ei_taylor(u, wp) + euler_fixed(wp)
t1 = from_man_exp(v, -wp)
t2 = mpf_log(xabs, wp)
v = mpf_add(t1, t2, prec, rnd)
else:
if x == fzero: v = fninf
elif x == finf: v = finf
elif x == fninf: v = fzero
else: v = fnan
if e1:
v = mpf_neg(v)
return v
def nthroot_fixed(y, n, prec, exp1):
start = 50
try:
y1 = rshift(y, prec - n*start)
r = MP_BASE(int(y1**(1.0/n)))
except OverflowError:
y1 = from_int(y1, start)
fn = from_int(n)
fn = mpf_rdiv_int(1, fn, start)
r = mpf_pow(y1, fn, start)
r = to_int(r)
extra = 10
extra1 = n
prevp = start
for p in giant_steps(start, prec+extra):
pm, pe = int_pow_fixed(r, n-1, prevp)
r2 = rshift(pm, (n-1)*prevp - p - pe - extra1)
B = lshift(y, 2*p-prec+extra1)//r2
r = (B + (n-1) * lshift(r, p-prevp))//n
prevp = p
return r
def nthroot_fixed(y, n, prec, exp1):
start = 50
try:
y1 = rshift(y, prec - n*start)
r = MP_BASE(y1**(1.0/n))
except OverflowError:
y1 = from_int(y1, start)
fn = from_int(n)
fn = mpf_rdiv_int(1, fn, start)
r = mpf_pow(y1, fn, start)
r = to_int(r)
extra = 10
extra1 = n
prevp = start
for p in giant_steps(start, prec+extra):
pm, pe = int_pow_fixed(r, n-1, prevp)
r2 = rshift(pm, (n-1)*prevp - p - pe - extra1)
B = lshift(y, 2*p-prec+extra1)//r2
r = (B + (n-1) * lshift(r, p-prevp))//n
prevp = p
return r
def mpf_erfc(x, prec, rnd=round_fast):
sign, man, exp, bc = x
if not man:
if x == fzero: return fone
if x == finf: return fzero
if x == fninf: return ftwo
return fnan
wp = prec + 20
mag = bc + exp
# Preserve full accuracy when exponent grows huge
wp += max(0, 2 * mag)
regular_erf = sign or mag < 2
if regular_erf or not erfc_check_series(x, wp):
if regular_erf:
return mpf_sub(fone, mpf_erf(x, prec + 10, negative_rnd[rnd]),
prec, rnd)
# 1-erf(x) ~ exp(-x^2), increase prec to deal with cancellation
n = to_int(x)
return mpf_sub(fone, mpf_erf(x, prec + int(n**2 * 1.44) + 10), prec,
rnd)
s = term = MP_ONE << wp
term_prev = 0
t = (2 * to_fixed(x, wp)**2) >> wp
k = 1
while 1:
term = ((term * (2 * k - 1)) << wp) // t
if k > 4 and term > term_prev or not term:
break
if k & 1:
s -= term
else:
s += term
term_prev = term
#print k, to_str(from_man_exp(term, -wp, 50), 10)
k += 1
s = (s << wp) // sqrt_fixed(pi_fixed(wp), wp)
s = from_man_exp(s, -wp, wp)
z = mpf_exp(mpf_neg(mpf_mul(x, x, wp), wp), wp)
y = mpf_div(mpf_mul(z, s, wp), x, prec, rnd)
return y
def bernfrac(n):
"""
Computes integers (p,q) such that p/q = B_n exactly, where
B_n denotes the nth Bernoulli number.
Use bernoulli(n) to get a floating-point approximation
instead of the exact fraction (much faster for large n).
"""
n = int(n)
if n < 3:
return [(1, 1), (-1, 2), (1, 6)][n]
if n & 1:
return (0, 1)
q = 1
for k in list_primes(n + 1):
if not (n % (k - 1)):
q *= k
prec = bernoulli_size(n) + int(math.log(q, 2)) + 20
b = mpf_bernoulli(n, prec)
p = mpf_mul(b, from_int(q))
pint = to_int(p, round_nearest)
return (pint, q)
def mpf_fibonacci(x, prec, rnd=round_fast):
sign, man, exp, bc = x
if not man:
if x == fninf:
return fnan
return x
# F(2^n) ~= 2^(2^n)
size = abs(exp+bc)
if exp >= 0:
# Exact
if size < 10 or size <= bitcount(prec):
return from_int(ifib(to_int(x)), prec, rnd)
# Use the modified Binet formula
wp = prec + size + 20
a = mpf_phi(wp)
b = mpf_add(mpf_shift(a, 1), fnone, wp)
u = mpf_pow(a, x, wp)
v = mpf_cos_pi(x, wp)
v = mpf_div(v, u, wp)
u = mpf_sub(u, v, wp)
u = mpf_div(u, b, prec, rnd)
return u
def bernfrac(n):
"""
Computes integers (p,q) such that p/q = B_n exactly, where
B_n denotes the nth Bernoulli number.
Use bernoulli(n) to get a floating-point approximation
instead of the exact fraction (much faster for large n).
"""
n = int(n)
if n < 3:
return [(1, 1), (-1, 2), (1, 6)][n]
if n & 1:
return (0, 1)
q = 1
for k in list_primes(n+1):
if not (n % (k-1)):
q *= k
prec = bernoulli_size(n) + int(math.log(q,2)) + 20
b = mpf_bernoulli(n, prec)
p = mpf_mul(b, from_int(q))
pint = to_int(p, round_nearest)
return (pint, q)
def mpf_erfc(x, prec, rnd=round_fast):
sign, man, exp, bc = x
if not man:
if x == fzero: return fone
if x == finf: return fzero
if x == fninf: return ftwo
return fnan
wp = prec + 20
mag = bc+exp
# Preserve full accuracy when exponent grows huge
wp += max(0, 2*mag)
regular_erf = sign or mag < 2
if regular_erf or not erfc_check_series(x, wp):
if regular_erf:
return mpf_sub(fone, mpf_erf(x, prec+10, negative_rnd[rnd]), prec, rnd)
# 1-erf(x) ~ exp(-x^2), increase prec to deal with cancellation
n = to_int(x)+1
return mpf_sub(fone, mpf_erf(x, prec + int(n**2*1.44) + 10), prec, rnd)
s = term = MPZ_ONE << wp
term_prev = 0
t = (2 * to_fixed(x, wp) ** 2) >> wp
k = 1
while 1:
term = ((term * (2*k - 1)) << wp) // t
if k > 4 and term > term_prev or not term:
break
if k & 1:
s -= term
else:
s += term
term_prev = term
#print k, to_str(from_man_exp(term, -wp, 50), 10)
k += 1
s = (s << wp) // sqrt_fixed(pi_fixed(wp), wp)
s = from_man_exp(s, -wp, wp)
z = mpf_exp(mpf_neg(mpf_mul(x,x,wp),wp),wp)
y = mpf_div(mpf_mul(z, s, wp), x, prec, rnd)
return y
def __int__(s): return int(to_int(s._mpf_)) def __long__(s): return long(to_int(s._mpf_))
def mpc_ei(z, prec, rnd=round_fast, e1=False):
if e1:
z = mpc_neg(z)
a, b = z
asign, aman, aexp, abc = a
bsign, bman, bexp, bbc = b
if b == fzero:
if e1:
x = mpf_neg(mpf_ei(a, prec, rnd))
if not asign:
y = mpf_neg(mpf_pi(prec, rnd))
else:
y = fzero
return x, y
else:
return mpf_ei(a, prec, rnd), fzero
if a != fzero:
if not aman or not bman:
return (fnan, fnan)
wp = prec + 40
amag = aexp + abc
bmag = bexp + bbc
zmag = max(amag, bmag)
can_use_asymp = zmag > wp
if not can_use_asymp:
zabsint = abs(to_int(a)) + abs(to_int(b))
can_use_asymp = zabsint > int(wp * 0.693) + 20
try:
if can_use_asymp:
if zmag > wp:
v = fone, fzero
else:
zre = to_fixed(a, wp)
zim = to_fixed(b, wp)
vre, vim = complex_ei_asymptotic(zre, zim, wp)
v = from_man_exp(vre, -wp), from_man_exp(vim, -wp)
v = mpc_mul(v, mpc_exp(z, wp), wp)
v = mpc_div(v, z, wp)
if e1:
v = mpc_neg(v, prec, rnd)
else:
x, y = v
if bsign:
v = mpf_pos(x, prec, rnd), mpf_sub(y, mpf_pi(wp), prec,
rnd)
else:
v = mpf_pos(x, prec, rnd), mpf_add(y, mpf_pi(wp), prec,
rnd)
return v
except NoConvergence:
pass
#wp += 2*max(0,zmag)
wp += 2 * int(to_int(mpc_abs(z, 5)))
zre = to_fixed(a, wp)
zim = to_fixed(b, wp)
vre, vim = complex_ei_taylor(zre, zim, wp)
vre += euler_fixed(wp)
v = from_man_exp(vre, -wp), from_man_exp(vim, -wp)
if e1:
u = mpc_log(mpc_neg(z), wp)
else:
u = mpc_log(z, wp)
v = mpc_add(v, u, prec, rnd)
if e1:
v = mpc_neg(v)
return v
def erfc_check_series(x, prec):
n = to_int(x)
if n**2 * 1.44 > prec:
return True
return False
def __long__(s): return long(to_int(s._mpf_)) def __float__(s): return to_float(s._mpf_)
def bernfrac(n):
r"""
Returns a tuple of integers `(p, q)` such that `p/q = B_n` exactly,
where `B_n` denotes the `n`-th Bernoulli number. The fraction is
always reduced to lowest terms. Note that for `n > 1` and `n` odd,
`B_n = 0`, and `(0, 1)` is returned.
**Examples**
The first few Bernoulli numbers are exactly::
>>> from mpmath import *
>>> for n in range(15):
... p, q = bernfrac(n)
... print n, "%s/%s" % (p, q)
...
0 1/1
1 -1/2
2 1/6
3 0/1
4 -1/30
5 0/1
6 1/42
7 0/1
8 -1/30
9 0/1
10 5/66
11 0/1
12 -691/2730
13 0/1
14 7/6
This function works for arbitrarily large `n`::
>>> p, q = bernfrac(10**4)
>>> print q
2338224387510
>>> print len(str(p))
27692
>>> mp.dps = 15
>>> print mpf(p) / q
-9.04942396360948e+27677
>>> print bernoulli(10**4)
-9.04942396360948e+27677
Note: :func:`bernoulli` computes a floating-point approximation
directly, without computing the exact fraction first.
This is much faster for large `n`.
**Algorithm**
:func:`bernfrac` works by computing the value of `B_n` numerically
and then using the von Staudt-Clausen theorem [1] to reconstruct
the exact fraction. For large `n`, this is significantly faster than
computing `B_1, B_2, \ldots, B_2` recursively with exact arithmetic.
The implementation has been tested for `n = 10^m` up to `m = 6`.
In practice, :func:`bernfrac` appears to be about three times
slower than the specialized program calcbn.exe [2]
**References**
1. MathWorld, von Staudt-Clausen Theorem:
http://mathworld.wolfram.com/vonStaudt-ClausenTheorem.html
2. The Bernoulli Number Page:
http://www.bernoulli.org/
"""
n = int(n)
if n < 3:
return [(1, 1), (-1, 2), (1, 6)][n]
if n & 1:
return (0, 1)
q = 1
for k in list_primes(n + 1):
if not (n % (k - 1)):
q *= k
prec = bernoulli_size(n) + int(math.log(q, 2)) + 20
b = mpf_bernoulli(n, prec)
p = mpf_mul(b, from_int(q))
pint = to_int(p, round_nearest)
return (pint, q)
def bernfrac(n):
r"""
Returns a tuple of integers `(p, q)` such that `p/q = B_n` exactly,
where `B_n` denotes the `n`-th Bernoulli number. The fraction is
always reduced to lowest terms. Note that for `n > 1` and `n` odd,
`B_n = 0`, and `(0, 1)` is returned.
**Examples**
The first few Bernoulli numbers are exactly::
>>> from mpmath import *
>>> for n in range(15):
... p, q = bernfrac(n)
... print n, "%s/%s" % (p, q)
...
0 1/1
1 -1/2
2 1/6
3 0/1
4 -1/30
5 0/1
6 1/42
7 0/1
8 -1/30
9 0/1
10 5/66
11 0/1
12 -691/2730
13 0/1
14 7/6
This function works for arbitrarily large `n`::
>>> p, q = bernfrac(10**4)
>>> print q
2338224387510
>>> print len(str(p))
27692
>>> mp.dps = 15
>>> print mpf(p) / q
-9.04942396360948e+27677
>>> print bernoulli(10**4)
-9.04942396360948e+27677
Note: :func:`bernoulli` computes a floating-point approximation
directly, without computing the exact fraction first.
This is much faster for large `n`.
**Algorithm**
:func:`bernfrac` works by computing the value of `B_n` numerically
and then using the von Staudt-Clausen theorem [1] to reconstruct
the exact fraction. For large `n`, this is significantly faster than
computing `B_1, B_2, \ldots, B_2` recursively with exact arithmetic.
The implementation has been tested for `n = 10^m` up to `m = 6`.
In practice, :func:`bernfrac` appears to be about three times
slower than the specialized program calcbn.exe [2]
**References**
1. MathWorld, von Staudt-Clausen Theorem:
http://mathworld.wolfram.com/vonStaudt-ClausenTheorem.html
2. The Bernoulli Number Page:
http://www.bernoulli.org/
"""
n = int(n)
if n < 3:
return [(1, 1), (-1, 2), (1, 6)][n]
if n & 1:
return (0, 1)
q = 1
for k in list_primes(n+1):
if not (n % (k-1)):
q *= k
prec = bernoulli_size(n) + int(math.log(q,2)) + 20
b = mpf_bernoulli(n, prec)
p = mpf_mul(b, from_int(q))
pint = to_int(p, round_nearest)
return (pint, q)
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 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 mpf_expint(n, x, prec, rnd=round_fast, gamma=False):
"""
E_n(x), n an integer, x real
With gamma=True, computes Gamma(n,x) (upper incomplete gamma function)
Returns (real, None) if real, otherwise (real, imag)
The imaginary part is an optional branch cut term
"""
sign, man, exp, bc = x
if not man:
if gamma:
if x == fzero:
# Actually gamma function pole
if n <= 0:
return finf, None
return mpf_gamma_int(n, prec, rnd), None
if x == finf:
return fzero, None
# TODO: could return finite imaginary value at -inf
return fnan, fnan
else:
if x == fzero:
if n > 1:
return from_rational(1, n-1, prec, rnd), None
else:
return finf, None
if x == finf:
return fzero, None
return fnan, fnan
n_orig = n
if gamma:
n = 1-n
wp = prec + 20
xmag = exp + bc
# Beware of near-poles
if xmag < -10:
raise NotImplementedError
nmag = bitcount(abs(n))
have_imag = n > 0 and sign
negx = mpf_neg(x)
# Skip series if direct convergence
if n == 0 or 2*nmag - xmag < -wp:
if gamma:
v = mpf_exp(negx, wp)
re = mpf_mul(v, mpf_pow_int(x, n_orig-1, wp), prec, rnd)
else:
v = mpf_exp(negx, wp)
re = mpf_div(v, x, prec, rnd)
else:
# Finite number of terms, or...
can_use_asymptotic_series = -3*wp < n <= 0
# ...large enough?
if not can_use_asymptotic_series:
xi = abs(to_int(x))
m = min(max(1, xi-n), 2*wp)
siz = -n*nmag + (m+n)*bitcount(abs(m+n)) - m*xmag - (144*m//100)
tol = -wp-10
can_use_asymptotic_series = siz < tol
if can_use_asymptotic_series:
r = ((-MPZ_ONE) << (wp+wp)) // to_fixed(x, wp)
m = n
t = r*m
s = MPZ_ONE << wp
while m and t:
s += t
m += 1
t = (m*r*t) >> wp
v = mpf_exp(negx, wp)
if gamma:
# ~ exp(-x) * x^(n-1) * (1 + ...)
v = mpf_mul(v, mpf_pow_int(x, n_orig-1, wp), wp)
else:
# ~ exp(-x)/x * (1 + ...)
v = mpf_div(v, x, wp)
re = mpf_mul(v, from_man_exp(s, -wp), prec, rnd)
elif n == 1:
re = mpf_neg(mpf_ei(negx, prec, rnd))
elif n > 0 and n < 3*wp:
T1 = mpf_neg(mpf_ei(negx, wp))
if gamma:
if n_orig & 1:
T1 = mpf_neg(T1)
else:
T1 = mpf_mul(T1, mpf_pow_int(negx, n-1, wp), wp)
r = t = to_fixed(x, wp)
facs = [1] * (n-1)
for k in range(1,n-1):
facs[k] = facs[k-1] * k
facs = facs[::-1]
s = facs[0] << wp
for k in range(1, n-1):
if k & 1:
s -= facs[k] * t
else:
s += facs[k] * t
t = (t*r) >> wp
T2 = from_man_exp(s, -wp, wp)
T2 = mpf_mul(T2, mpf_exp(negx, wp))
if gamma:
T2 = mpf_mul(T2, mpf_pow_int(x, n_orig, wp), wp)
R = mpf_add(T1, T2)
re = mpf_div(R, from_int(ifac(n-1)), prec, rnd)
else:
raise NotImplementedError
if have_imag:
M = from_int(-ifac(n-1))
if gamma:
im = mpf_div(mpf_pi(wp), M, prec, rnd)
else:
im = mpf_div(mpf_mul(mpf_pi(wp), mpf_pow_int(negx, n_orig-1, wp), wp), M, prec, rnd)
return re, im
else:
return re, None
def __long__(s): return long(to_int(s._mpf_)) def __float__(s): return to_float(s._mpf_)
def mpc_zeta(s, prec, rnd=round_fast, alt=0):
re, im = s
if im == fzero:
return mpf_zeta(re, prec, rnd, alt), fzero
wp = prec + 20
# 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 = MP_ZERO
tim = MP_ZERO
one = MP_ONE << wp
one_2wp = MP_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 = cos_sin(
from_man_exp(-imf * log_int_fixed(k + 1, wp), -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, mpc_sub(mpc_one, s, wp), wp), wp)
return mpc_div((tre, tim), q, prec, rnd)
def mpf_zeta(s, prec, rnd=round_fast, alt=0):
sign, man, exp, bc = s
if not man:
if s == fzero:
if alt:
return fhalf
else:
return mpf_neg(fhalf)
if s == finf:
return fone
return fnan
wp = prec + 20
# First term vanishes?
if (not sign) and (exp + bc > (math.log(wp,2) + 2)):
return mpf_perturb(fone, alt, prec, rnd)
# Optimize for integer arguments
elif exp >= 0:
if alt:
if s == fone:
return mpf_ln2(prec, rnd)
z = mpf_zeta_int(to_int(s), wp, negative_rnd[rnd])
q = mpf_sub(fone, mpf_pow(ftwo, mpf_sub(fone, s, wp), wp), wp)
return mpf_mul(z, q, prec, rnd)
else:
return mpf_zeta_int(to_int(s), prec, rnd)
# Negative: use the reflection formula
# Borwein only proves the accuracy bound for x >= 1/2. However, based on
# tests, the accuracy without reflection is quite good even some distance
# to the left of 1/2. XXX: verify this.
if sign:
# XXX: could use the separate refl. formula for Dirichlet eta
if alt:
q = mpf_sub(fone, mpf_pow(ftwo, mpf_sub(fone, s, wp), wp), wp)
return mpf_mul(mpf_zeta(s, wp), q, prec, rnd)
# XXX: -1 should be done exactly
y = mpf_sub(fone, s, 10*wp)
a = mpf_gamma(y, wp)
b = mpf_zeta(y, wp)
c = mpf_sin_pi(mpf_shift(s, -1), wp)
wp2 = wp + (exp+bc)
pi = mpf_pi(wp+wp2)
d = mpf_div(mpf_pow(mpf_shift(pi, 1), s, wp2), pi, wp2)
return mpf_mul(a,mpf_mul(b,mpf_mul(c,d,wp),wp),prec,rnd)
# Near pole
r = mpf_sub(fone, s, wp)
asign, aman, aexp, abc = mpf_abs(r)
pole_dist = -2*(aexp+abc)
if pole_dist > wp:
if alt:
return mpf_ln2(prec, rnd)
else:
q = mpf_neg(mpf_div(fone, r, wp))
return mpf_add(q, mpf_euler(wp), prec, rnd)
else:
wp += max(0, pole_dist)
t = MPZ_ZERO
#wp += 16 - (prec & 15)
# Use Borwein's algorithm
n = int(wp/2.54 + 5)
d = borwein_coefficients(n)
t = MPZ_ZERO
sf = to_fixed(s, wp)
for k in xrange(n):
u = from_man_exp(-sf*log_int_fixed(k+1, wp), -2*wp, wp)
esign, eman, eexp, ebc = mpf_exp(u, wp)
offset = eexp + wp
if offset >= 0:
w = ((d[k] - d[n]) * eman) << offset
else:
w = ((d[k] - d[n]) * eman) >> (-offset)
if k & 1:
t -= w
else:
t += w
t = t // (-d[n])
t = from_man_exp(t, -wp, wp)
if alt:
return mpf_pos(t, prec, rnd)
else:
q = mpf_sub(fone, mpf_pow(ftwo, mpf_sub(fone, s, wp), wp), wp)
return mpf_div(t, q, prec, rnd)
def mpf_zeta(s, prec, rnd=round_fast, alt=0):
sign, man, exp, bc = s
if not man:
if s == fzero:
if alt:
return fhalf
else:
return mpf_neg(fhalf)
if s == finf:
return fone
return fnan
wp = prec + 20
# First term vanishes?
if (not sign) and (exp + bc > (math.log(wp, 2) + 2)):
return mpf_perturb(fone, alt, prec, rnd)
# Optimize for integer arguments
elif exp >= 0:
if alt:
if s == fone:
return mpf_ln2(prec, rnd)
z = mpf_zeta_int(to_int(s), wp, negative_rnd[rnd])
q = mpf_sub(fone, mpf_pow(ftwo, mpf_sub(fone, s, wp), wp), wp)
return mpf_mul(z, q, prec, rnd)
else:
return mpf_zeta_int(to_int(s), prec, rnd)
# Negative: use the reflection formula
# Borwein only proves the accuracy bound for x >= 1/2. However, based on
# tests, the accuracy without reflection is quite good even some distance
# to the left of 1/2. XXX: verify this.
if sign:
# XXX: could use the separate refl. formula for Dirichlet eta
if alt:
q = mpf_sub(fone, mpf_pow(ftwo, mpf_sub(fone, s, wp), wp), wp)
return mpf_mul(mpf_zeta(s, wp), q, prec, rnd)
# XXX: -1 should be done exactly
y = mpf_sub(fone, s, 10 * wp)
a = mpf_gamma(y, wp)
b = mpf_zeta(y, wp)
c = mpf_sin_pi(mpf_shift(s, -1), wp)
wp2 = wp + (exp + bc)
pi = mpf_pi(wp + wp2)
d = mpf_div(mpf_pow(mpf_shift(pi, 1), s, wp2), pi, wp2)
return mpf_mul(a, mpf_mul(b, mpf_mul(c, d, wp), wp), prec, rnd)
t = MP_ZERO
#wp += 16 - (prec & 15)
# Use Borwein's algorithm
n = int(wp / 2.54 + 5)
d = borwein_coefficients(n)
t = MP_ZERO
sf = to_fixed(s, wp)
for k in xrange(n):
u = from_man_exp(-sf * log_int_fixed(k + 1, wp), -2 * wp, wp)
esign, eman, eexp, ebc = mpf_exp(u, wp)
offset = eexp + wp
if offset >= 0:
w = ((d[k] - d[n]) * eman) << offset
else:
w = ((d[k] - d[n]) * eman) >> (-offset)
if k & 1:
t -= w
else:
t += w
t = t // (-d[n])
t = from_man_exp(t, -wp, wp)
if alt:
return mpf_pos(t, prec, rnd)
else:
q = mpf_sub(fone, mpf_pow(ftwo, mpf_sub(fone, s, wp), wp), wp)
return mpf_div(t, 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 mpf_expint(n, x, prec, rnd=round_fast, gamma=False):
"""
E_n(x), n an integer, x real
With gamma=True, computes Gamma(n,x) (upper incomplete gamma function)
Returns (real, None) if real, otherwise (real, imag)
The imaginary part is an optional branch cut term
"""
sign, man, exp, bc = x
if not man:
if gamma:
if x == fzero:
# Actually gamma function pole
if n <= 0:
return finf
return mpf_gamma_int(n, prec, rnd)
if x == finf:
return fzero, None
# TODO: could return finite imaginary value at -inf
return fnan, fnan
else:
if x == fzero:
if n > 1:
return from_rational(1, n - 1, prec, rnd), None
else:
return finf, None
if x == finf:
return fzero, None
return fnan, fnan
n_orig = n
if gamma:
n = 1 - n
wp = prec + 20
xmag = exp + bc
# Beware of near-poles
if xmag < -10:
raise NotImplementedError
nmag = bitcount(abs(n))
have_imag = n > 0 and sign
negx = mpf_neg(x)
# Skip series if direct convergence
if n == 0 or 2 * nmag - xmag < -wp:
if gamma:
v = mpf_exp(negx, wp)
re = mpf_mul(v, mpf_pow_int(x, n_orig - 1, wp), prec, rnd)
else:
v = mpf_exp(negx, wp)
re = mpf_div(v, x, prec, rnd)
else:
# Finite number of terms, or...
can_use_asymptotic_series = -3 * wp < n <= 0
# ...large enough?
if not can_use_asymptotic_series:
xi = abs(to_int(x))
m = min(max(1, xi - n), 2 * wp)
siz = -n * nmag + (m + n) * bitcount(abs(m + n)) - m * xmag - (
144 * m // 100)
tol = -wp - 10
can_use_asymptotic_series = siz < tol
if can_use_asymptotic_series:
r = ((-MP_ONE) << (wp + wp)) // to_fixed(x, wp)
m = n
t = r * m
s = MP_ONE << wp
while m and t:
s += t
m += 1
t = (m * r * t) >> wp
v = mpf_exp(negx, wp)
if gamma:
# ~ exp(-x) * x^(n-1) * (1 + ...)
v = mpf_mul(v, mpf_pow_int(x, n_orig - 1, wp), wp)
else:
# ~ exp(-x)/x * (1 + ...)
v = mpf_div(v, x, wp)
re = mpf_mul(v, from_man_exp(s, -wp), prec, rnd)
elif n == 1:
re = mpf_neg(mpf_ei(negx, prec, rnd))
elif n > 0 and n < 3 * wp:
T1 = mpf_neg(mpf_ei(negx, wp))
if gamma:
if n_orig & 1:
T1 = mpf_neg(T1)
else:
T1 = mpf_mul(T1, mpf_pow_int(negx, n - 1, wp), wp)
r = t = to_fixed(x, wp)
facs = [1] * (n - 1)
for k in range(1, n - 1):
facs[k] = facs[k - 1] * k
facs = facs[::-1]
s = facs[0] << wp
for k in range(1, n - 1):
if k & 1:
s -= facs[k] * t
else:
s += facs[k] * t
t = (t * r) >> wp
T2 = from_man_exp(s, -wp, wp)
T2 = mpf_mul(T2, mpf_exp(negx, wp))
if gamma:
T2 = mpf_mul(T2, mpf_pow_int(x, n_orig, wp), wp)
R = mpf_add(T1, T2)
re = mpf_div(R, from_int(int_fac(n - 1)), prec, rnd)
else:
raise NotImplementedError
if have_imag:
M = from_int(-int_fac(n - 1))
if gamma:
im = mpf_div(mpf_pi(wp), M, prec, rnd)
else:
im = mpf_div(
mpf_mul(mpf_pi(wp), mpf_pow_int(negx, n_orig - 1, wp), wp), M,
prec, rnd)
return re, im
else:
return re, None
def mpc_ei(z, prec, rnd=round_fast, e1=False):
if e1:
z = mpc_neg(z)
a, b = z
asign, aman, aexp, abc = a
bsign, bman, bexp, bbc = b
if b == fzero:
if e1:
x = mpf_neg(mpf_ei(a, prec, rnd))
if not asign:
y = mpf_neg(mpf_pi(prec, rnd))
else:
y = fzero
return x, y
else:
return mpf_ei(a, prec, rnd), fzero
if a != fzero:
if not aman or not bman:
return (fnan, fnan)
wp = prec + 40
amag = aexp+abc
bmag = bexp+bbc
zmag = max(amag, bmag)
can_use_asymp = zmag > wp
if not can_use_asymp:
zabsint = abs(to_int(a)) + abs(to_int(b))
can_use_asymp = zabsint > int(wp*0.693) + 20
try:
if can_use_asymp:
if zmag > wp:
v = fone, fzero
else:
zre = to_fixed(a, wp)
zim = to_fixed(b, wp)
vre, vim = complex_ei_asymptotic(zre, zim, wp)
v = from_man_exp(vre, -wp), from_man_exp(vim, -wp)
v = mpc_mul(v, mpc_exp(z, wp), wp)
v = mpc_div(v, z, wp)
if e1:
v = mpc_neg(v, prec, rnd)
else:
x, y = v
if bsign:
v = mpf_pos(x, prec, rnd), mpf_sub(y, mpf_pi(wp), prec, rnd)
else:
v = mpf_pos(x, prec, rnd), mpf_add(y, mpf_pi(wp), prec, rnd)
return v
except NoConvergence:
pass
#wp += 2*max(0,zmag)
wp += 2*int(to_int(mpc_abs(z, 5)))
zre = to_fixed(a, wp)
zim = to_fixed(b, wp)
vre, vim = complex_ei_taylor(zre, zim, wp)
vre += euler_fixed(wp)
v = from_man_exp(vre,-wp), from_man_exp(vim,-wp)
if e1:
u = mpc_log(mpc_neg(z),wp)
else:
u = mpc_log(z,wp)
v = mpc_add(v, u, prec, rnd)
if e1:
v = mpc_neg(v)
return v
def erfc_check_series(x, prec):
n = to_int(x)
if n**2 * 1.44 > prec:
return True
return False
def __int__(s): return int(to_int(s._mpf_)) def __long__(s): return long(to_int(s._mpf_))
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)
