def zeta(s):
"""
zeta(s) -- calculate the Riemann zeta function of a real or complex
argument s.
"""
Float.store()
Float._prec += 8
si = s
s = ComplexFloat(s)
if s.real < 0:
# Reflection formula (XXX: gets bad around the zeros)
pi = pi_float()
y = power(2, s) * power(pi, s-1) * sin(pi*s/2) * gamma(1-s) * zeta(1-s)
else:
p = Float._prec
n = int((p + 2.28*abs(float(s.imag)))/2.54) + 3
d = _zeta_coefs(n)
if isinstance(si, (int, long)):
t = 0
for k in range(n):
t += (((-1)**k * (d[k] - d[n])) << p) // (k+1)**si
y = (Float((t, -p)) / -d[n]) / (Float(1) - Float(2)**(1-si))
else:
t = Float(0)
for k in range(n):
t += (-1)**k * Float(d[k]-d[n]) * exp(-_logk(k+1)*s)
y = (t / -d[n]) / (Float(1) - exp(log(2)*(1-s)))
Float.revert()
if isinstance(y, ComplexFloat) and s.imag == 0:
return +y.real
else:
return +y
def jsn(u, m):
"""
Computes of the Jacobi elliptic sn function in terms
of Jacobi theta functions.
`u` is any complex number, `m` must be in the unit disk
The sn-function is doubly periodic in the complex
plane with periods `4 K(m)` and `2 i K(1-m)`
(see :func:`ellipk`)::
>>> from mpmath import *
>>> mp.dps = 25
>>> print jsn(2, 0.25)
0.9628981775982774425751399
>>> print jsn(2+4*ellipk(0.25), 0.25)
0.9628981775982774425751399
>>> print chop(jsn(2+2*j*ellipk(1-0.25), 0.25))
0.9628981775982774425751399
"""
if abs(m) < eps:
return sin(u)
elif m == one:
return tanh(u)
else:
extra = 10
try:
mp.prec += extra
q = calculate_nome(sqrt(m))
v3 = jtheta(3, zero, q)
v2 = jtheta(2, zero, q) # mathworld says v4
arg1 = u / (v3*v3)
v1 = jtheta(1, arg1, q)
v4 = jtheta(4, arg1, q)
sn = (v3/v2)*(v1/v4)
finally:
mp.prec -= extra
return sn
def gamma(x):
"""
gamma(x) -- calculate the gamma function of a real or complex
number x.
x must not be a negative integer or 0
"""
Float.store()
Float._prec += 2
if isinstance(x, complex):
x = ComplexFloat(x)
elif not isinstance(x, (Float, ComplexFloat, Rational, int, long)):
x = Float(x)
if isinstance(x, (ComplexFloat, complex)):
re, im = x.real, x.imag
else:
re, im = x, 0
# For negative x (or positive x close to the pole at x = 0),
# we use the reflection formula
if re < 0.25:
if re == int(re) and im == 0:
raise ZeroDivisionError, "gamma function pole"
Float._prec += 3
p = pi_float()
g = p / (sin(p*x) * gamma(1-x))
else:
x -= 1
prec, a, c = _get_spouge_coefficients(Float.getprec()+7)
s = _spouge_sum(x, prec, a, c)
if not isinstance(x, (Float, ComplexFloat)):
x = Float(x)
# TODO: higher precision may be needed here when the precision
# and/or size of x are extremely large
Float._prec += 10
g = exp(log(x+a)*(x+Float(0.5))) * exp(-x-a) * s
Float.revert()
return +g
def _jacobi_theta2(z, q):
extra1 = 10
extra2 = 20
# the loops below break when the fixed precision quantities
# a and b go to zero;
# right shifting small negative numbers by wp one obtains -1, not zero,
# so the condition a**2 + b**2 > MIN is used to break the loops.
MIN = 2
if z == zero:
if isinstance(q, mpf):
wp = mp.prec + extra1
x = to_fixed(q._mpf_, wp)
x2 = (x*x) >> wp
a = b = x2
s = x2
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
s += a
s = (1 << (wp+1)) + (s << 1)
s = mpf(from_man_exp(s, -wp, mp.prec, 'n'))
else:
wp = mp.prec + extra1
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = x2re
aim = bim = x2im
sre = (1<<wp) + are
sim = aim
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
sre += are
sim += aim
sre = (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
else:
if isinstance(q, mpf) and isinstance(z, mpf):
wp = mp.prec + extra1
x = to_fixed(q._mpf_, wp)
x2 = (x*x) >> wp
a = b = x2
c1, s1 = cos_sin(z._mpf_, wp)
cn = c1 = to_fixed(c1, wp)
sn = s1 = to_fixed(s1, wp)
c2 = (c1*c1 - s1*s1) >> wp
s2 = (c1 * s1) >> (wp - 1)
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
s = c1 + ((a * cn) >> wp)
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
s += (a * cn) >> wp
s = (s << 1)
s = mpf(from_man_exp(s, -wp, mp.prec, 'n'))
s *= nthroot(q, 4)
return s
# case z real, q complex
elif isinstance(z, mpf):
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = x2re
aim = bim = x2im
c1, s1 = cos_sin(z._mpf_, wp)
cn = c1 = to_fixed(c1, wp)
sn = s1 = to_fixed(s1, wp)
c2 = (c1*c1 - s1*s1) >> wp
s2 = (c1 * s1) >> (wp - 1)
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
sre = c1 + ((are * cn) >> wp)
sim = ((aim * cn) >> wp)
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
sre += ((are * cn) >> wp)
sim += ((aim * cn) >> wp)
sre = (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
#case z complex, q real
elif isinstance(q, mpf):
wp = mp.prec + extra2
x = to_fixed(q._mpf_, wp)
x2 = (x*x) >> wp
a = b = x2
prec0 = mp.prec
mp.prec = wp
c1 = cos(z)
s1 = sin(z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
#c2 = (c1*c1 - s1*s1) >> wp
c2re = (c1re*c1re - c1im*c1im - s1re*s1re + s1im*s1im) >> wp
c2im = (c1re*c1im - s1re*s1im) >> (wp - 1)
#s2 = (c1 * s1) >> (wp - 1)
s2re = (c1re*s1re - c1im*s1im) >> (wp - 1)
s2im = (c1re*s1im + c1im*s1re) >> (wp - 1)
#cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
sre = c1re + ((a * cnre) >> wp)
sim = c1im + ((a * cnim) >> wp)
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
sre += ((a * cnre) >> wp)
sim += ((a * cnim) >> wp)
sre = (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
# case z and q complex
else:
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = x2re
aim = bim = x2im
prec0 = mp.prec
mp.prec = wp
# cos(z), siz(z) with z complex
c1 = cos(z)
s1 = sin(z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
c2re = (c1re*c1re - c1im*c1im - s1re*s1re + s1im*s1im) >> wp
c2im = (c1re*c1im - s1re*s1im) >> (wp - 1)
s2re = (c1re*s1re - c1im*s1im) >> (wp - 1)
s2im = (c1re*s1im + c1im*s1re) >> (wp - 1)
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
n = 1
termre = c1re
termim = c1im
sre = c1re + ((are * cnre - aim * cnim) >> wp)
sim = c1im + ((are * cnim + aim * cnre) >> wp)
n = 3
termre = ((are * cnre - aim * cnim) >> wp)
termim = ((are * cnim + aim * cnre) >> wp)
sre = c1re + ((are * cnre - aim * cnim) >> wp)
sim = c1im + ((are * cnim + aim * cnre) >> wp)
n = 5
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
#cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
termre = ((are * cnre - aim * cnim) >> wp)
termim = ((aim * cnre + are * cnim) >> wp)
sre += ((are * cnre - aim * cnim) >> wp)
sim += ((aim * cnre + are * cnim) >> wp)
n += 2
sre = (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
s *= nthroot(q, 4)
return s
def _djacobi_theta3(z, q, nd):
"""nd=1,2,3 order of the derivative with respect to z"""
MIN = 2
extra1 = 10
extra2 = 20
if isinstance(q, mpf) and isinstance(z, mpf):
s = MP_ZERO
wp = mp.prec + extra1
x = to_fixed(q._mpf_, wp)
a = b = x
x2 = (x*x) >> wp
c1, s1 = cos_sin(mpf_shift(z._mpf_, 1), wp)
c1 = to_fixed(c1, wp)
s1 = to_fixed(s1, wp)
cn = c1
sn = s1
if (nd&1):
s += (a * sn) >> wp
else:
s += (a * cn) >> wp
n = 2
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
if nd&1:
s += (a * sn * n**nd) >> wp
else:
s += (a * cn * n**nd) >> wp
n += 1
s = -(s << (nd+1))
s = mpf(from_man_exp(s, -wp, mp.prec, 'n'))
# case z real, q complex
elif isinstance(z, mpf):
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = xre
aim = bim = xim
c1, s1 = cos_sin(mpf_shift(z._mpf_, 1), wp)
c1 = to_fixed(c1, wp)
s1 = to_fixed(s1, wp)
cn = c1
sn = s1
if (nd&1):
sre = (are * sn) >> wp
sim = (aim * sn) >> wp
else:
sre = (are * cn) >> wp
sim = (aim * cn) >> wp
n = 2
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
if nd&1:
sre += (are * sn * n**nd) >> wp
sim += (aim * sn * n**nd) >> wp
else:
sre += (are * cn * n**nd) >> wp
sim += (aim * cn * n**nd) >> wp
n += 1
sre = -(sre << (nd+1))
sim = -(sim << (nd+1))
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
#case z complex, q real
elif isinstance(q, mpf):
wp = mp.prec + extra2
x = to_fixed(q._mpf_, wp)
a = b = x
x2 = (x*x) >> wp
prec0 = mp.prec
mp.prec = wp
c1 = cos(2*z)
s1 = sin(2*z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
if (nd&1):
sre = (a * snre) >> wp
sim = (a * snim) >> wp
else:
sre = (a * cnre) >> wp
sim = (a * cnim) >> wp
n = 2
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
t1 = (cnre*c1re - cnim*c1im - snre*s1re + snim*s1im) >> wp
t2 = (cnre*c1im + cnim*c1re - snre*s1im - snim*s1re) >> wp
t3 = (snre*c1re - snim*c1im + cnre*s1re - cnim*s1im) >> wp
t4 = (snre*c1im + snim*c1re + cnre*s1im + cnim*s1re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if (nd&1):
sre += (a * snre * n**nd) >> wp
sim += (a * snim * n**nd) >> wp
else:
sre += (a * cnre * n**nd) >> wp
sim += (a * cnim * n**nd) >> wp
n += 1
sre = -(sre << (nd+1))
sim = -(sim << (nd+1))
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
# case z and q complex
else:
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = xre
aim = bim = xim
prec0 = mp.prec
mp.prec = wp
# cos(2*z), sin(2*z) with z complex
c1 = cos(2*z)
s1 = sin(2*z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
if (nd&1):
sre = (are * snre - aim * snim) >> wp
sim = (aim * snre + are * snim) >> wp
else:
sre = (are * cnre - aim * cnim) >> wp
sim = (aim * cnre + are * cnim) >> wp
n = 2
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
t1 = (cnre*c1re - cnim*c1im - snre*s1re + snim*s1im) >> wp
t2 = (cnre*c1im + cnim*c1re - snre*s1im - snim*s1re) >> wp
t3 = (snre*c1re - snim*c1im + cnre*s1re - cnim*s1im) >> wp
t4 = (snre*c1im + snim*c1re + cnre*s1im + cnim*s1re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if(nd&1):
sre += ((are * snre - aim * snim) * n**nd) >> wp
sim += ((aim * snre + are * snim) * n**nd) >> wp
else:
sre += ((are * cnre - aim * cnim) * n**nd) >> wp
sim += ((aim * cnre + are * cnim) * n**nd) >> wp
n += 1
sre = -(sre << (nd+1))
sim = -(sim << (nd+1))
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
if (nd&1):
return (-1)**(nd//2) * s
else:
return (-1)**(1 + nd//2) * s
def _jacobi_theta3(z, q):
extra1 = 10
extra2 = 20
MIN = 2
if z == zero:
if isinstance(q, mpf):
wp = mp.prec + extra1
x = to_fixed(q._mpf_, wp)
s = x
a = b = x
x2 = (x*x) >> wp
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
s += a
s = (1 << wp) + (s << 1)
s = mpf(from_man_exp(s, -wp, mp.prec, 'n'))
return s
else:
wp = mp.prec + extra1
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
sre = are = bre = xre
sim = aim = bim = xim
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
sre += are
sim += aim
sre = (1 << wp) + (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
return s
else:
if isinstance(q, mpf) and isinstance(z, mpf):
s = MP_ZERO
wp = mp.prec + extra1
x = to_fixed(q._mpf_, wp)
a = b = x
x2 = (x*x) >> wp
c1, s1 = cos_sin(mpf_shift(z._mpf_, 1), wp)
c1 = to_fixed(c1, wp)
s1 = to_fixed(s1, wp)
cn = c1
sn = s1
s += (a * cn) >> wp
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
s += (a * cn) >> wp
s = (1 << wp) + (s << 1)
s = mpf(from_man_exp(s, -wp, mp.prec, 'n'))
return s
# case z real, q complex
elif isinstance(z, mpf):
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = xre
aim = bim = xim
c1, s1 = cos_sin(mpf_shift(z._mpf_, 1), wp)
c1 = to_fixed(c1, wp)
s1 = to_fixed(s1, wp)
cn = c1
sn = s1
sre = (are * cn) >> wp
sim = (aim * cn) >> wp
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
sre += (are * cn) >> wp
sim += (aim * cn) >> wp
sre = (1 << wp) + (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
return s
#case z complex, q real
elif isinstance(q, mpf):
wp = mp.prec + extra2
x = to_fixed(q._mpf_, wp)
a = b = x
x2 = (x*x) >> wp
prec0 = mp.prec
mp.prec = wp
c1 = cos(2*z)
s1 = sin(2*z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
sre = (a * cnre) >> wp
sim = (a * cnim) >> wp
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
t1 = (cnre*c1re - cnim*c1im - snre*s1re + snim*s1im) >> wp
t2 = (cnre*c1im + cnim*c1re - snre*s1im - snim*s1re) >> wp
t3 = (snre*c1re - snim*c1im + cnre*s1re - cnim*s1im) >> wp
t4 = (snre*c1im + snim*c1re + cnre*s1im + cnim*s1re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
sre += (a * cnre) >> wp
sim += (a * cnim) >> wp
sre = (1 << wp) + (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
return s
# case z and q complex
else:
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = xre
aim = bim = xim
prec0 = mp.prec
mp.prec = wp
# cos(2*z), sin(2*z) with z complex
c1 = cos(2*z)
s1 = sin(2*z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
sre = (are * cnre - aim * cnim) >> wp
sim = (aim * cnre + are * cnim) >> wp
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
t1 = (cnre*c1re - cnim*c1im - snre*s1re + snim*s1im) >> wp
t2 = (cnre*c1im + cnim*c1re - snre*s1im - snim*s1re) >> wp
t3 = (snre*c1re - snim*c1im + cnre*s1re - cnim*s1im) >> wp
t4 = (snre*c1im + snim*c1re + cnre*s1im + cnim*s1re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
sre += (are * cnre - aim * cnim) >> wp
sim += (aim * cnre + are * cnim) >> wp
sre = (1 << wp) + (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
return s
def _djacobi_theta2(z, q, nd):
MIN = 2
extra1 = 10
extra2 = 20
if isinstance(q, mpf) and isinstance(z, mpf):
wp = mp.prec + extra1
x = to_fixed(q._mpf_, wp)
x2 = (x*x) >> wp
a = b = x2
c1, s1 = cos_sin(z._mpf_, wp)
cn = c1 = to_fixed(c1, wp)
sn = s1 = to_fixed(s1, wp)
c2 = (c1*c1 - s1*s1) >> wp
s2 = (c1 * s1) >> (wp - 1)
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
if (nd&1):
s = s1 + ((a * sn * 3**nd) >> wp)
else:
s = c1 + ((a * cn * 3**nd) >> wp)
n = 2
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
if nd&1:
s += (a * sn * (2*n+1)**nd) >> wp
else:
s += (a * cn * (2*n+1)**nd) >> wp
n += 1
s = -(s << 1)
s = mpf(from_man_exp(s, -wp, mp.prec, 'n'))
# case z real, q complex
elif isinstance(z, mpf):
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = x2re
aim = bim = x2im
c1, s1 = cos_sin(z._mpf_, wp)
cn = c1 = to_fixed(c1, wp)
sn = s1 = to_fixed(s1, wp)
c2 = (c1*c1 - s1*s1) >> wp
s2 = (c1 * s1) >> (wp - 1)
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
if (nd&1):
sre = s1 + ((are * sn * 3**nd) >> wp)
sim = ((aim * sn * 3**nd) >> wp)
else:
sre = c1 + ((are * cn * 3**nd) >> wp)
sim = ((aim * cn * 3**nd) >> wp)
n = 5
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
if (nd&1):
sre += ((are * sn * n**nd) >> wp)
sim += ((aim * sn * n**nd) >> wp)
else:
sre += ((are * cn * n**nd) >> wp)
sim += ((aim * cn * n**nd) >> wp)
n += 2
sre = -(sre << 1)
sim = -(sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
#case z complex, q real
elif isinstance(q, mpf):
wp = mp.prec + extra2
x = to_fixed(q._mpf_, wp)
x2 = (x*x) >> wp
a = b = x2
prec0 = mp.prec
mp.prec = wp
c1 = cos(z)
s1 = sin(z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
#c2 = (c1*c1 - s1*s1) >> wp
c2re = (c1re*c1re - c1im*c1im - s1re*s1re + s1im*s1im) >> wp
c2im = (c1re*c1im - s1re*s1im) >> (wp - 1)
#s2 = (c1 * s1) >> (wp - 1)
s2re = (c1re*s1re - c1im*s1im) >> (wp - 1)
s2im = (c1re*s1im + c1im*s1re) >> (wp - 1)
#cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if (nd&1):
sre = s1re + ((a * snre * 3**nd) >> wp)
sim = s1im + ((a * snim * 3**nd) >> wp)
else:
sre = c1re + ((a * cnre * 3**nd) >> wp)
sim = c1im + ((a * cnim * 3**nd) >> wp)
n = 5
while abs(a) > MIN:
b = (b*x2) >> wp
a = (a*b) >> wp
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if (nd&1):
sre += ((a * snre * n**nd) >> wp)
sim += ((a * snim * n**nd) >> wp)
else:
sre += ((a * cnre * n**nd) >> wp)
sim += ((a * cnim * n**nd) >> wp)
n += 2
sre = -(sre << 1)
sim = -(sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
# case z and q complex
else:
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre*xre - xim*xim) >> wp
x2im = (xre*xim) >> (wp - 1)
are = bre = x2re
aim = bim = x2im
prec0 = mp.prec
mp.prec = wp
# cos(2*z), siz(2*z) with z complex
c1 = cos(z)
s1 = sin(z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
c2re = (c1re*c1re - c1im*c1im - s1re*s1re + s1im*s1im) >> wp
c2im = (c1re*c1im - s1re*s1im) >> (wp - 1)
s2re = (c1re*s1re - c1im*s1im) >> (wp - 1)
s2im = (c1re*s1im + c1im*s1re) >> (wp - 1)
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if (nd&1):
sre = s1re + (((are * snre - aim * snim) * 3**nd) >> wp)
sim = s1im + (((are * snim + aim * snre)* 3**nd) >> wp)
else:
sre = c1re + (((are * cnre - aim * cnim) * 3**nd) >> wp)
sim = c1im + (((are * cnim + aim * cnre)* 3**nd) >> wp)
n = 5
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
#cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
t1 = (cnre*c2re - cnim*c2im - snre*s2re + snim*s2im) >> wp
t2 = (cnre*c2im + cnim*c2re - snre*s2im - snim*s2re) >> wp
t3 = (snre*c2re - snim*c2im + cnre*s2re - cnim*s2im) >> wp
t4 = (snre*c2im + snim*c2re + cnre*s2im + cnim*s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if (nd&1):
sre += (((are * snre - aim * snim) * n**nd) >> wp)
sim += (((aim * snre + are * snim) * n**nd) >> wp)
else:
sre += (((are * cnre - aim * cnim) * n**nd) >> wp)
sim += (((aim * cnre + are * cnim) * n**nd) >> wp)
n += 2
sre = -(sre << 1)
sim = -(sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
s *= nthroot(q, 4)
if (nd&1):
return (-1)**(nd//2) * s
else:
return (-1)**(1 + nd//2) * s
def jacobi_elliptic_sn(u, m, verbose=False):
"""
Implements the jacobi elliptic sn function, using the expansion in
terms of q, from Abramowitz 16.23.1.
u is any complex number, m is the parameter
Alternative implementation:
Expansion in terms of jacobi theta functions appears to fail with
round off error, despite I also think that the expansion in
terms of q is much faster than four expansions in terms of q.
**********************************
Previous implementation kept here:
if not isinstance(u, mpf):
raise TypeError
if not isinstance(m, mpf):
raise TypeError
zero = mpf('0')
if u == zero and m == 0:
return zero
else:
q = calculate_nome(sqrt(m))
v3 = jacobi_theta_3(zero, q)
v2 = jacobi_theta_2(zero, q) # mathworld says v4
arg1 = u / (v3*v3)
v1 = jacobi_theta_1(arg1, q)
v4 = jacobi_theta_4(arg1, q)
sn = (v3/v2)*(v1/v4)
return sn
**********************************
"""
u = convert_lossless(u)
m = convert_lossless(m)
if verbose:
print >> sys.stderr, '\nelliptic.jacobi_elliptic_sn'
print >> sys.stderr, '\tu: %1.12f' % u
print >> sys.stderr, '\tm: %1.12f' % m
zero = mpf('0')
onehalf = mpf('0.5')
one = mpf('1')
two = mpf('2')
if m == zero: # sn collapes to sin(u)
if verbose:
print >> sys.stderr, '\nsn: special case, m == 0'
return sin(u)
elif m == one: # sn collapses to tanh(u)
if verbose:
print >> sys.stderr, '\nsn: special case, m == 1'
return tanh(u)
else:
k = sqrt(m) # convert m to k
q = calculate_nome(k)
v = (pi * u) / (two*ellipk(k**2))
if v == pi or v == zero: # sin factor always zero
return zero
sum = zero
term = zero # series starts at zero
while True:
if verbose:
print >> sys.stderr, 'elliptic.jacobi_elliptic_sn: calculating'
factor1 = (q**(term + onehalf)) / (one - q**(two*term + one))
factor2 = sin((two*term + one)*v)
term_n = factor1*factor2
sum = sum + term_n
if verbose:
print >> sys.stderr, '\tTerm: %d' % term,
print >> sys.stderr, '\tterm_n: %e' % term_n,
print >> sys.stderr, '\tsum: %e' % sum
if not factor2 == zero:
#if log(term_n, '10') < -1*mpf.dps:
if abs(term_n) < eps:
break
term = term + one
answer = (two*pi) / (sqrt(m) * ellipk(k**2)) * sum
return answer
def jacobi_theta_1(z, m, verbose=False):
"""
Implements the jacobi theta function 1, using the series expansion
found in Abramowitz & Stegun [1].
z is any complex number, but only reals here?
m is the parameter, which must be converted to the nome
"""
if verbose:
print >> sys.stderr, 'elliptic.jacobi_theta_1'
m = convert_lossless(m)
z = convert_lossless(z)
k = sqrt(m)
q = calculate_nome(k)
if verbose:
print >> sys.stderr, '\tk: %f ' % k
print >> sys.stderr, '\tq: %f ' % q
if abs(q) >= mpf('1'):
raise ValueError
sum = 0
term = 0 # series starts at 0
minusone = mpf('-1')
zero = mpf('0')
one = mpf('1')
if z == zero:
if verbose:
print >> sys.stderr, 'elliptic.jacobi_theta_1: z == 0, return 0'
return zero
elif q == zero:
if verbose:
print >> sys.stderr, 'elliptic.jacobi_theta_1: q == 0, return 0'
return zero
else:
if verbose:
print >> sys.stderr, 'ellipticl.jacobi_theta_1: calculating'
while True:
if (term % 2) == 0:
factor0 = one
else:
factor0 = minusone
factor1 = q**(term*(term +1))
factor2 = sin((2*term + 1)*z)
term_n = factor0 * factor1 * factor2
sum = sum + term_n
if verbose:
print >> sys.stderr, '\tTerm: %d' % term,
print >> sys.stderr, '\tterm_n: %e' % term_n,
print >> sys.stderr, '\tsum: %e' % sum
if factor1 == zero: # all further terms will be zero
break
if factor2 != zero: # check precision iff cos != 0
#if log(term_n, '10') < -1*mpf.dps:
if abs(term_n) < eps:
break
term = term + 1
return (2*q**(0.25))*sum
# can't get here
print >> sys.stderr, 'elliptic.jacobi_theta_1 in impossible state'
sys.exit(2)
def sine_dist(self, t):
return self.v0 + self.k * functions.sin(self.phi + self.w * t)
def _jacobi_theta2(z, q):
extra1 = 10
extra2 = 20
# the loops below break when the fixed precision quantities
# a and b go to zero;
# right shifting small negative numbers by wp one obtains -1, not zero,
# so the condition a**2 + b**2 > MIN is used to break the loops.
MIN = 2
if z == zero:
if isinstance(q, mpf):
wp = mp.prec + extra1
x = to_fixed(q._mpf_, wp)
x2 = (x * x) >> wp
a = b = x2
s = x2
while abs(a) > MIN:
b = (b * x2) >> wp
a = (a * b) >> wp
s += a
s = (1 << (wp + 1)) + (s << 1)
s = mpf(from_man_exp(s, -wp, mp.prec, 'n'))
else:
wp = mp.prec + extra1
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre * xre - xim * xim) >> wp
x2im = (xre * xim) >> (wp - 1)
are = bre = x2re
aim = bim = x2im
sre = (1 << wp) + are
sim = aim
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
sre += are
sim += aim
sre = (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
else:
if isinstance(q, mpf) and isinstance(z, mpf):
wp = mp.prec + extra1
x = to_fixed(q._mpf_, wp)
x2 = (x * x) >> wp
a = b = x2
c1, s1 = cos_sin(z._mpf_, wp)
cn = c1 = to_fixed(c1, wp)
sn = s1 = to_fixed(s1, wp)
c2 = (c1 * c1 - s1 * s1) >> wp
s2 = (c1 * s1) >> (wp - 1)
cn, sn = (cn * c2 - sn * s2) >> wp, (sn * c2 + cn * s2) >> wp
s = c1 + ((a * cn) >> wp)
while abs(a) > MIN:
b = (b * x2) >> wp
a = (a * b) >> wp
cn, sn = (cn * c2 - sn * s2) >> wp, (sn * c2 + cn * s2) >> wp
s += (a * cn) >> wp
s = (s << 1)
s = mpf(from_man_exp(s, -wp, mp.prec, 'n'))
s *= nthroot(q, 4)
return s
# case z real, q complex
elif isinstance(z, mpf):
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre * xre - xim * xim) >> wp
x2im = (xre * xim) >> (wp - 1)
are = bre = x2re
aim = bim = x2im
c1, s1 = cos_sin(z._mpf_, wp)
cn = c1 = to_fixed(c1, wp)
sn = s1 = to_fixed(s1, wp)
c2 = (c1 * c1 - s1 * s1) >> wp
s2 = (c1 * s1) >> (wp - 1)
cn, sn = (cn * c2 - sn * s2) >> wp, (sn * c2 + cn * s2) >> wp
sre = c1 + ((are * cn) >> wp)
sim = ((aim * cn) >> wp)
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
cn, sn = (cn * c2 - sn * s2) >> wp, (sn * c2 + cn * s2) >> wp
sre += ((are * cn) >> wp)
sim += ((aim * cn) >> wp)
sre = (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
#case z complex, q real
elif isinstance(q, mpf):
wp = mp.prec + extra2
x = to_fixed(q._mpf_, wp)
x2 = (x * x) >> wp
a = b = x2
prec0 = mp.prec
mp.prec = wp
c1 = cos(z)
s1 = sin(z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
#c2 = (c1*c1 - s1*s1) >> wp
c2re = (c1re * c1re - c1im * c1im - s1re * s1re +
s1im * s1im) >> wp
c2im = (c1re * c1im - s1re * s1im) >> (wp - 1)
#s2 = (c1 * s1) >> (wp - 1)
s2re = (c1re * s1re - c1im * s1im) >> (wp - 1)
s2im = (c1re * s1im + c1im * s1re) >> (wp - 1)
#cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
t1 = (cnre * c2re - cnim * c2im - snre * s2re + snim * s2im) >> wp
t2 = (cnre * c2im + cnim * c2re - snre * s2im - snim * s2re) >> wp
t3 = (snre * c2re - snim * c2im + cnre * s2re - cnim * s2im) >> wp
t4 = (snre * c2im + snim * c2re + cnre * s2im + cnim * s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
sre = c1re + ((a * cnre) >> wp)
sim = c1im + ((a * cnim) >> wp)
while abs(a) > MIN:
b = (b * x2) >> wp
a = (a * b) >> wp
t1 = (cnre * c2re - cnim * c2im - snre * s2re +
snim * s2im) >> wp
t2 = (cnre * c2im + cnim * c2re - snre * s2im -
snim * s2re) >> wp
t3 = (snre * c2re - snim * c2im + cnre * s2re -
cnim * s2im) >> wp
t4 = (snre * c2im + snim * c2re + cnre * s2im +
cnim * s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
sre += ((a * cnre) >> wp)
sim += ((a * cnim) >> wp)
sre = (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
# case z and q complex
else:
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre * xre - xim * xim) >> wp
x2im = (xre * xim) >> (wp - 1)
are = bre = x2re
aim = bim = x2im
prec0 = mp.prec
mp.prec = wp
# cos(z), siz(z) with z complex
c1 = cos(z)
s1 = sin(z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
c2re = (c1re * c1re - c1im * c1im - s1re * s1re +
s1im * s1im) >> wp
c2im = (c1re * c1im - s1re * s1im) >> (wp - 1)
s2re = (c1re * s1re - c1im * s1im) >> (wp - 1)
s2im = (c1re * s1im + c1im * s1re) >> (wp - 1)
t1 = (cnre * c2re - cnim * c2im - snre * s2re + snim * s2im) >> wp
t2 = (cnre * c2im + cnim * c2re - snre * s2im - snim * s2re) >> wp
t3 = (snre * c2re - snim * c2im + cnre * s2re - cnim * s2im) >> wp
t4 = (snre * c2im + snim * c2re + cnre * s2im + cnim * s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
n = 1
termre = c1re
termim = c1im
sre = c1re + ((are * cnre - aim * cnim) >> wp)
sim = c1im + ((are * cnim + aim * cnre) >> wp)
n = 3
termre = ((are * cnre - aim * cnim) >> wp)
termim = ((are * cnim + aim * cnre) >> wp)
sre = c1re + ((are * cnre - aim * cnim) >> wp)
sim = c1im + ((are * cnim + aim * cnre) >> wp)
n = 5
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
#cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
t1 = (cnre * c2re - cnim * c2im - snre * s2re +
snim * s2im) >> wp
t2 = (cnre * c2im + cnim * c2re - snre * s2im -
snim * s2re) >> wp
t3 = (snre * c2re - snim * c2im + cnre * s2re -
cnim * s2im) >> wp
t4 = (snre * c2im + snim * c2re + cnre * s2im +
cnim * s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
termre = ((are * cnre - aim * cnim) >> wp)
termim = ((aim * cnre + are * cnim) >> wp)
sre += ((are * cnre - aim * cnim) >> wp)
sim += ((aim * cnre + are * cnim) >> wp)
n += 2
sre = (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
s *= nthroot(q, 4)
return s
def _djacobi_theta3(z, q, nd):
"""nd=1,2,3 order of the derivative with respect to z"""
MIN = 2
extra1 = 10
extra2 = 20
if isinstance(q, mpf) and isinstance(z, mpf):
s = MP_ZERO
wp = mp.prec + extra1
x = to_fixed(q._mpf_, wp)
a = b = x
x2 = (x * x) >> wp
c1, s1 = cos_sin(mpf_shift(z._mpf_, 1), wp)
c1 = to_fixed(c1, wp)
s1 = to_fixed(s1, wp)
cn = c1
sn = s1
if (nd & 1):
s += (a * sn) >> wp
else:
s += (a * cn) >> wp
n = 2
while abs(a) > MIN:
b = (b * x2) >> wp
a = (a * b) >> wp
cn, sn = (cn * c1 - sn * s1) >> wp, (sn * c1 + cn * s1) >> wp
if nd & 1:
s += (a * sn * n**nd) >> wp
else:
s += (a * cn * n**nd) >> wp
n += 1
s = -(s << (nd + 1))
s = mpf(from_man_exp(s, -wp, mp.prec, 'n'))
# case z real, q complex
elif isinstance(z, mpf):
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre * xre - xim * xim) >> wp
x2im = (xre * xim) >> (wp - 1)
are = bre = xre
aim = bim = xim
c1, s1 = cos_sin(mpf_shift(z._mpf_, 1), wp)
c1 = to_fixed(c1, wp)
s1 = to_fixed(s1, wp)
cn = c1
sn = s1
if (nd & 1):
sre = (are * sn) >> wp
sim = (aim * sn) >> wp
else:
sre = (are * cn) >> wp
sim = (aim * cn) >> wp
n = 2
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
cn, sn = (cn * c1 - sn * s1) >> wp, (sn * c1 + cn * s1) >> wp
if nd & 1:
sre += (are * sn * n**nd) >> wp
sim += (aim * sn * n**nd) >> wp
else:
sre += (are * cn * n**nd) >> wp
sim += (aim * cn * n**nd) >> wp
n += 1
sre = -(sre << (nd + 1))
sim = -(sim << (nd + 1))
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
#case z complex, q real
elif isinstance(q, mpf):
wp = mp.prec + extra2
x = to_fixed(q._mpf_, wp)
a = b = x
x2 = (x * x) >> wp
prec0 = mp.prec
mp.prec = wp
c1 = cos(2 * z)
s1 = sin(2 * z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
if (nd & 1):
sre = (a * snre) >> wp
sim = (a * snim) >> wp
else:
sre = (a * cnre) >> wp
sim = (a * cnim) >> wp
n = 2
while abs(a) > MIN:
b = (b * x2) >> wp
a = (a * b) >> wp
t1 = (cnre * c1re - cnim * c1im - snre * s1re + snim * s1im) >> wp
t2 = (cnre * c1im + cnim * c1re - snre * s1im - snim * s1re) >> wp
t3 = (snre * c1re - snim * c1im + cnre * s1re - cnim * s1im) >> wp
t4 = (snre * c1im + snim * c1re + cnre * s1im + cnim * s1re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if (nd & 1):
sre += (a * snre * n**nd) >> wp
sim += (a * snim * n**nd) >> wp
else:
sre += (a * cnre * n**nd) >> wp
sim += (a * cnim * n**nd) >> wp
n += 1
sre = -(sre << (nd + 1))
sim = -(sim << (nd + 1))
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
# case z and q complex
else:
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre * xre - xim * xim) >> wp
x2im = (xre * xim) >> (wp - 1)
are = bre = xre
aim = bim = xim
prec0 = mp.prec
mp.prec = wp
# cos(2*z), sin(2*z) with z complex
c1 = cos(2 * z)
s1 = sin(2 * z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
if (nd & 1):
sre = (are * snre - aim * snim) >> wp
sim = (aim * snre + are * snim) >> wp
else:
sre = (are * cnre - aim * cnim) >> wp
sim = (aim * cnre + are * cnim) >> wp
n = 2
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
t1 = (cnre * c1re - cnim * c1im - snre * s1re + snim * s1im) >> wp
t2 = (cnre * c1im + cnim * c1re - snre * s1im - snim * s1re) >> wp
t3 = (snre * c1re - snim * c1im + cnre * s1re - cnim * s1im) >> wp
t4 = (snre * c1im + snim * c1re + cnre * s1im + cnim * s1re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if (nd & 1):
sre += ((are * snre - aim * snim) * n**nd) >> wp
sim += ((aim * snre + are * snim) * n**nd) >> wp
else:
sre += ((are * cnre - aim * cnim) * n**nd) >> wp
sim += ((aim * cnre + are * cnim) * n**nd) >> wp
n += 1
sre = -(sre << (nd + 1))
sim = -(sim << (nd + 1))
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
if (nd & 1):
return (-1)**(nd // 2) * s
else:
return (-1)**(1 + nd // 2) * s
def _jacobi_theta3(z, q):
extra1 = 10
extra2 = 20
MIN = 2
if z == zero:
if isinstance(q, mpf):
wp = mp.prec + extra1
x = to_fixed(q._mpf_, wp)
s = x
a = b = x
x2 = (x * x) >> wp
while abs(a) > MIN:
b = (b * x2) >> wp
a = (a * b) >> wp
s += a
s = (1 << wp) + (s << 1)
s = mpf(from_man_exp(s, -wp, mp.prec, 'n'))
return s
else:
wp = mp.prec + extra1
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre * xre - xim * xim) >> wp
x2im = (xre * xim) >> (wp - 1)
sre = are = bre = xre
sim = aim = bim = xim
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
sre += are
sim += aim
sre = (1 << wp) + (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
return s
else:
if isinstance(q, mpf) and isinstance(z, mpf):
s = MP_ZERO
wp = mp.prec + extra1
x = to_fixed(q._mpf_, wp)
a = b = x
x2 = (x * x) >> wp
c1, s1 = cos_sin(mpf_shift(z._mpf_, 1), wp)
c1 = to_fixed(c1, wp)
s1 = to_fixed(s1, wp)
cn = c1
sn = s1
s += (a * cn) >> wp
while abs(a) > MIN:
b = (b * x2) >> wp
a = (a * b) >> wp
cn, sn = (cn * c1 - sn * s1) >> wp, (sn * c1 + cn * s1) >> wp
s += (a * cn) >> wp
s = (1 << wp) + (s << 1)
s = mpf(from_man_exp(s, -wp, mp.prec, 'n'))
return s
# case z real, q complex
elif isinstance(z, mpf):
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre * xre - xim * xim) >> wp
x2im = (xre * xim) >> (wp - 1)
are = bre = xre
aim = bim = xim
c1, s1 = cos_sin(mpf_shift(z._mpf_, 1), wp)
c1 = to_fixed(c1, wp)
s1 = to_fixed(s1, wp)
cn = c1
sn = s1
sre = (are * cn) >> wp
sim = (aim * cn) >> wp
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
cn, sn = (cn * c1 - sn * s1) >> wp, (sn * c1 + cn * s1) >> wp
sre += (are * cn) >> wp
sim += (aim * cn) >> wp
sre = (1 << wp) + (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
return s
#case z complex, q real
elif isinstance(q, mpf):
wp = mp.prec + extra2
x = to_fixed(q._mpf_, wp)
a = b = x
x2 = (x * x) >> wp
prec0 = mp.prec
mp.prec = wp
c1 = cos(2 * z)
s1 = sin(2 * z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
sre = (a * cnre) >> wp
sim = (a * cnim) >> wp
while abs(a) > MIN:
b = (b * x2) >> wp
a = (a * b) >> wp
t1 = (cnre * c1re - cnim * c1im - snre * s1re +
snim * s1im) >> wp
t2 = (cnre * c1im + cnim * c1re - snre * s1im -
snim * s1re) >> wp
t3 = (snre * c1re - snim * c1im + cnre * s1re -
cnim * s1im) >> wp
t4 = (snre * c1im + snim * c1re + cnre * s1im +
cnim * s1re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
sre += (a * cnre) >> wp
sim += (a * cnim) >> wp
sre = (1 << wp) + (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
return s
# case z and q complex
else:
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre * xre - xim * xim) >> wp
x2im = (xre * xim) >> (wp - 1)
are = bre = xre
aim = bim = xim
prec0 = mp.prec
mp.prec = wp
# cos(2*z), sin(2*z) with z complex
c1 = cos(2 * z)
s1 = sin(2 * z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
sre = (are * cnre - aim * cnim) >> wp
sim = (aim * cnre + are * cnim) >> wp
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
t1 = (cnre * c1re - cnim * c1im - snre * s1re +
snim * s1im) >> wp
t2 = (cnre * c1im + cnim * c1re - snre * s1im -
snim * s1re) >> wp
t3 = (snre * c1re - snim * c1im + cnre * s1re -
cnim * s1im) >> wp
t4 = (snre * c1im + snim * c1re + cnre * s1im +
cnim * s1re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
sre += (are * cnre - aim * cnim) >> wp
sim += (aim * cnre + are * cnim) >> wp
sre = (1 << wp) + (sre << 1)
sim = (sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
return s
def _djacobi_theta2(z, q, nd):
MIN = 2
extra1 = 10
extra2 = 20
if isinstance(q, mpf) and isinstance(z, mpf):
wp = mp.prec + extra1
x = to_fixed(q._mpf_, wp)
x2 = (x * x) >> wp
a = b = x2
c1, s1 = cos_sin(z._mpf_, wp)
cn = c1 = to_fixed(c1, wp)
sn = s1 = to_fixed(s1, wp)
c2 = (c1 * c1 - s1 * s1) >> wp
s2 = (c1 * s1) >> (wp - 1)
cn, sn = (cn * c2 - sn * s2) >> wp, (sn * c2 + cn * s2) >> wp
if (nd & 1):
s = s1 + ((a * sn * 3**nd) >> wp)
else:
s = c1 + ((a * cn * 3**nd) >> wp)
n = 2
while abs(a) > MIN:
b = (b * x2) >> wp
a = (a * b) >> wp
cn, sn = (cn * c2 - sn * s2) >> wp, (sn * c2 + cn * s2) >> wp
if nd & 1:
s += (a * sn * (2 * n + 1)**nd) >> wp
else:
s += (a * cn * (2 * n + 1)**nd) >> wp
n += 1
s = -(s << 1)
s = mpf(from_man_exp(s, -wp, mp.prec, 'n'))
# case z real, q complex
elif isinstance(z, mpf):
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre * xre - xim * xim) >> wp
x2im = (xre * xim) >> (wp - 1)
are = bre = x2re
aim = bim = x2im
c1, s1 = cos_sin(z._mpf_, wp)
cn = c1 = to_fixed(c1, wp)
sn = s1 = to_fixed(s1, wp)
c2 = (c1 * c1 - s1 * s1) >> wp
s2 = (c1 * s1) >> (wp - 1)
cn, sn = (cn * c2 - sn * s2) >> wp, (sn * c2 + cn * s2) >> wp
if (nd & 1):
sre = s1 + ((are * sn * 3**nd) >> wp)
sim = ((aim * sn * 3**nd) >> wp)
else:
sre = c1 + ((are * cn * 3**nd) >> wp)
sim = ((aim * cn * 3**nd) >> wp)
n = 5
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
cn, sn = (cn * c2 - sn * s2) >> wp, (sn * c2 + cn * s2) >> wp
if (nd & 1):
sre += ((are * sn * n**nd) >> wp)
sim += ((aim * sn * n**nd) >> wp)
else:
sre += ((are * cn * n**nd) >> wp)
sim += ((aim * cn * n**nd) >> wp)
n += 2
sre = -(sre << 1)
sim = -(sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
#case z complex, q real
elif isinstance(q, mpf):
wp = mp.prec + extra2
x = to_fixed(q._mpf_, wp)
x2 = (x * x) >> wp
a = b = x2
prec0 = mp.prec
mp.prec = wp
c1 = cos(z)
s1 = sin(z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
#c2 = (c1*c1 - s1*s1) >> wp
c2re = (c1re * c1re - c1im * c1im - s1re * s1re + s1im * s1im) >> wp
c2im = (c1re * c1im - s1re * s1im) >> (wp - 1)
#s2 = (c1 * s1) >> (wp - 1)
s2re = (c1re * s1re - c1im * s1im) >> (wp - 1)
s2im = (c1re * s1im + c1im * s1re) >> (wp - 1)
#cn, sn = (cn*c2 - sn*s2) >> wp, (sn*c2 + cn*s2) >> wp
t1 = (cnre * c2re - cnim * c2im - snre * s2re + snim * s2im) >> wp
t2 = (cnre * c2im + cnim * c2re - snre * s2im - snim * s2re) >> wp
t3 = (snre * c2re - snim * c2im + cnre * s2re - cnim * s2im) >> wp
t4 = (snre * c2im + snim * c2re + cnre * s2im + cnim * s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if (nd & 1):
sre = s1re + ((a * snre * 3**nd) >> wp)
sim = s1im + ((a * snim * 3**nd) >> wp)
else:
sre = c1re + ((a * cnre * 3**nd) >> wp)
sim = c1im + ((a * cnim * 3**nd) >> wp)
n = 5
while abs(a) > MIN:
b = (b * x2) >> wp
a = (a * b) >> wp
t1 = (cnre * c2re - cnim * c2im - snre * s2re + snim * s2im) >> wp
t2 = (cnre * c2im + cnim * c2re - snre * s2im - snim * s2re) >> wp
t3 = (snre * c2re - snim * c2im + cnre * s2re - cnim * s2im) >> wp
t4 = (snre * c2im + snim * c2re + cnre * s2im + cnim * s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if (nd & 1):
sre += ((a * snre * n**nd) >> wp)
sim += ((a * snim * n**nd) >> wp)
else:
sre += ((a * cnre * n**nd) >> wp)
sim += ((a * cnim * n**nd) >> wp)
n += 2
sre = -(sre << 1)
sim = -(sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
# case z and q complex
else:
wp = mp.prec + extra2
xre, xim = q._mpc_
xre = to_fixed(xre, wp)
xim = to_fixed(xim, wp)
x2re = (xre * xre - xim * xim) >> wp
x2im = (xre * xim) >> (wp - 1)
are = bre = x2re
aim = bim = x2im
prec0 = mp.prec
mp.prec = wp
# cos(2*z), siz(2*z) with z complex
c1 = cos(z)
s1 = sin(z)
mp.prec = prec0
cnre = c1re = to_fixed(c1.real._mpf_, wp)
cnim = c1im = to_fixed(c1.imag._mpf_, wp)
snre = s1re = to_fixed(s1.real._mpf_, wp)
snim = s1im = to_fixed(s1.imag._mpf_, wp)
c2re = (c1re * c1re - c1im * c1im - s1re * s1re + s1im * s1im) >> wp
c2im = (c1re * c1im - s1re * s1im) >> (wp - 1)
s2re = (c1re * s1re - c1im * s1im) >> (wp - 1)
s2im = (c1re * s1im + c1im * s1re) >> (wp - 1)
t1 = (cnre * c2re - cnim * c2im - snre * s2re + snim * s2im) >> wp
t2 = (cnre * c2im + cnim * c2re - snre * s2im - snim * s2re) >> wp
t3 = (snre * c2re - snim * c2im + cnre * s2re - cnim * s2im) >> wp
t4 = (snre * c2im + snim * c2re + cnre * s2im + cnim * s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if (nd & 1):
sre = s1re + (((are * snre - aim * snim) * 3**nd) >> wp)
sim = s1im + (((are * snim + aim * snre) * 3**nd) >> wp)
else:
sre = c1re + (((are * cnre - aim * cnim) * 3**nd) >> wp)
sim = c1im + (((are * cnim + aim * cnre) * 3**nd) >> wp)
n = 5
while are**2 + aim**2 > MIN:
bre, bim = (bre * x2re - bim * x2im) >> wp, \
(bre * x2im + bim * x2re) >> wp
are, aim = (are * bre - aim * bim) >> wp, \
(are * bim + aim * bre) >> wp
#cn, sn = (cn*c1 - sn*s1) >> wp, (sn*c1 + cn*s1) >> wp
t1 = (cnre * c2re - cnim * c2im - snre * s2re + snim * s2im) >> wp
t2 = (cnre * c2im + cnim * c2re - snre * s2im - snim * s2re) >> wp
t3 = (snre * c2re - snim * c2im + cnre * s2re - cnim * s2im) >> wp
t4 = (snre * c2im + snim * c2re + cnre * s2im + cnim * s2re) >> wp
cnre = t1
cnim = t2
snre = t3
snim = t4
if (nd & 1):
sre += (((are * snre - aim * snim) * n**nd) >> wp)
sim += (((aim * snre + are * snim) * n**nd) >> wp)
else:
sre += (((are * cnre - aim * cnim) * n**nd) >> wp)
sim += (((aim * cnre + are * cnim) * n**nd) >> wp)
n += 2
sre = -(sre << 1)
sim = -(sim << 1)
sre = from_man_exp(sre, -wp, mp.prec, 'n')
sim = from_man_exp(sim, -wp, mp.prec, 'n')
s = mpc(sre, sim)
s *= nthroot(q, 4)
if (nd & 1):
return (-1)**(nd // 2) * s
else:
return (-1)**(1 + nd // 2) * s
def sin(self):
return F.sin(self)