def __compute_periods(self):
# A+S 18.9.
from mpmath import sqrt, ellipk, mpc, pi, mpf
Delta = self.Delta
e1, e2, e3 = self.__roots
if Delta > 0:
m = (e2 - e3) / (e1 - e3)
Km = ellipk(m)
Kpm = ellipk(1 - m)
om = Km / sqrt(e1 - e3)
omp = mpc(0,1) * om * Kpm / Km
elif Delta < 0:
# NOTE: the expression in the sqrt has to be real and positive, as e1 and e3 are
# complex conjugate and e2 is real.
H2 = (sqrt((e2 - e3) * (e2 - e1))).real
assert(H2 > 0)
m = mpf(1) / mpf(2) - 3 * e2 / (4 * H2)
Km = ellipk(m)
Kpm = ellipk(1 - m)
om2 = Km / sqrt(H2)
om2p = mpc(0,1) * Kpm * om2 / Km
om = (om2 - om2p) / 2
omp = (om2 + om2p) / 2
else:
g2, g3 = self.__invariants
if g2 == 0 and g3 == 0:
om = mpf('+inf')
omp = mpc(0,'+inf')
else:
# NOTE: here there is no need for the dichotomy on the sign of g3 because
# we are already working in a regime in which g3 >= 0 by definition.
c = e1 / 2
om = 1 / sqrt(12 * c) * pi()
omp = mpc(0,'+inf')
return 2 * om, 2 * omp
def __compute_periods(self):
# A+S 18.9.
from mpmath import sqrt, ellipk, mpc, pi, mpf
Delta = self.Delta
e1, e2, e3 = self.__roots
if Delta > 0:
m = (e2 - e3) / (e1 - e3)
Km = ellipk(m)
Kpm = ellipk(1 - m)
om = Km / sqrt(e1 - e3)
omp = mpc(0, 1) * om * Kpm / Km
elif Delta < 0:
# NOTE: the expression in the sqrt has to be real and positive, as e1 and e3 are
# complex conjugate and e2 is real.
H2 = (sqrt((e2 - e3) * (e2 - e1))).real
assert (H2 > 0)
m = mpf(1) / mpf(2) - 3 * e2 / (4 * H2)
Km = ellipk(m)
Kpm = ellipk(1 - m)
om2 = Km / sqrt(H2)
om2p = mpc(0, 1) * Kpm * om2 / Km
om = (om2 - om2p) / 2
omp = (om2 + om2p) / 2
else:
g2, g3 = self.__invariants
if g2 == 0 and g3 == 0:
om = mpf('+inf')
omp = mpc(0, '+inf')
else:
# NOTE: here there is no need for the dichotomy on the sign of g3 because
# we are already working in a regime in which g3 >= 0 by definition.
c = e1 / 2
om = 1 / sqrt(12 * c) * pi()
omp = mpc(0, '+inf')
return 2 * om, 2 * omp
def test_value3( self):
k = np.sqrt( 0.50)
m = mp.mfrom( k = k)
K = mp.ellipk( m)
Kp = mp.ellipk( 1 - m)
self.assertAlmostEqual( 0.50, m)
self.assertAlmostEqual( mp.mpf( 1.854074677301), K)
self.assertAlmostEqual( mp.mpf( 1.854074677301), Kp)
def test_value4( self):
k = np.sqrt( 0.90)
m = mp.mfrom( k = k)
K = mp.ellipk( m)
Kp = mp.ellipk( 1 - m)
self.assertAlmostEqual( 0.90, m)
self.assertAlmostEqual( mp.mpf( 2.578092113348), K)
self.assertAlmostEqual( mp.mpf( 1.612441348720), Kp)
def test_value5( self):
k = np.sqrt( 0.99)
m = mp.mfrom( k = k)
K = mp.ellipk( m)
Kp = mp.ellipk( 1 - m)
self.assertEqual( 0.99, m)
self.assertAlmostEqual( mp.mpf( 3.695637362989), K)
self.assertAlmostEqual( mp.mpf( 1.574745561317), Kp)
def test_value0( self):
k = np.sqrt( 0.00)
m = mp.mfrom( k = k)
K = mp.ellipk( m)
Kp = mp.ellipk( 1 - m)
self.assertAlmostEqual( 0.00, m)
self.assertAlmostEqual( mp.mpf( 1.570796326794), K)
self.assertEqual( mp.mpf( '+inf'), Kp)
def DOS(gamfactor, E):
kred = redE(gamfactor, E)
if abs(E) < 1 - gamfactor:
return 1 / sqrt(gamfactor) * kred * (mpmath.ellipk(kred +
10**(-3) * 1j)).real
elif abs(E) < 1 + gamfactor:
return sqrt(gamfactor) * (mpmath.ellipk(kred + 10**(-3) * 1j)).real
else:
return 0
def band_stop_obj(wp, ind, passb, stopb, gpass, gstop, type):
"""
Band Stop Objective Function for order minimization.
Returns the non-integer order for an analog band stop filter.
Parameters
----------
wp : scalar
Edge of passband `passb`.
ind : int, {0, 1}
Index specifying which `passb` edge to vary (0 or 1).
passb : ndarray
Two element sequence of fixed passband edges.
stopb : ndarray
Two element sequence of fixed stopband edges.
gstop : float
Amount of attenuation in stopband in dB.
gpass : float
Amount of ripple in the passband in dB.
type : {'butter', 'cheby', 'ellip'}
Type of filter.
Returns
-------
n : scalar
Filter order (possibly non-integer).
"""
passbC = passb.copy()
passbC[ind] = wp
nat = (stopb * (passbC[0] - passbC[1]) /
(stopb**2 - passbC[0] * passbC[1]))
nat = min(abs(nat))
if type == 'butter':
GSTOP = 10**(0.1 * abs(gstop))
GPASS = 10**(0.1 * abs(gpass))
n = (log10((GSTOP - 1.0) / (GPASS - 1.0)) / (2 * log10(nat)))
elif type == 'cheby':
GSTOP = 10**(0.1 * abs(gstop))
GPASS = 10**(0.1 * abs(gpass))
n = arccosh(sqrt((GSTOP - 1.0) / (GPASS - 1.0))) / arccosh(nat)
elif type == 'ellip':
GSTOP = 10**(0.1 * gstop)
GPASS = 10**(0.1 * gpass)
arg1 = sqrt((GPASS - 1.0) / (GSTOP - 1.0))
arg0 = 1.0 / nat
d0 = ellipk([arg0**2, 1 - arg0**2])
d1 = ellipk([arg1**2, 1 - arg1**2])
n = (d0[0] * d1[1] / (d0[1] * d1[0]))
else:
raise ValueError("Incorrect type: %s" % type)
return n
def mino_freqs_sc(slr, ecc, x):
"""
Mino frequencies for the SC case (aa = 0)
Parameters:
slr (float): semi-latus rectum
ecc (float): eccentricity
x (float): inclincation
Returns:
ups_r (float): radial Mino frequency
ups_theta (float): polar Mino frequency
ups_phi (float): azimuthal Mino frequency
gamma (float): time Mino frequency
"""
pi = mp.pi
ups_r = (pi * sqrt(-((slr * (-6 + 2 * ecc + slr)) /
(3 + ecc**2 - slr)))) / (2 * ellipk(
(4 * ecc) / (-6 + 2 * ecc + slr)))
ups_theta = slr / sqrt(-3 - ecc**2 + slr)
ups_phi = (slr * x) / (sqrt(-3 - ecc**2 + slr) * abs(x))
gamma = (sqrt(
(-4 * ecc**2 + (-2 + slr)**2) / (slr * (-3 - ecc**2 + slr))) *
(8 + (-(((-4 + slr) * slr**2 * (-6 + 2 * ecc + slr) * ellipe(
(4 * ecc) / (-6 + 2 * ecc + slr))) / (-1 + ecc**2)) +
(slr**2 * (28 + 4 * ecc**2 - 12 * slr + slr**2) * ellipk(
(4 * ecc) / (-6 + 2 * ecc + slr))) / (-1 + ecc**2) -
(2 * (6 + 2 * ecc - slr) *
(3 + ecc**2 - slr) * slr**2 * ellippi(
(2 * ecc * (-4 + slr)) / ((1 + ecc) *
(-6 + 2 * ecc + slr)),
(4 * ecc) / (-6 + 2 * ecc + slr),
)) / ((-1 + ecc) * (1 + ecc)**2) +
(4 * (-4 + slr) * slr * (2 * (1 + ecc) * ellipk(
(4 * ecc) /
(-6 + 2 * ecc + slr)) + (-6 - 2 * ecc + slr) * ellippi(
(2 * ecc * (-4 + slr)) / ((1 + ecc) *
(-6 + 2 * ecc + slr)),
(4 * ecc) / (-6 + 2 * ecc + slr),
))) / (1 + ecc) + 2 * (-4 + slr)**2 *
((-4 + slr) * ellipk((4 * ecc) / (-6 + 2 * ecc + slr)) -
((6 + 2 * ecc - slr) * slr * ellippi(
(16 * ecc) /
(12 + 8 * ecc - 4 * ecc**2 - 8 * slr + slr**2),
(4 * ecc) / (-6 + 2 * ecc + slr),
)) / (2 + 2 * ecc - slr))) / ((-4 + slr)**2 * ellipk(
(4 * ecc) / (-6 + 2 * ecc + slr))))) / 2.0
return ups_r, ups_theta, ups_phi, gamma
def Zolotarev2(p, q, m):
"""Another way to generate Zolotarev polynomial. It is not done yet."""
n = p + q
u0 = (p / (p + q)) * ellipk(m)
wp = 2 * (ellipfun('cd', u0, m)) ** 2 - 1
ws = 2 * (ellipfun('cn', u0, m)) ** 2 - 1
wq = (wp + ws) / 2
sn = ellipfun('sn', u0, m)
cn = ellipfun('cn', u0, m)
dn = ellipfun('dn', u0, m)
wm = ws + 2 * z_zn(u0, m) * ((sn * cn) / (dn))
beta = np.zeros((n + 5, 1))
beta[n] = 1
d = np.zeros((6, 1))
# Implementation of the main recursive algorithm
for m1 in range(n + 2, 2, -1):
d[0] = (m1 + 2) * (m1 + 1) * wp * ws * wm
d[1] = -(m1 + 1) * (m1 - 1) * wp * ws - (m1 + 1) * (2 * m1 + 1) * wm * wq
d[2] = wm * (n ** 2 * wm ** 2 - m1 ** 2 * wp * ws) + m1 ** 2 * (wm - wq) + 3 * m1 * (m1 - 1) * wq
d[3] = (m1 - 1) * (m1 - 2) * (wp * ws - wm * wq - 1) - 3 * wm * (n ** 2 * wm - (m1 - 1) ** 2 * wq)
d[4] = (2 * m1 - 5) * (m1 - 2) * (wm - wq) + 3 * wm * (n ** 2 - (m1 - 2) ** 2)
d[5] = n ** 2 - (m1 - 3) ** 2
tmp = 0
for mu in range(0, 5):
tmp = tmp + d[mu] * beta[m1 + 3 - (mu + 1)]
beta[m1 - 3] = tmp / d[5]
tmp = 0
for m1 in range(0, n + 1):
tmp = tmp + beta[m1]
b = np.zeros((n + 1, 1))
for m1 in range(0, n + 1):
b[m1] = (-1) ** p * (beta[m1] / tmp)
return b
def z_eta(u, m):
"""Jacobi eta function (eq 16.31.3, [Abramowitz]_)."""
q = qfrom(m=m)
DM = ellipk(m)
z = mp.pi * u / (2 * DM)
eta = jtheta(n=1, z=z, q=q)
return eta
def calc_lambda_0(chi, zp, zm, En, Lz, aa, slr, x):
"""
Mino time as a function of polar angle, chi
Parameters:
chi (float): polar angle
zp (float): polar root
zm (float): polar root
En (float): energy
Lz (float): angular momentum
aa (float): spin
slr (float): semi-latus rectum
x (float): inclination
Returns:
lambda_0 (float)
"""
pi = mp.pi
beta = aa * aa * (1 - En * En)
k = sqrt(zm / zp)
k2 = k * k
prefactor = 1 / sqrt(beta * zp)
ellipticK_k = ellipk(k2)
ellipticF = ellipf(pi / 2 - chi, k2)
return prefactor * (ellipticK_k - ellipticF)
def z_eta(u, m):
"""Jacobi eta function (eq 16.31.3, [Abramowitz]_)."""
q = qfrom(m=m)
DM = ellipk(m)
z = mp.pi * u / (2 * DM)
eta = jtheta(n=1, z=z, q=q)
return eta
def phi_inf(P, M):
Qvar = Q(P, M)
ksq = (Qvar - P + 6. * M) / (2. * Qvar)
zinf = zeta_inf(P, M)
phi = 2. * (mpmath.sqrt(
P / Qvar)) * (mpmath.ellipk(ksq) - mpmath.ellipf(zinf, ksq))
return phi
def z_Zolotarev(N, x, m):
r"""
Function to evaluate the Zolotarev polynomial (eq 1, [McNamara93]_).
:param N: Order of the Zolotarev polynomial
:param x: The argument at which one would like to evaluate the Zolotarev polynomial
:param m: m is the elliptic parameter (not the modulus k and not the nome q)
:rtype: Returns a float, the value of Zolotarev polynomial at x
"""
M = -ellipk(m) / N
x3 = ellipfun('sn', u=-M, m=m)
xbar = x3 * mp.sqrt(
(x**2 - 1) / (x**2 - x3**2)) # rearranged eq 21, [Levy70]_
u = ellipf(mp.asin(xbar),
m) # rearranged eq 20, [Levy70]_, asn(x) = F(asin(x)|m)
f = mp.cosh((N / 2) * mp.log(z_eta(M + u, m) / z_eta(M - u, m)))
if f.imag / f.real > 1e-10:
print("imaginary part of the Zolotarev function is not negligible!")
print("f_imaginary = ", f.imag)
else:
if (x > 0): # no idea why I am doing this ... anyhow, it seems working
f = -f.real
else:
f = f.real
return f
def energy_density_at_origin(k):
K = complex64(ellipk(k**2))
E = complex64(ellipe(k**2))
k1 = sqrt(1-k**2)
A = 32*(k**2 *(-K**2 * k**2 +E**2-4*E*K+3* K**2 + k**2)-2*(E-K)**2)**2/(k**8 * K**4 * k1**2)
return A.real
def energy_density_at_origin(k):
K = complex64(ellipk(k**2))
E = complex64(ellipe(k**2))
k1 = sqrt(1-k**2)
A = 32*(k**2 *(-K**2 * k**2 +E**2-4*E*K+3* K**2 + k**2)-2*(E-K)**2)**2/(k**8 * K**4 * k1**2)
return A.real
def test_ellipk(self):
assert_mpmath_equal(sc.ellipk,
mpmath.ellipk,
[Arg(b=1.0)])
assert_mpmath_equal(sc.ellipkm1,
lambda m: mpmath.ellipk(1 - m),
[Arg(a=0.0)],
dps=400)
def testJacobiEllipticCn(self):
"""
Test some special cases of the cn(z, q) function.
This is an intensive test, so precision turned down during
development.
"""
mpmath.mpf.dps = 100
mpmath.mp.dps = 100
#mpmath.mpf.dps = 20 # testing version
#mpmath.mp.dps = 20
testlimit = mpmath.mpf('10')**(-1*(mpmath.mpf.dps - 4))
#print >> sys.stderr, testlimit
zero = mpmath.mpf('0')
one = mpmath.mpf('1')
# Abramowitz Table 16.5
#
# cn(0, q) = 1
for i in range(10):
qstring = str(random.random())
q = mpmath.mpf(qstring)
cn = jacobi_elliptic_cn(zero, q)
equality = one - cn
if equality < testlimit:
self.assertEquals(True, True)
else:
print >> sys.stderr, 'Cn (~ 1): %e' % cn
print >> sys.stderr, 'Equality (~ zero): %e' % equality
self.assertEquals(False, True)
# Abramowitz Table 16.5
#
# cn(K, q) = 0; K is K(k), first complete elliptic integral
for i in range(10):
mstring = str(random.random())
m = mpmath.mpf(mstring)
k = m.sqrt()
K = mpmath.ellipk(k**2)
equality = jacobi_elliptic_cn(K, m)
if equality < testlimit:
self.assertEquals(True, True)
else:
print >> sys.stderr, '\n**** Cn failure ****'
print >> sys.stderr, '\nK: %e' % K,
print >> sys.stderr, '\tm: %f' % m,
print >> sys.stderr, '\tcn: %e' % equality
equality = jacobi_elliptic_cn(K, k, True)
self.assertEquals(False, True)
def testJacobiEllipticCn(self):
"""
Test some special cases of the cn(z, q) function.
This is an intensive test, so precision turned down during
development.
"""
mpmath.mpf.dps = 100
mpmath.mp.dps = 100
#mpmath.mpf.dps = 20 # testing version
#mpmath.mp.dps = 20
testlimit = mpmath.mpf('10')**(-1*(mpmath.mpf.dps - 4))
#print >> sys.stderr, testlimit
zero = mpmath.mpf('0')
one = mpmath.mpf('1')
# Abramowitz Table 16.5
#
# cn(0, q) = 1
for i in range(10):
qstring = str(random.random())
q = mpmath.mpf(qstring)
cn = jacobi_elliptic_cn(zero, q)
equality = one - cn
if equality < testlimit:
self.assertEquals(True, True)
else:
print >> sys.stderr, 'Cn (~ 1): %e' % cn
print >> sys.stderr, 'Equality (~ zero): %e' % equality
self.assertEquals(False, True)
# Abramowitz Table 16.5
#
# cn(K, q) = 0; K is K(k), first complete elliptic integral
for i in range(10):
mstring = str(random.random())
m = mpmath.mpf(mstring)
k = m.sqrt()
K = mpmath.ellipk(k**2)
equality = jacobi_elliptic_cn(K, m)
if equality < testlimit:
self.assertEquals(True, True)
else:
print >> sys.stderr, '\n**** Cn failure ****'
print >> sys.stderr, '\nK: %e' % K,
print >> sys.stderr, '\tm: %f' % m,
print >> sys.stderr, '\tcn: %e' % equality
equality = jacobi_elliptic_cn(K, k, True)
self.assertEquals(False, True)
def testing_components_energy_density(k, x1, x2, x3):
if (is_awc_multiple_root(k, x1, x2, x3)):
return 4
if (is_awc_branch_point(k, x1, x2, x3)):
return 4
zeta = calc_zeta(k, x1, x2, x3)
eta = calc_eta(k, x1, x2, x3)
abel = calc_abel(k, zeta, eta)
mu = calc_mu(k, x1, x2, x3, zeta, abel)
x = [x1, x2, x3]
K = complex64(ellipk(k**2))
E = complex64(ellipe(k**2))
cm = (2 * E - K) / K
k1 = sqrt(1 - k**2)
xp = x[0] + complex(0, 1) * x[1]
xm = x[0] - complex(0, 1) * x[1]
S = sqrt(K**2 - 4 * xp * xm)
SP = sqrt(K**2 - 4 * xp**2)
SM = sqrt(K**2 - 4 * xm**2)
SPM = sqrt(-k1**2 * (K**2 * k**2 - 4 * xm * xp) + (xm - xp)**2)
R = 2 * K**2 * k1**2 - S**2 - 8 * x[2]**2
RM = complex(0, 1) * SM**2 * (xm * (2 * k1**2 - 1) +
xp) - (16 * complex(0, 1)) * xm * x[2]**2
RP = complex(0, 1) * SM**2 * (xp * (2 * k1**2 - 1) +
xm) + (16 * complex(0, 1)) * xp * x[2]**2
RMBAR = -complex(0, 1) * SP**2 * (xp *
(2 * k1**2 - 1) + xm) + 16 * complex(
0, 1) * xp * x[2]**2
RPBAR = -complex(0, 1) * SP**2 * (xm *
(2 * k1**2 - 1) + xp) - 16 * complex(
0, 1) * xm * x[2]**2
r = sqrt(x[0]**2 + x[1]**2 + x[2]**2)
DM = dmus(zeta, x, k)
DZ = dzetas(zeta, x, k)
DDM = ddmus(zeta, x, k)
DDZ = ddzetas(zeta, x, k)
GNUM = grams(zeta, mu, [x1, x2, x3], k)
# DGS1 = dgrams1(zeta, mu, DM, DZ, x, k)
#
# DGS2 = dgrams2(zeta, mu, DM, DZ, x, k)
#
# DGS3 = dgrams3(zeta, mu, DM, DZ, x, k)
# return exp(-6 * mu[0])
print zeta, '\n', abel, '\n', mu, '\n' #mu[0], abel[0], exp(-6*mu[0])
return
def density_of_states_simple(s):
density_of_states = 0
gamma = 1
if s > 1 and s < 3:
density_of_states = (1 / math.pi)**2 * integrate.quad(
lambda x: mpmath.ellipk(
(k_simple(s, gamma, x)**2 - 1)**0.5 / k_simple(s, gamma, x)),
0, math.acos((s - 2) / gamma))[0]
elif s > 0 and s < 1:
density_of_states = (1 / math.pi)**2 * integrate.quad(
lambda x: mpmath.ellipk(
((k_simple(s, gamma, x)**2 - 1)**0.5) / k_simple(s, gamma, x)),
0,
math.pi,
limit=10000,
points=[math.acos(s / gamma)])[0]
return density_of_states
def quartic_roots(k, x1, x2, x3):
K = complex128(ellipk(k**2))
e0 = complex128((x1*j - x2)**2 + .25 * K**2)
e1 = complex128(4*(x1*j-x2)*x3)
e2 = complex128(4*(x3**2) - 2 * (x1**2) - 2 * (x2**2) + (K**2) * (k**2 - 0.5))
e3 = complex128(4*x3*(x2 + j*x1))
e4 = complex128(x2**2 - x1**2 + 2*j*x1*x2 + 0.25*K**2)
return sort_complex(roots([e4, e3, e2, e1, e0])) # I put the sort_complex to have a canonical form, so that when we order them they will vary continuously
def is_awc_branch_point(k, x1, x2, x3): # This will test if we get a branch point as a roots; these are numerically unstable
K = complex64(ellipk(k**2))
k1 = sqrt(1-k**2)
tol = 0.001
if ( (abs(k1 * x1- k * x2) < tol) and x3==0):
return True
return False
def is_awc_branch_point(k, x1, x2, x3): # This will test if we get a branch point as a roots; these are numerically unstable
K = complex64(ellipk(k**2))
k1 = sqrt(1-k**2)
tol = 0.001
if ( (abs(k1 * x1- k * x2) < tol) and x3==0):
return True
return False
def quartic_roots(k, x1, x2, x3):
K = complex128(ellipk(k**2))
e0 = complex128((x1*j - x2)**2 + .25 * K**2)
e1 = complex128(4*(x1*j-x2)*x3)
e2 = complex128(4*(x3**2) - 2 * (x1**2) - 2 * (x2**2) + (K**2) * (k**2 - 0.5))
e3 = complex128(4*x3*(x2 + j*x1))
e4 = complex128(x2**2 - x1**2 + 2*j*x1*x2 + 0.25*K**2)
return sort_complex(roots([e4, e3, e2, e1, e0])) # I put the sort_complex to have a canonical form, so that when we order them they will vary continuously
def dmus(zeta, x, k):
K = complex64(ellipk(k**2))
E = complex64(ellipe(k**2))
cm= (2*E-K)/K
k1 = sqrt(1-k**2)
xp = x[0]+complex(0,1)*x[1]
xm = x[0]-complex(0,1)*x[1]
S = sqrt(K**2-4*xp*xm)
SP = sqrt(K**2-4*xp**2)
SM = sqrt(K**2-4*xm**2)
SPM = sqrt(-k1**2*(K**2*k**2-4*xm*xp)+(xm-xp)**2)
R = 2*K**2*k1**2-S**2-8*x[2]**2
RM = complex(0,1)*SM**2*(xm*(2*k1**2-1)+xp)-(16*complex(0,1))*xm*x[2]**2
RP = complex(0,1)*SM**2*(xp*(2*k1**2-1)+xm)+(16*complex(0,1))*xp*x[2]**2
RMBAR=-complex(0,1)*SP**2*( xp*(2*k1**2-1)+xm ) +16*complex(0,1)*xp*x[2]**2
RPBAR=-complex(0,1)*SP**2*( xm*(2*k1**2-1)+xp ) -16*complex(0,1)*xm*x[2]**2
r=sqrt(x[0]**2+x[1]**2+x[2]**2)
DM = [None,None,None,None]
DM[0] = [None]*3
DM[1] = [None]*3
DM[2] = [None]*3
DM[3] = [None]*3
DM[0][0] = complex(0, 1) * (E * K * zeta[0] ** 2 - K ** 2 * zeta[0] ** 2 + 4 * zeta[0] ** 2 * x[0] ** 2 + complex(0, -4) * zeta[0] ** 2 * x[1] * x[0] + E * K - K ** 2 + 4 * x[0] ** 2 + complex(0, -8) * zeta[0] * x[2] * x[0] + complex(0, 4) * x[0] * x[1]) / (4 * zeta[0] ** 3 * x[0] ** 2 + complex(0, -8) * zeta[0] ** 3 * x[0] * x[1] + 2 * K ** 2 * k1 ** 2 * zeta[0] - K ** 2 * zeta[0] ** 3 + 4 * x[0] ** 2 * zeta[0] + complex(0, -12) * zeta[0] ** 2 * x[0] * x[2] - 4 * zeta[0] ** 3 * x[1] ** 2 - 12 * zeta[0] ** 2 * x[1] * x[2] - K ** 2 * zeta[0] + complex(0, -4) * x[0] * x[2] + 4 * x[1] ** 2 * zeta[0] - 8 * zeta[0] * x[2] ** 2 + 4 * x[1] * x[2])
DM[1][0] = complex(0, 1) * (E * K * zeta[1] ** 2 - K ** 2 * zeta[1] ** 2 + 4 * zeta[1] ** 2 * x[0] ** 2 + complex(0, -4) * zeta[1] ** 2 * x[1] * x[0] + E * K - K ** 2 + 4 * x[0] ** 2 + complex(0, -8) * zeta[1] * x[2] * x[0] + complex(0, 4) * x[0] * x[1]) / (4 * zeta[1] ** 3 * x[0] ** 2 + complex(0, -8) * zeta[1] ** 3 * x[0] * x[1] + 2 * K ** 2 * k1 ** 2 * zeta[1] - K ** 2 * zeta[1] ** 3 + 4 * x[0] ** 2 * zeta[1] + complex(0, -12) * zeta[1] ** 2 * x[0] * x[2] - 4 * zeta[1] ** 3 * x[1] ** 2 - 12 * zeta[1] ** 2 * x[1] * x[2] - K ** 2 * zeta[1] + complex(0, -4) * x[0] * x[2] + 4 * x[1] ** 2 * zeta[1] - 8 * zeta[1] * x[2] ** 2 + 4 * x[1] * x[2])
DM[2][0] = complex(0, 1) * (E * K * zeta[2] ** 2 - K ** 2 * zeta[2] ** 2 + 4 * zeta[2] ** 2 * x[0] ** 2 + complex(0, -4) * zeta[2] ** 2 * x[1] * x[0] + E * K - K ** 2 + 4 * x[0] ** 2 + complex(0, -8) * zeta[2] * x[2] * x[0] + complex(0, 4) * x[0] * x[1]) / (4 * zeta[2] ** 3 * x[0] ** 2 + complex(0, -8) * zeta[2] ** 3 * x[0] * x[1] + 2 * K ** 2 * k1 ** 2 * zeta[2] - K ** 2 * zeta[2] ** 3 + 4 * x[0] ** 2 * zeta[2] + complex(0, -12) * zeta[2] ** 2 * x[0] * x[2] - 4 * zeta[2] ** 3 * x[1] ** 2 - 12 * zeta[2] ** 2 * x[1] * x[2] - K ** 2 * zeta[2] + complex(0, -4) * x[0] * x[2] + 4 * x[1] ** 2 * zeta[2] - 8 * zeta[2] * x[2] ** 2 + 4 * x[1] * x[2])
DM[3][0] = complex(0, 1) * (E * K * zeta[3] ** 2 - K ** 2 * zeta[3] ** 2 + 4 * zeta[3] ** 2 * x[0] ** 2 + complex(0, -4) * zeta[3] ** 2 * x[1] * x[0] + E * K - K ** 2 + 4 * x[0] ** 2 + complex(0, -8) * zeta[3] * x[2] * x[0] + complex(0, 4) * x[0] * x[1]) / (4 * zeta[3] ** 3 * x[0] ** 2 + complex(0, -8) * zeta[3] ** 3 * x[0] * x[1] + 2 * K ** 2 * k1 ** 2 * zeta[3] - K ** 2 * zeta[3] ** 3 + 4 * x[0] ** 2 * zeta[3] + complex(0, -12) * zeta[3] ** 2 * x[0] * x[2] - 4 * zeta[3] ** 3 * x[1] ** 2 - 12 * zeta[3] ** 2 * x[1] * x[2] - K ** 2 * zeta[3] + complex(0, -4) * x[0] * x[2] + 4 * x[1] ** 2 * zeta[3] - 8 * zeta[3] * x[2] ** 2 + 4 * x[1] * x[2])
DM[0][1] = (complex(0, 4) * zeta[0] ** 2 * x[1] * x[0] + E * K * zeta[0] ** 2 + 4 * zeta[0] ** 2 * x[1] ** 2 + complex(0, 4) * x[0] * x[1] + 8 * zeta[0] * x[1] * x[2] - E * K - 4 * x[1] ** 2) / (4 * zeta[0] ** 3 * x[0] ** 2 + complex(0, -8) * zeta[0] ** 3 * x[0] * x[1] + 2 * K ** 2 * k1 ** 2 * zeta[0] - K ** 2 * zeta[0] ** 3 + 4 * x[0] ** 2 * zeta[0] + complex(0, -12) * zeta[0] ** 2 * x[0] * x[2] - 4 * zeta[0] ** 3 * x[1] ** 2 - 12 * zeta[0] ** 2 * x[1] * x[2] - K ** 2 * zeta[0] + complex(0, -4) * x[0] * x[2] + 4 * x[1] ** 2 * zeta[0] - 8 * zeta[0] * x[2] ** 2 + 4 * x[1] * x[2])
DM[1][1] = (complex(0, 4) * zeta[1] ** 2 * x[1] * x[0] + E * K * zeta[1] ** 2 + 4 * zeta[1] ** 2 * x[1] ** 2 + complex(0, 4) * x[0] * x[1] + 8 * zeta[1] * x[1] * x[2] - E * K - 4 * x[1] ** 2) / (4 * zeta[1] ** 3 * x[0] ** 2 + complex(0, -8) * zeta[1] ** 3 * x[0] * x[1] + 2 * K ** 2 * k1 ** 2 * zeta[1] - K ** 2 * zeta[1] ** 3 + 4 * x[0] ** 2 * zeta[1] + complex(0, -12) * zeta[1] ** 2 * x[0] * x[2] - 4 * zeta[1] ** 3 * x[1] ** 2 - 12 * zeta[1] ** 2 * x[1] * x[2] - K ** 2 * zeta[1] + complex(0, -4) * x[0] * x[2] + 4 * x[1] ** 2 * zeta[1] - 8 * zeta[1] * x[2] ** 2 + 4 * x[1] * x[2])
DM[2][1] = (complex(0, 4) * zeta[2] ** 2 * x[1] * x[0] + E * K * zeta[2] ** 2 + 4 * zeta[2] ** 2 * x[1] ** 2 + complex(0, 4) * x[0] * x[1] + 8 * zeta[2] * x[1] * x[2] - E * K - 4 * x[1] ** 2) / (4 * zeta[2] ** 3 * x[0] ** 2 + complex(0, -8) * zeta[2] ** 3 * x[0] * x[1] + 2 * K ** 2 * k1 ** 2 * zeta[2] - K ** 2 * zeta[2] ** 3 + 4 * x[0] ** 2 * zeta[2] + complex(0, -12) * zeta[2] ** 2 * x[0] * x[2] - 4 * zeta[2] ** 3 * x[1] ** 2 - 12 * zeta[2] ** 2 * x[1] * x[2] - K ** 2 * zeta[2] + complex(0, -4) * x[0] * x[2] + 4 * x[1] ** 2 * zeta[2] - 8 * zeta[2] * x[2] ** 2 + 4 * x[1] * x[2])
DM[3][1] = (complex(0, 4) * zeta[3] ** 2 * x[1] * x[0] + E * K * zeta[3] ** 2 + 4 * zeta[3] ** 2 * x[1] ** 2 + complex(0, 4) * x[0] * x[1] + 8 * zeta[3] * x[1] * x[2] - E * K - 4 * x[1] ** 2) / (4 * zeta[3] ** 3 * x[0] ** 2 + complex(0, -8) * zeta[3] ** 3 * x[0] * x[1] + 2 * K ** 2 * k1 ** 2 * zeta[3] - K ** 2 * zeta[3] ** 3 + 4 * x[0] ** 2 * zeta[3] + complex(0, -12) * zeta[3] ** 2 * x[0] * x[2] - 4 * zeta[3] ** 3 * x[1] ** 2 - 12 * zeta[3] ** 2 * x[1] * x[2] - K ** 2 * zeta[3] + complex(0, -4) * x[0] * x[2] + 4 * x[1] ** 2 * zeta[3] - 8 * zeta[3] * x[2] ** 2 + 4 * x[1] * x[2])
DM[0][2] = 2 * (-K ** 2 * k1 ** 2 * zeta[0] + complex(0, 2) * zeta[0] ** 2 * x[0] * x[2] + 2 * zeta[0] ** 2 * x[1] * x[2] + E * K * zeta[0] + complex(0, 2) * x[0] * x[2] + 4 * zeta[0] * x[2] ** 2 - 2 * x[1] * x[2]) / (4 * zeta[0] ** 3 * x[0] ** 2 + complex(0, -8) * zeta[0] ** 3 * x[0] * x[1] + 2 * K ** 2 * k1 ** 2 * zeta[0] - K ** 2 * zeta[0] ** 3 + 4 * x[0] ** 2 * zeta[0] + complex(0, -12) * zeta[0] ** 2 * x[0] * x[2] - 4 * zeta[0] ** 3 * x[1] ** 2 - 12 * zeta[0] ** 2 * x[1] * x[2] - K ** 2 * zeta[0] + complex(0, -4) * x[0] * x[2] + 4 * x[1] ** 2 * zeta[0] - 8 * zeta[0] * x[2] ** 2 + 4 * x[1] * x[2])
DM[1][2] = 2 * (-K ** 2 * k1 ** 2 * zeta[1] + complex(0, 2) * zeta[1] ** 2 * x[0] * x[2] + 2 * zeta[1] ** 2 * x[1] * x[2] + E * K * zeta[1] + complex(0, 2) * x[0] * x[2] + 4 * zeta[1] * x[2] ** 2 - 2 * x[1] * x[2]) / (4 * zeta[1] ** 3 * x[0] ** 2 + complex(0, -8) * zeta[1] ** 3 * x[0] * x[1] + 2 * K ** 2 * k1 ** 2 * zeta[1] - K ** 2 * zeta[1] ** 3 + 4 * x[0] ** 2 * zeta[1] + complex(0, -12) * zeta[1] ** 2 * x[0] * x[2] - 4 * zeta[1] ** 3 * x[1] ** 2 - 12 * zeta[1] ** 2 * x[1] * x[2] - K ** 2 * zeta[1] + complex(0, -4) * x[0] * x[2] + 4 * x[1] ** 2 * zeta[1] - 8 * zeta[1] * x[2] ** 2 + 4 * x[1] * x[2])
DM[2][2] = 2 * (-K ** 2 * k1 ** 2 * zeta[2] + complex(0, 2) * zeta[2] ** 2 * x[0] * x[2] + 2 * zeta[2] ** 2 * x[1] * x[2] + E * K * zeta[2] + complex(0, 2) * x[0] * x[2] + 4 * zeta[2] * x[2] ** 2 - 2 * x[1] * x[2]) / (4 * zeta[2] ** 3 * x[0] ** 2 + complex(0, -8) * zeta[2] ** 3 * x[0] * x[1] + 2 * K ** 2 * k1 ** 2 * zeta[2] - K ** 2 * zeta[2] ** 3 + 4 * x[0] ** 2 * zeta[2] + complex(0, -12) * zeta[2] ** 2 * x[0] * x[2] - 4 * zeta[2] ** 3 * x[1] ** 2 - 12 * zeta[2] ** 2 * x[1] * x[2] - K ** 2 * zeta[2] + complex(0, -4) * x[0] * x[2] + 4 * x[1] ** 2 * zeta[2] - 8 * zeta[2] * x[2] ** 2 + 4 * x[1] * x[2])
DM[3][2] = 2 * (-K ** 2 * k1 ** 2 * zeta[3] + complex(0, 2) * zeta[3] ** 2 * x[0] * x[2] + 2 * zeta[3] ** 2 * x[1] * x[2] + E * K * zeta[3] + complex(0, 2) * x[0] * x[2] + 4 * zeta[3] * x[2] ** 2 - 2 * x[1] * x[2]) / (4 * zeta[3] ** 3 * x[0] ** 2 + complex(0, -8) * zeta[3] ** 3 * x[0] * x[1] + 2 * K ** 2 * k1 ** 2 * zeta[3] - K ** 2 * zeta[3] ** 3 + 4 * x[0] ** 2 * zeta[3] + complex(0, -12) * zeta[3] ** 2 * x[0] * x[2] - 4 * zeta[3] ** 3 * x[1] ** 2 - 12 * zeta[3] ** 2 * x[1] * x[2] - K ** 2 * zeta[3] + complex(0, -4) * x[0] * x[2] + 4 * x[1] ** 2 * zeta[3] - 8 * zeta[3] * x[2] ** 2 + 4 * x[1] * x[2])
return DM
def dzetas(zeta, x, k):
K = complex64(ellipk(k**2))
E = complex64(ellipe(k**2))
cm= (2*E-K)/K
k1 = sqrt(1-k**2)
xp = x[0]+complex(0,1)*x[1]
xm = x[0]-complex(0,1)*x[1]
S = sqrt(K**2-4*xp*xm)
SP = sqrt(K**2-4*xp**2)
SM = sqrt(K**2-4*xm**2)
SPM = sqrt(-k1**2*(K**2*k**2-4*xm*xp)+(xm-xp)**2)
R = 2*K**2*k1**2-S**2-8*x[2]**2
RM = complex(0,1)*SM**2*(xm*(2*k1**2-1)+xp)-(16*complex(0,1))*xm*x[2]**2
RP = complex(0,1)*SM**2*(xp*(2*k1**2-1)+xm)+(16*complex(0,1))*xp*x[2]**2
RMBAR=-complex(0,1)*SP**2*( xp*(2*k1**2-1)+xm ) +16*complex(0,1)*xp*x[2]**2
RPBAR=-complex(0,1)*SP**2*( xm*(2*k1**2-1)+xp ) -16*complex(0,1)*xm*x[2]**2
r=sqrt(x[0]**2+x[1]**2+x[2]**2)
DZ = [None]*4
DZ[0] = [None]*3
DZ[1] = [None]*3
DZ[2] = [None]*3
DZ[3] = [None]*3
DZ[0][0] = -2 * ((x[1] + complex(0, 1) * x[0]) * zeta[0] ** 2 + 2 * x[2] * zeta[0] - x[1] + complex(0, 1) * x[0]) * (complex(0, 1) * zeta[0] ** 2 + complex(0, 1)) / (2 * ((x[1] + complex(0, 1) * x[0]) * zeta[0] ** 2 + 2 * x[2] * zeta[0] - x[1] + complex(0, 1) * x[0]) * (2 * (x[1] + complex(0, 1) * x[0]) * zeta[0] + 2 * x[2]) + K ** 2 * (4 * zeta[0] ** 3 + 4 * (-2 * k1 ** 2 + 1) * zeta[0]) / 4)
DZ[1][0] = -2 * ((x[1] + complex(0, 1) * x[0]) * zeta[1] ** 2 + 2 * zeta[1] * x[2] - x[1] + complex(0, 1) * x[0]) * (complex(0, 1) * zeta[1] ** 2 + complex(0, 1)) / (2 * ((x[1] + complex(0, 1) * x[0]) * zeta[1] ** 2 + 2 * zeta[1] * x[2] - x[1] + complex(0, 1) * x[0]) * (2 * (x[1] + complex(0, 1) * x[0]) * zeta[1] + 2 * x[2]) + K ** 2 * (4 * zeta[1] ** 3 + 4 * (-2 * k1 ** 2 + 1) * zeta[1]) / 4)
DZ[2][0] = -2 * ((x[1] + complex(0, 1) * x[0]) * zeta[2] ** 2 + 2 * x[2] * zeta[2] - x[1] + complex(0, 1) * x[0]) * (complex(0, 1) * zeta[2] ** 2 + complex(0, 1)) / (2 * ((x[1] + complex(0, 1) * x[0]) * zeta[2] ** 2 + 2 * x[2] * zeta[2] - x[1] + complex(0, 1) * x[0]) * (2 * (x[1] + complex(0, 1) * x[0]) * zeta[2] + 2 * x[2]) + K ** 2 * (4 * zeta[2] ** 3 + 4 * (-2 * k1 ** 2 + 1) * zeta[2]) / 4)
DZ[3][0] = -2 * ((x[1] + complex(0, 1) * x[0]) * zeta[3] ** 2 + 2 * zeta[3] * x[2] - x[1] + complex(0, 1) * x[0]) * (complex(0, 1) * zeta[3] ** 2 + complex(0, 1)) / (2 * ((x[1] + complex(0, 1) * x[0]) * zeta[3] ** 2 + 2 * zeta[3] * x[2] - x[1] + complex(0, 1) * x[0]) * (2 * (x[1] + complex(0, 1) * x[0]) * zeta[3] + 2 * x[2]) + K ** 2 * (4 * zeta[3] ** 3 + 4 * (-2 * k1 ** 2 + 1) * zeta[3]) / 4)
DZ[0][1] = -2 * ((x[1] + complex(0, 1) * x[0]) * zeta[0] ** 2 + 2 * x[2] * zeta[0] - x[1] + complex(0, 1) * x[0]) * (zeta[0] ** 2 - 1) / (2 * ((x[1] + complex(0, 1) * x[0]) * zeta[0] ** 2 + 2 * x[2] * zeta[0] - x[1] + complex(0, 1) * x[0]) * (2 * (x[1] + complex(0, 1) * x[0]) * zeta[0] + 2 * x[2]) + K ** 2 * (4 * zeta[0] ** 3 + 4 * (-2 * k1 ** 2 + 1) * zeta[0]) / 4)
DZ[1][1] = -2 * ((x[1] + complex(0, 1) * x[0]) * zeta[1] ** 2 + 2 * zeta[1] * x[2] - x[1] + complex(0, 1) * x[0]) * (zeta[1] ** 2 - 1) / (2 * ((x[1] + complex(0, 1) * x[0]) * zeta[1] ** 2 + 2 * zeta[1] * x[2] - x[1] + complex(0, 1) * x[0]) * (2 * (x[1] + complex(0, 1) * x[0]) * zeta[1] + 2 * x[2]) + K ** 2 * (4 * zeta[1] ** 3 + 4 * (-2 * k1 ** 2 + 1) * zeta[1]) / 4)
DZ[2][1] = -2 * ((x[1] + complex(0, 1) * x[0]) * zeta[2] ** 2 + 2 * x[2] * zeta[2] - x[1] + complex(0, 1) * x[0]) * (zeta[2] ** 2 - 1) / (2 * ((x[1] + complex(0, 1) * x[0]) * zeta[2] ** 2 + 2 * x[2] * zeta[2] - x[1] + complex(0, 1) * x[0]) * (2 * (x[1] + complex(0, 1) * x[0]) * zeta[2] + 2 * x[2]) + K ** 2 * (4 * zeta[2] ** 3 + 4 * (-2 * k1 ** 2 + 1) * zeta[2]) / 4)
DZ[3][1] = -2 * ((x[1] + complex(0, 1) * x[0]) * zeta[3] ** 2 + 2 * zeta[3] * x[2] - x[1] + complex(0, 1) * x[0]) * (zeta[3] ** 2 - 1) / (2 * ((x[1] + complex(0, 1) * x[0]) * zeta[3] ** 2 + 2 * zeta[3] * x[2] - x[1] + complex(0, 1) * x[0]) * (2 * (x[1] + complex(0, 1) * x[0]) * zeta[3] + 2 * x[2]) + K ** 2 * (4 * zeta[3] ** 3 + 4 * (-2 * k1 ** 2 + 1) * zeta[3]) / 4)
DZ[0][2] = -4 * ((x[1] + complex(0, 1) * x[0]) * zeta[0] ** 2 + 2 * x[2] * zeta[0] - x[1] + complex(0, 1) * x[0]) * zeta[0] / (2 * ((x[1] + complex(0, 1) * x[0]) * zeta[0] ** 2 + 2 * x[2] * zeta[0] - x[1] + complex(0, 1) * x[0]) * (2 * (x[1] + complex(0, 1) * x[0]) * zeta[0] + 2 * x[2]) + K ** 2 * (4 * zeta[0] ** 3 + 4 * (-2 * k1 ** 2 + 1) * zeta[0]) / 4)
DZ[1][2] = -4 * ((x[1] + complex(0, 1) * x[0]) * zeta[1] ** 2 + 2 * zeta[1] * x[2] - x[1] + complex(0, 1) * x[0]) * zeta[1] / (2 * ((x[1] + complex(0, 1) * x[0]) * zeta[1] ** 2 + 2 * zeta[1] * x[2] - x[1] + complex(0, 1) * x[0]) * (2 * (x[1] + complex(0, 1) * x[0]) * zeta[1] + 2 * x[2]) + K ** 2 * (4 * zeta[1] ** 3 + 4 * (-2 * k1 ** 2 + 1) * zeta[1]) / 4)
DZ[2][2] = -4 * ((x[1] + complex(0, 1) * x[0]) * zeta[2] ** 2 + 2 * x[2] * zeta[2] - x[1] + complex(0, 1) * x[0]) * zeta[2] / (2 * ((x[1] + complex(0, 1) * x[0]) * zeta[2] ** 2 + 2 * x[2] * zeta[2] - x[1] + complex(0, 1) * x[0]) * (2 * (x[1] + complex(0, 1) * x[0]) * zeta[2] + 2 * x[2]) + K ** 2 * (4 * zeta[2] ** 3 + 4 * (-2 * k1 ** 2 + 1) * zeta[2]) / 4)
DZ[3][2] = -4 * ((x[1] + complex(0, 1) * x[0]) * zeta[3] ** 2 + 2 * zeta[3] * x[2] - x[1] + complex(0, 1) * x[0]) * zeta[3] / (2 * ((x[1] + complex(0, 1) * x[0]) * zeta[3] ** 2 + 2 * zeta[3] * x[2] - x[1] + complex(0, 1) * x[0]) * (2 * (x[1] + complex(0, 1) * x[0]) * zeta[3] + 2 * x[2]) + K ** 2 * (4 * zeta[3] ** 3 + 4 * (-2 * k1 ** 2 + 1) * zeta[3]) / 4)
return DZ
def density_of_states_bcc(s):
density_of_states = 0
if s <= 0 or s > 1:
raise ValueError("s must be between 0 and 1.")
else:
density_of_states = 4 / (math.pi**2 * s) * integrate.quad(
lambda x: 1 / k_bcc(s, x) * mpmath.ellipk(
(k_bcc(s, x)**2 - 1)**0.5 / k_bcc(s, x)), 0, math.acos(s))[0]
return density_of_states
def is_awc_multiple_root(k, x1, x2, x3): # This will test if there are multiple roots; the analytic derivation assumes they are distinct
K = complex64(ellipk(k**2))
k1 = sqrt(1-k**2)
tol = 0.001
if ( (abs(4 * x1**2 * k1**2 - k**2 *( K**2 *k1**2 + 4* x2**2)) < tol) and x3==0):
return True
elif ( (abs(4 * x1**2 * k1**2 - K**2 *k1**2 + 4* x3**2 )< tol) and x2==0):
return True
return False
def z_x123_frm_m(N, m):
"""Function to get x1, x2 and x3 (eq 3, 5 and 6, [McNamara93]_)."""
M = -ellipk(m) / N
snMM = ellipfun('sn', u= -M, m=m)
snM = ellipfun('sn', u=M, m=m)
cnM = ellipfun('cn', u=M, m=m)
dnM = ellipfun('dn', u=M, m=m)
znM = z_zn(M, m)
x3 = snMM
x1 = x3 * mp.sqrt(1 - m) / dnM
x2 = x3 * mp.sqrt(1 - (cnM * znM) / (snM * dnM))
return x1, x2, x3
def z_x123_frm_m(N, m):
"""Function to get x1, x2 and x3 (eq 3, 5 and 6, [McNamara93]_)."""
M = -ellipk(m) / N
snMM = ellipfun('sn', u=-M, m=m)
snM = ellipfun('sn', u=M, m=m)
cnM = ellipfun('cn', u=M, m=m)
dnM = ellipfun('dn', u=M, m=m)
znM = z_zn(M, m)
x3 = snMM
x1 = x3 * mp.sqrt(1 - m) / dnM
x2 = x3 * mp.sqrt(1 - (cnM * znM) / (snM * dnM))
return x1, x2, x3
def z_zn(u, m):
r"""
A function evaluate Jacobi Zeta (zn(u,m)) function (eq 16.26.12, [Abramowitz]_).
:param u: Argument u
:param m: m is the elliptic parameter (not the modulus k and not the nome q)
:rtype: Returns a float, Jacobi Zeta function evaluated for the argument `u` and parameter `m`
"""
phi = z_am(u, m)
zn = ellipe(phi, m) - ellipe(m) * u / ellipk(m)
return zn
def calc_rq(qr, r1, r2, r3, r4):
"""
function used in computing radial geodesic coordinates
Parameters:
qr (float)
r1 (float): radial root
r2 (float): radial root
r3 (float): radial root
r4 (float): radial root
Returns:
rq (float)
"""
pi = mp.pi
kr = ((r1 - r2) * (r3 - r4)) / ((r1 - r3) * (r2 - r4))
sn = ellipfun("sn")
return (-(r2 * (r1 - r3)) + (r1 - r2) * r3 * sn(
(qr * ellipk(kr)) / pi, kr)**2) / (-r1 + r3 + (r1 - r2) * sn(
(qr * ellipk(kr)) / pi, kr)**2)
def z_zn(u, m):
r"""
A function evaluate Jacobi Zeta (zn(u,m)) function (eq 16.26.12, [Abramowitz]_).
:param u: Argument u
:param m: m is the elliptic parameter (not the modulus k and not the nome q)
:rtype: Returns a float, Jacobi Zeta function evaluated for the argument `u` and parameter `m`
"""
phi = z_am(u, m)
zn = ellipe(phi, m) - ellipe(m) * u / ellipk(m)
return zn
def evaluate(self, time):
"""
Calculate a light curve according to the analytical models
given by Pal 2008.
Parameters
----------
time : array
An array of time points at which the light curve
shall be calculated.
.. note:: time = 0 -> Planet is exactly in the line of sight (phase = 0).
Returns
-------
Model : array
The analytical light curve is stored in the property `lightcurve`.
"""
# Translate the given parameters into an orbit and, finally,
# into a projected, normalized distance (z-parameter)
self._calcZList(time - self["T0"])
# 'W' parameters corresponding to notation in Pal '08
w = 6. - 2. * self["linLimb"] - self["quadLimb"]
w0 = (6. - 6. * self["linLimb"] - 12. * self["quadLimb"]) / w
w1 = (6. * self["linLimb"] + 12. * self["quadLimb"]) / w
w2 = 6. * self["quadLimb"] / w
# Initialize flux decrease array
df = numpy.zeros(len(time))
# Get a list of 'cases' (according to Pal '08). Depends on radius ratio and 'z'-parameter along the orbit
ca = self._selectCases()
# Loop through z-list, and calculate the light curve at each point in z (->time)
for i in self._intrans:
# Calculate the coefficients to be substituted into the Pal '08 equation
c = self._returnCoeff(ca[i].step, self._zlist[i])
# Substitute the coefficients and get 'flux decrease'
if ca[i].step != 12:
# Calculate flux decrease only if there is an occultation
if not self.useBoost:
df[i] = w0 * c[0] + w2 * c[5] + w1 * (
c[1] + c[2] * mpmath.ellipk(c[6]**2) +
c[3] * mpmath.ellipe(c[6]**2) +
c[4] * mpmath.ellippi(c[7], c[6]**2))
else:
df[i] = w0 * c[0] + w2 * c[5] + w1 * (
c[1] + c[2] * self.ell.ell1(c[6]) + c[3] *
self.ell.ell2(c[6]) + c[4] * self.ell.ell3(c[7], c[6]))
self.lightcurve = (1. - df) * 1. / \
(1. + self["b"]) + self["b"] / (1.0 + self["b"])
return self.lightcurve
def testing_components_energy_density(k, x1, x2, x3):
if (is_awc_multiple_root(k, x1, x2, x3) ):
return 4
if (is_awc_branch_point(k, x1, x2, x3) ):
return 4
zeta = calc_zeta(k ,x1, x2, x3)
eta = calc_eta(k, x1, x2, x3)
abel = calc_abel(k, zeta, eta)
mu = calc_mu(k, x1, x2, x3, zeta, abel)
x=[x1,x2,x3]
K = complex64(ellipk(k**2))
E = complex64(ellipe(k**2))
cm= (2*E-K)/K
k1 = sqrt(1-k**2)
xp = x[0]+complex(0,1)*x[1]
xm = x[0]-complex(0,1)*x[1]
S = sqrt(K**2-4*xp*xm)
SP = sqrt(K**2-4*xp**2)
SM = sqrt(K**2-4*xm**2)
SPM = sqrt(-k1**2*(K**2*k**2-4*xm*xp)+(xm-xp)**2)
R = 2*K**2*k1**2-S**2-8*x[2]**2
RM = complex(0,1)*SM**2*(xm*(2*k1**2-1)+xp)-(16*complex(0,1))*xm*x[2]**2
RP = complex(0,1)*SM**2*(xp*(2*k1**2-1)+xm)+(16*complex(0,1))*xp*x[2]**2
RMBAR=-complex(0,1)*SP**2*( xp*(2*k1**2-1)+xm ) +16*complex(0,1)*xp*x[2]**2
RPBAR=-complex(0,1)*SP**2*( xm*(2*k1**2-1)+xp ) -16*complex(0,1)*xm*x[2]**2
r=sqrt(x[0]**2+x[1]**2+x[2]**2)
DM = dmus(zeta, x, k)
DZ = dzetas(zeta, x,k)
DDM = ddmus(zeta, x, k)
DDZ = ddzetas(zeta, x,k)
GNUM = grams(zeta, mu, [x1, x2, x3], k)
# DGS1 = dgrams1(zeta, mu, DM, DZ, x, k)
#
# DGS2 = dgrams2(zeta, mu, DM, DZ, x, k)
#
# DGS3 = dgrams3(zeta, mu, DM, DZ, x, k)
# return exp(-6 * mu[0])
print zeta, '\n', abel, '\n', mu, '\n' #mu[0], abel[0], exp(-6*mu[0])
return
def z_eta(u, m):
r"""
A function evaluate Jacobi eta function (eq 16.31.3, [Abramowitz]_).
:param u: Argument u
:param m: m is the elliptic parameter (not the modulus k and not the nome q)
:rtype: Returns a float, Jacobi eta function evaluated for the argument `u` and parameter `m`
"""
q = qfrom(m=m)
DM = ellipk(m)
z = mp.pi * u / (2 * DM)
eta = jtheta(n=1, z=z, q=q)
return eta
def z_eta(u, m):
r"""
A function evaluate Jacobi eta function (eq 16.31.3, [Abramowitz]_).
:param u: Argument u
:param m: m is the elliptic parameter (not the modulus k and not the nome q)
:rtype: Returns a float, Jacobi eta function evaluated for the argument `u` and parameter `m`
"""
q = qfrom(m=m)
DM = ellipk(m)
z = mp.pi * u / (2 * DM)
eta = jtheta(n=1, z=z, q=q)
return eta
def evaluate(self, time):
"""
Calculate a light curve according to the analytical models
given by Pal 2008.
Parameters
----------
time : array
An array of time points at which the light curve
shall be calculated.
.. note:: time = 0 -> Planet is exactly in the line of sight (phase = 0).
Returns
-------
Model : array
The analytical light curve is stored in the property `lightcurve`.
"""
# Translate the given parameters into an orbit and, finally,
# into a projected, normalized distance (z-parameter)
self._calcZList(time - self["T0"])
# 'W' parameters corresponding to notation in Pal '08
w = 6. - 2. * self["linLimb"] - self["quadLimb"]
w0 = (6. - 6. * self["linLimb"] - 12. * self["quadLimb"]) / w
w1 = (6. * self["linLimb"] + 12. * self["quadLimb"]) / w
w2 = 6. * self["quadLimb"] / w
# Initialize flux decrease array
df = numpy.zeros(len(time))
# Get a list of 'cases' (according to Pal '08). Depends on radius ratio and 'z'-parameter along the orbit
ca = self._selectCases()
# Loop through z-list, and calculate the light curve at each point in z (->time)
for i in self._intrans:
# Calculate the coefficients to be substituted into the Pal '08 equation
c = self._returnCoeff(ca[i].step, self._zlist[i])
# Substitute the coefficients and get 'flux decrease'
if ca[i].step != 12:
# Calculate flux decrease only if there is an occultation
if not self.useBoost:
df[i] = w0 * c[0] + w2 * c[5] + w1 * (c[1] + c[2] * mpmath.ellipk(
c[6]**2) + c[3] * mpmath.ellipe(c[6]**2) + c[4] * mpmath.ellippi(c[7], c[6]**2))
else:
df[i] = w0 * c[0] + w2 * c[5] + w1 * (c[1] + c[2] * self.ell.ell1(
c[6]) + c[3] * self.ell.ell2(c[6]) + c[4] * self.ell.ell3(c[7], c[6]))
self.lightcurve = (1. - df) * 1. / \
(1. + self["b"]) + self["b"] / (1.0 + self["b"])
return self.lightcurve
def higgs_squared(k, x1, x2, x3):
if (is_awc_multiple_root(k, x1, x2, x3)):
return float(255) / float(256)
if (is_awc_branch_point(k, x1, x2, x3)):
return float(255) / float(256)
zeta = calc_zeta(k, x1, x2, x3)
eta = calc_eta(k, x1, x2, x3)
abel = calc_abel(k, zeta, eta)
mu = calc_mu(k, x1, x2, x3, zeta, abel)
x = [x1, x2, x3]
K = complex64(ellipk(k**2))
E = complex64(ellipe(k**2))
cm = (2 * E - K) / K
k1 = sqrt(1 - k**2)
xp = x[0] + complex(0, 1) * x[1]
xm = x[0] - complex(0, 1) * x[1]
S = sqrt(K**2 - 4 * xp * xm)
SP = sqrt(K**2 - 4 * xp**2)
SM = sqrt(K**2 - 4 * xm**2)
SPM = sqrt(-k1**2 * (K**2 * k**2 - 4 * xm * xp) + (xm - xp)**2)
R = 2 * K**2 * k1**2 - S**2 - 8 * x[2]**2
RM = complex(0, 1) * SM**2 * (xm * (2 * k1**2 - 1) +
xp) - (16 * complex(0, 1)) * xm * x[2]**2
RP = complex(0, 1) * SM**2 * (xp * (2 * k1**2 - 1) +
xm) + (16 * complex(0, 1)) * xp * x[2]**2
RMBAR = -complex(0, 1) * SP**2 * (xp *
(2 * k1**2 - 1) + xm) + 16 * complex(
0, 1) * xp * x[2]**2
RPBAR = -complex(0, 1) * SP**2 * (xm *
(2 * k1**2 - 1) + xp) - 16 * complex(
0, 1) * xm * x[2]**2
r = sqrt(x[0]**2 + x[1]**2 + x[2]**2)
GNUM = grams(zeta, mu, [x1, x2, x3], k)
inv_gram = matrix(GNUM).I
higgs = phis(zeta, mu, [x1, x2, x3], k)
return -(trace(matmul(matmul(higgs, inv_gram), matmul(higgs,
inv_gram))).real) / 2
def is_awc_multiple_root(
k, x1, x2, x3
): # This will test if there are multiple roots; the analytic derivation assumes they are distinct
K = complex64(ellipk(k**2))
k1 = sqrt(1 - k**2)
tol = 0.001
if ((abs(4 * x1**2 * k1**2 - k**2 * (K**2 * k1**2 + 4 * x2**2)) < tol)
and x3 == 0):
return True
elif ((abs(4 * x1**2 * k1**2 - K**2 * k1**2 + 4 * x3**2) < tol)
and x2 == 0):
return True
return False
def elliptic_core_g(x,y): x = x/2 y = y/2 factor = (cos(x)/sin(y) + sin(y)/cos(x) - (cos(y)/tan(y)/cos(x) + sin(x)*tan(x)/sin(y)))/pi k = tan(x)/tan(y) m = k*k n = (sin(x)/sin(y))*(sin(x)/sin(y)) u = asin(tan(y)/tan(x)) complete = ellipk(m) - ellippi(n, m) incomplete = ellipf(u,m) - ellippi(n/k/k,1/m)/k #incomplete = ellipf(u,m) - ellippi(n,u,m) return re(1.0 - factor*(incomplete + complete))
def calc_dwtheta_dchi(chi, zp, zm):
"""
derivative of w_theta
Parameters:
chi (float): polar angle
zp (float): polar root
zm (float): polar root
Returns:
dw_dtheta (float)
"""
pi = mp.pi
k = sqrt(zm / zp)
ellipticK_k = ellipk(k**2)
return pi / (2 * ellipticK_k) * (1 / (1 - k * k * cos(chi)**2))
def z_Zolotarev(N, x, m):
"""Function to evaluate the Zolotarev polynomial (eq 1, [McNamara93]_)."""
M = -ellipk(m) / N
x3 = ellipfun('sn', u= -M, m=m)
xbar = x3 * mp.sqrt((x ** 2 - 1) / (x ** 2 - x3 ** 2)) # rearranged eq 21, [Levy70]_
u = ellipf(mp.asin(xbar), m) # rearranged eq 20, [Levy70]_, asn(x) = F(asin(x)|m)
f = mp.cosh((N / 2) * mp.log(z_eta(M + u, m) / z_eta(M - u, m)))
if (f.imag / f.real > 1e-10):
print "imaginary part of the Zolotarev function is not negligible!"
print "f_imaginary = ", f.imag
else:
if (x > 0): # no idea why I am doing this ... anyhow, it seems working
f = -f.real
else:
f = f.real
return f
def calc_psi_z(qz, zp, zm, En, aa):
"""
angle used in polar geodesic calculations
Parameters:
qz (float)
zp (float): polar root
zm (float): polar root
En (float): energy
aa (float): spin
Returns:
psi_z (float)
"""
ktheta = (aa**2 * (1 - En**2) * zm**2) / zp**2
return am((2 * (pi / 2.0 + qz) * ellipk(ktheta)) / pi, ktheta)
def Zolotarev2(p, q, m):
r"""
Another way to generate Zolotarev polynomial. It is not done yet.
:param p:
:param q:
:param m:
:rtype: Returns
"""
n = p + q
u0 = (p / (p + q)) * ellipk(m)
wp = 2 * (ellipfun('cd', u0, m))**2 - 1
ws = 2 * (ellipfun('cn', u0, m))**2 - 1
wq = (wp + ws) / 2
sn = ellipfun('sn', u0, m)
cn = ellipfun('cn', u0, m)
dn = ellipfun('dn', u0, m)
wm = ws + 2 * z_zn(u0, m) * ((sn * cn) / (dn))
beta = np.zeros((n + 5, 1))
beta[n] = 1
d = np.zeros((6, 1))
# Implementation of the main recursive algorithm
for m1 in range(n + 2, 2, -1):
d[0] = (m1 + 2) * (m1 + 1) * wp * ws * wm
d[1] = -(m1 + 1) * (m1 - 1) * wp * ws - (m1 + 1) * (2 * m1 +
1) * wm * wq
d[2] = wm * (n**2 * wm**2 - m1**2 * wp * ws) + m1**2 * (
wm - wq) + 3 * m1 * (m1 - 1) * wq
d[3] = (m1 - 1) * (m1 - 2) * (wp * ws - wm * wq -
1) - 3 * wm * (n**2 * wm -
(m1 - 1)**2 * wq)
d[4] = (2 * m1 - 5) * (m1 - 2) * (wm - wq) + 3 * wm * (n**2 -
(m1 - 2)**2)
d[5] = n**2 - (m1 - 3)**2
tmp = 0
for mu in range(0, 5):
tmp = tmp + d[mu] * beta[m1 + 3 - (mu + 1)]
beta[m1 - 3] = tmp / d[5]
tmp = 0
for m1 in range(0, n + 1):
tmp = tmp + beta[m1]
b = np.zeros((n + 1, 1))
for m1 in range(0, n + 1):
b[m1] = (-1)**p * (beta[m1] / tmp)
return b
def calc_psi_r(qr, r1, r2, r3, r4):
"""
radial geodesic angle
Parameters:
qr (float)
r1 (float): radial root
r2 (float): radial root
r3 (float): radial root
r4 (float): radial root
Returns:
psi_r (float)
"""
pi = mp.pi
kr = ((r1 - r2) * (r3 - r4)) / ((r1 - r3) * (r2 - r4))
return am((qr * ellipk(kr)) / pi, kr)
def calc_abel(k, zeta, eta):
k1 = sqrt(1-k**2)
a=k1+complex(0, 1)*k
b=k1-complex(0, 1)*k
abel_tmp = map(lambda zetai : \
complex(0, 1) * 1/(complex64(ellipk(k**2))*2*b) \
* complex64(ellipf( asin( (zetai )/a), mfrom(k=a/b))) \
- taufrom(k=k)/2,
zeta)
abel = []
for i in range(0, 4, 1):
abel.append(abel_select(k, abel_tmp[i], eta[i]))
return abel
def z_Zolotarev(N, x, m):
"""Function to evaluate the Zolotarev polynomial (eq 1, [McNamara93]_)."""
M = -ellipk(m) / N
x3 = ellipfun('sn', u=-M, m=m)
xbar = x3 * mp.sqrt(
(x**2 - 1) / (x**2 - x3**2)) # rearranged eq 21, [Levy70]_
u = ellipf(mp.asin(xbar),
m) # rearranged eq 20, [Levy70]_, asn(x) = F(asin(x)|m)
f = mp.cosh((N / 2) * mp.log(z_eta(M + u, m) / z_eta(M - u, m)))
if (f.imag / f.real > 1e-10):
print("imaginary part of the Zolotarev function is not negligible!")
print("f_imaginary = ", f.imag)
else:
if (x > 0): # no idea why I am doing this ... anyhow, it seems working
f = -f.real
else:
f = f.real
return f
def calc_abel(k, zeta, eta):
k1 = sqrt(1 - k**2)
a = k1 + complex(0, 1) * k
b = k1 - complex(0, 1) * k
abel_tmp = map(lambda zetai : \
complex(0, 1) * 1/(complex64(ellipk(k**2))*2*b) \
* complex64(ellipf( asin( (zetai )/a), mfrom(k=a/b))) \
- taufrom(k=k)/2,
zeta)
abel = []
for i in range(0, 4, 1):
abel.append(abel_select(k, abel_tmp[i], eta[i]))
return abel
def higgs_squared(k, x1, x2, x3):
if (is_awc_multiple_root(k, x1, x2, x3) ):
return float(255)/float(256)
if (is_awc_branch_point(k, x1, x2, x3) ):
return float(255)/float(256)
zeta = calc_zeta(k ,x1, x2, x3)
eta = calc_eta(k, x1, x2, x3)
abel = calc_abel(k, zeta, eta)
mu = calc_mu(k, x1, x2, x3, zeta, abel)
x=[x1,x2,x3]
K = complex64(ellipk(k**2))
E = complex64(ellipe(k**2))
cm= (2*E-K)/K
k1 = sqrt(1-k**2)
xp = x[0]+complex(0,1)*x[1]
xm = x[0]-complex(0,1)*x[1]
S = sqrt(K**2-4*xp*xm)
SP = sqrt(K**2-4*xp**2)
SM = sqrt(K**2-4*xm**2)
SPM = sqrt(-k1**2*(K**2*k**2-4*xm*xp)+(xm-xp)**2)
R = 2*K**2*k1**2-S**2-8*x[2]**2
RM = complex(0,1)*SM**2*(xm*(2*k1**2-1)+xp)-(16*complex(0,1))*xm*x[2]**2
RP = complex(0,1)*SM**2*(xp*(2*k1**2-1)+xm)+(16*complex(0,1))*xp*x[2]**2
RMBAR=-complex(0,1)*SP**2*( xp*(2*k1**2-1)+xm ) +16*complex(0,1)*xp*x[2]**2
RPBAR=-complex(0,1)*SP**2*( xm*(2*k1**2-1)+xp ) -16*complex(0,1)*xm*x[2]**2
r=sqrt(x[0]**2+x[1]**2+x[2]**2)
GNUM = grams(zeta, mu, [x1, x2, x3], k)
inv_gram = matrix(GNUM).I
higgs = phis(zeta, mu, [x1, x2, x3], k)
# print GNUM, inv_gram, higgs # for testing
return -(trace(matmul( matmul(higgs, inv_gram), matmul(higgs, inv_gram) )).real)/2
def z_x123_frm_m(N, m):
r"""
Function to get x1, x2 and x3 (eq 3, 5 and 6, [McNamara93]_).
:param N: Order of the Zolotarev polynomial
:param m: m is the elliptic parameter (not the modulus k and not the nome q)
:rtype: Returns [x1,x2,x3] ... where x1, x2, and x3 are defined in Fig. 1, [McNamara93]_
"""
M = -ellipk(m) / N
snMM = ellipfun('sn', u=-M, m=m)
snM = ellipfun('sn', u=M, m=m)
cnM = ellipfun('cn', u=M, m=m)
dnM = ellipfun('dn', u=M, m=m)
znM = z_zn(M, m)
x3 = snMM
x1 = x3 * mp.sqrt(1 - m) / dnM
x2 = x3 * mp.sqrt(1 - (cnM * znM) / (snM * dnM))
return x1, x2, x3
def z_x123_frm_m(N, m):
r"""
Function to get x1, x2 and x3 (eq 3, 5 and 6, [McNamara93]_).
:param N: Order of the Zolotarev polynomial
:param m: m is the elliptic parameter (not the modulus k and not the nome q)
:rtype: Returns [x1,x2,x3] ... where x1, x2, and x3 are defined in Fig. 1, [McNamara93]_
"""
M = -ellipk(m) / N
snMM = ellipfun('sn', u= -M, m=m)
snM = ellipfun('sn', u=M, m=m)
cnM = ellipfun('cn', u=M, m=m)
dnM = ellipfun('dn', u=M, m=m)
znM = z_zn(M, m)
x3 = snMM
x1 = x3 * mp.sqrt(1 - m) / dnM
x2 = x3 * mp.sqrt(1 - (cnM * znM) / (snM * dnM))
return x1, x2, x3
def test_on_line(k, x0, x1, y, z, partition_size):
k1 = sqrt(1-k**2)
a=k1+complex(0, 1)*k
b=k1-complex(0, 1)*k
x_step = (x1 - x0) / partition_size
for i in range(0, partition_size):
x=x0+i*x_step
zeta = calc_zeta(k ,x, y, z)
eta = calc_eta(k, x, y, z)
abel = calc_abel(k, zeta, eta)
mu = calc_mu(k, x, y, z, zeta, abel)
print "zeta/a", zeta[0]/a, zeta[1]/a, zeta[2]/a, zeta[3]/a, '\n'
print "eta", eta[0], eta[1], eta[2], eta[3], '\n'
print "abel", abel[0], abel[1],abel[2],abel[3],'\n'
abel_tmp= map(lambda zetai : \
complex(0, 1) * 1/(complex64(ellipk(k**2))*2*b) \
* complex64(ellipf( asin( (zetai )/a), mfrom(k=a/b))) \
- taufrom(k=k)/2,
zeta)
print "abel_tmp", abel_tmp[0], abel_tmp[1],abel_tmp[2],abel_tmp[3],'\n'
print abel[0]+conj(abel[2]), abel[1]+conj(abel[3]), '\n'
print - taufrom(k=k)/2, '\n'
for l in range(0,4):
tmp= abs(complex64(calc_eta_by_theta(k, abel[l])) - eta[l])
print tmp
value = higgs_squared(k, x, y, z)
print "higgs", higgs_squared(k,x,y,z)
if (value > 1.0 or value <0.0):
print "Exception"
print '\n'
# print mu[0]+mu[2], mu[1]+mu[3], '\n'
return
def is_awc_multiple_root(k, x1, x2, x3): # This will test if there are multiple roots; the analytic derivation assumes they are distinct
K = complex64(ellipk(k**2))
k1 = sqrt(1-k**2)
tol1 = 0.007
tol2 = 0.026
# Two smoothings here. This one is the better
if ( (abs(4 * x1**2 * k1**2 - k**2 *( K**2 *k1**2 + 4* x2**2)) < tol1) and abs(x3)<tol2):
return True
elif ( (abs(4 * x1**2 * k1**2 - K**2 *k1**2 + 4* x3**2 )< tol1) and abs(x2)<tol2):
return True
# Second smoothing
# tol = 0.01
#
# if ( (abs(4 * x1**2 * k1**2 - k**2 *( K**2 *k1**2 + 4* x2**2)) < tol) and x3==0):
# return True
# elif ( (abs(4 * x1**2 * k1**2 - K**2 *k1**2 + 4* x3**2 )< tol) and x2==0):
# return True
return False
def grams(zeta, mu, x, k):
K = complex64(ellipk(k**2))
E = complex64(ellipe(k**2))
cm= (2*E-K)/K
k1 = sqrt(1-k**2)
xp = x[0]+complex(0,1)*x[1]
xm = x[0]-complex(0,1)*x[1]
S = sqrt(K**2-4*xp*xm)
SP = sqrt(K**2-4*xp**2)
SM = sqrt(K**2-4*xm**2)
SPM = sqrt(-k1**2*(K**2*k**2-4*xm*xp)+(xm-xp)**2)
R = 2*K**2*k1**2-S**2-8*x[2]**2
RM = complex(0,1)*SM**2*(xm*(2*k1**2-1)+xp)-(16*complex(0,1))*xm*x[2]**2
RP = complex(0,1)*SM**2*(xp*(2*k1**2-1)+xm)+(16*complex(0,1))*xp*x[2]**2
RMBAR=-complex(0,1)*SP**2*( xp*(2*k1**2-1)+xm ) +16*complex(0,1)*xp*x[2]**2
RPBAR=-complex(0,1)*SP**2*( xm*(2*k1**2-1)+xp ) -16*complex(0,1)*xm*x[2]**2
r=sqrt(x[0]**2+x[1]**2+x[2]**2)
return mat([[(complex(0, -1) * (-exp(2 * mu[2]) + exp(2 * mu[1])) * zeta[0] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[0] ** 3 - R / zeta[0] ** 2 + complex(0, 12) * xm * x[2] / zeta[0] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[0]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[2] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[0] - (-x[0] + complex(0, -1) * x[1]) / zeta[0] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[0]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[0] - x[2] / zeta[0] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[2] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[0] + complex(0, 2) * pi - 2 * mu[2])) / K ** 2 / (-RMBAR / zeta[0] ** 3 + 2 * R * x[2] / zeta[0] ** 2 + RPBAR / zeta[0] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (exp(2 * mu[2]) - exp(2 * mu[0])) * zeta[1] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[1] ** 3 - R / zeta[1] ** 2 + complex(0, 12) * xm * x[2] / zeta[1] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[1]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[2] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[1] - (-x[0] + complex(0, -1) * x[1]) / zeta[1] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[1]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[1] - x[2] / zeta[1] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[2] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[1] + complex(0, 2) * pi - 2 * mu[2])) / K ** 2 / (-RMBAR / zeta[1] ** 3 + 2 * R * x[2] / zeta[1] ** 2 + RPBAR / zeta[1] - x[2] * (SM ** 2 + SP ** 2))) * (exp(-2 * mu[0]) - exp(-2 * mu[3])) + (complex(0, -1) * (-exp(2 * mu[1]) + exp(2 * mu[0])) * zeta[2] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[2] ** 3 - R / zeta[2] ** 2 + complex(0, 12) * xm * x[2] / zeta[2] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[2]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[3] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[2] - (-x[0] + complex(0, -1) * x[1]) / zeta[2] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[3] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[3] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[2]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[3] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[3] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[2] - x[2] / zeta[2] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[3] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[2] - 2 * mu[3])) / K ** 2 / (-RMBAR / zeta[2] ** 3 + 2 * R * x[2] / zeta[2] ** 2 + RPBAR / zeta[2] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (-exp(2 * mu[2]) + exp(2 * mu[1])) * zeta[0] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[0] ** 3 - R / zeta[0] ** 2 + complex(0, 12) * xm * x[2] / zeta[0] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[0]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[3] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[0] - (-x[0] + complex(0, -1) * x[1]) / zeta[0] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[3] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[3] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[0]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[3] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[3] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[0] - x[2] / zeta[0] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[3] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[0] + complex(0, 2) * pi - 2 * mu[3])) / K ** 2 / (-RMBAR / zeta[0] ** 3 + 2 * R * x[2] / zeta[0] ** 2 + RPBAR / zeta[0] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (exp(2 * mu[2]) - exp(2 * mu[0])) * zeta[1] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[1] ** 3 - R / zeta[1] ** 2 + complex(0, 12) * xm * x[2] / zeta[1] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[1]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[3] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[1] - (-x[0] + complex(0, -1) * x[1]) / zeta[1] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[3] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[3] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[1]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[3] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[3] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[1] - x[2] / zeta[1] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[3] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[1] + complex(0, 2) * pi - 2 * mu[3])) / K ** 2 / (-RMBAR / zeta[1] ** 3 + 2 * R * x[2] / zeta[1] ** 2 + RPBAR / zeta[1] - x[2] * (SM ** 2 + SP ** 2))) * (-exp(-2 * mu[0]) + exp(-2 * mu[2])) + (complex(0, -1) * (-exp(2 * mu[1]) + exp(2 * mu[0])) * zeta[2] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[2] ** 3 - R / zeta[2] ** 2 + complex(0, 12) * xm * x[2] / zeta[2] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[2]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[0] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[2] - (-x[0] + complex(0, -1) * x[1]) / zeta[2] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[2]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[2] - x[2] / zeta[2] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[0] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[2] + complex(0, -2) * pi - 2 * mu[0])) / K ** 2 / (-RMBAR / zeta[2] ** 3 + 2 * R * x[2] / zeta[2] ** 2 + RPBAR / zeta[2] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (exp(2 * mu[2]) - exp(2 * mu[0])) * zeta[1] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[1] ** 3 - R / zeta[1] ** 2 + complex(0, 12) * xm * x[2] / zeta[1] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[1]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[0] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[1] - (-x[0] + complex(0, -1) * x[1]) / zeta[1] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[1]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[1] - x[2] / zeta[1] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[0] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[1] - 2 * mu[0])) / K ** 2 / (-RMBAR / zeta[1] ** 3 + 2 * R * x[2] / zeta[1] ** 2 + RPBAR / zeta[1] - x[2] * (SM ** 2 + SP ** 2))) * (exp(-2 * mu[3]) - exp(-2 * mu[2])),(complex(0, -1) * (-exp(2 * mu[2]) + exp(2 * mu[1])) * zeta[0] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[0] ** 3 - R / zeta[0] ** 2 + complex(0, 12) * xm * x[2] / zeta[0] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[0]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[2] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[0] - (-x[0] + complex(0, -1) * x[1]) / zeta[0] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[0]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[0] - x[2] / zeta[0] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[2] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[0] + complex(0, 2) * pi - 2 * mu[2])) / K ** 2 / (-RMBAR / zeta[0] ** 3 + 2 * R * x[2] / zeta[0] ** 2 + RPBAR / zeta[0] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (exp(2 * mu[2]) - exp(2 * mu[0])) * zeta[1] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[1] ** 3 - R / zeta[1] ** 2 + complex(0, 12) * xm * x[2] / zeta[1] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[1]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[2] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[1] - (-x[0] + complex(0, -1) * x[1]) / zeta[1] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[1]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[1] - x[2] / zeta[1] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[2] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[1] + complex(0, 2) * pi - 2 * mu[2])) / K ** 2 / (-RMBAR / zeta[1] ** 3 + 2 * R * x[2] / zeta[1] ** 2 + RPBAR / zeta[1] - x[2] * (SM ** 2 + SP ** 2))) * (exp(-2 * mu[1]) - exp(-2 * mu[0])) + (complex(0, -1) * (-exp(2 * mu[1]) + exp(2 * mu[0])) * zeta[2] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[2] ** 3 - R / zeta[2] ** 2 + complex(0, 12) * xm * x[2] / zeta[2] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[2]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[0] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[2] - (-x[0] + complex(0, -1) * x[1]) / zeta[2] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[2]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[2] - x[2] / zeta[2] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[0] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[2] + complex(0, -2) * pi - 2 * mu[0])) / K ** 2 / (-RMBAR / zeta[2] ** 3 + 2 * R * x[2] / zeta[2] ** 2 + RPBAR / zeta[2] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (exp(2 * mu[2]) - exp(2 * mu[0])) * zeta[1] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[1] ** 3 - R / zeta[1] ** 2 + complex(0, 12) * xm * x[2] / zeta[1] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[1]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[0] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[1] - (-x[0] + complex(0, -1) * x[1]) / zeta[1] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[1]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[1] - x[2] / zeta[1] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[0] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[1] - 2 * mu[0])) / K ** 2 / (-RMBAR / zeta[1] ** 3 + 2 * R * x[2] / zeta[1] ** 2 + RPBAR / zeta[1] - x[2] * (SM ** 2 + SP ** 2))) * (exp(-2 * mu[2]) - exp(-2 * mu[1])) + (complex(0, -1) * (-exp(2 * mu[1]) + exp(2 * mu[0])) * zeta[2] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[2] ** 3 - R / zeta[2] ** 2 + complex(0, 12) * xm * x[2] / zeta[2] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[2]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[1] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[2] - (-x[0] + complex(0, -1) * x[1]) / zeta[2] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[1] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[1] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[2]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[1] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[1] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[2] - x[2] / zeta[2] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[1] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[2] + complex(0, -2) * pi - 2 * mu[1])) / K ** 2 / (-RMBAR / zeta[2] ** 3 + 2 * R * x[2] / zeta[2] ** 2 + RPBAR / zeta[2] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (-exp(2 * mu[2]) + exp(2 * mu[1])) * zeta[0] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[0] ** 3 - R / zeta[0] ** 2 + complex(0, 12) * xm * x[2] / zeta[0] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[0]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[1] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[0] - (-x[0] + complex(0, -1) * x[1]) / zeta[0] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[1] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[1] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[0]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[1] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[1] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[0] - x[2] / zeta[0] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[1] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[0] - 2 * mu[1])) / K ** 2 / (-RMBAR / zeta[0] ** 3 + 2 * R * x[2] / zeta[0] ** 2 + RPBAR / zeta[0] - x[2] * (SM ** 2 + SP ** 2))) * (-exp(-2 * mu[2]) + exp(-2 * mu[0]))],[(complex(0, -1) * (exp(2 * mu[0]) - exp(2 * mu[2])) * zeta[3] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[3] ** 3 - R / zeta[3] ** 2 + complex(0, 12) * xm * x[2] / zeta[3] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[3]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[2] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[3] - (-x[0] + complex(0, -1) * x[1]) / zeta[3] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[3]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[3] - x[2] / zeta[3] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[2] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[3] - 2 * mu[2])) / K ** 2 / (-RMBAR / zeta[3] ** 3 + 2 * R * x[2] / zeta[3] ** 2 + RPBAR / zeta[3] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (-exp(2 * mu[3]) + exp(2 * mu[2])) * zeta[0] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[0] ** 3 - R / zeta[0] ** 2 + complex(0, 12) * xm * x[2] / zeta[0] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[0]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[2] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[0] - (-x[0] + complex(0, -1) * x[1]) / zeta[0] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[0]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[0] - x[2] / zeta[0] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[2] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[0] + complex(0, 2) * pi - 2 * mu[2])) / K ** 2 / (-RMBAR / zeta[0] ** 3 + 2 * R * x[2] / zeta[0] ** 2 + RPBAR / zeta[0] - x[2] * (SM ** 2 + SP ** 2))) * (exp(-2 * mu[0]) - exp(-2 * mu[3])) + (complex(0, -1) * (-exp(2 * mu[0]) + exp(2 * mu[3])) * zeta[2] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[2] ** 3 - R / zeta[2] ** 2 + complex(0, 12) * xm * x[2] / zeta[2] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[2]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[3] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[2] - (-x[0] + complex(0, -1) * x[1]) / zeta[2] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[3] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[3] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[2]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[3] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[3] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[2] - x[2] / zeta[2] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[3] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[2] - 2 * mu[3])) / K ** 2 / (-RMBAR / zeta[2] ** 3 + 2 * R * x[2] / zeta[2] ** 2 + RPBAR / zeta[2] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (-exp(2 * mu[3]) + exp(2 * mu[2])) * zeta[0] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[0] ** 3 - R / zeta[0] ** 2 + complex(0, 12) * xm * x[2] / zeta[0] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[0]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[3] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[0] - (-x[0] + complex(0, -1) * x[1]) / zeta[0] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[3] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[3] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[0]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[3] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[3] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[0] - x[2] / zeta[0] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[3] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[0] + complex(0, 2) * pi - 2 * mu[3])) / K ** 2 / (-RMBAR / zeta[0] ** 3 + 2 * R * x[2] / zeta[0] ** 2 + RPBAR / zeta[0] - x[2] * (SM ** 2 + SP ** 2))) * (-exp(-2 * mu[0]) + exp(-2 * mu[2])) + (complex(0, -1) * (-exp(2 * mu[0]) + exp(2 * mu[3])) * zeta[2] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[2] ** 3 - R / zeta[2] ** 2 + complex(0, 12) * xm * x[2] / zeta[2] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[2]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[0] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[2] - (-x[0] + complex(0, -1) * x[1]) / zeta[2] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[2]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[2] - x[2] / zeta[2] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[0] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[2] + complex(0, -2) * pi - 2 * mu[0])) / K ** 2 / (-RMBAR / zeta[2] ** 3 + 2 * R * x[2] / zeta[2] ** 2 + RPBAR / zeta[2] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (exp(2 * mu[0]) - exp(2 * mu[2])) * zeta[3] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[3] ** 3 - R / zeta[3] ** 2 + complex(0, 12) * xm * x[2] / zeta[3] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[3]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[0] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[3] - (-x[0] + complex(0, -1) * x[1]) / zeta[3] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[3]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[3] - x[2] / zeta[3] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[0] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[3] + complex(0, -2) * pi - 2 * mu[0])) / K ** 2 / (-RMBAR / zeta[3] ** 3 + 2 * R * x[2] / zeta[3] ** 2 + RPBAR / zeta[3] - x[2] * (SM ** 2 + SP ** 2))) * (exp(-2 * mu[3]) - exp(-2 * mu[2])),(complex(0, -1) * (exp(2 * mu[0]) - exp(2 * mu[2])) * zeta[3] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[3] ** 3 - R / zeta[3] ** 2 + complex(0, 12) * xm * x[2] / zeta[3] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[3]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[2] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[3] - (-x[0] + complex(0, -1) * x[1]) / zeta[3] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[3]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[3] - x[2] / zeta[3] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[2] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[3] - 2 * mu[2])) / K ** 2 / (-RMBAR / zeta[3] ** 3 + 2 * R * x[2] / zeta[3] ** 2 + RPBAR / zeta[3] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (-exp(2 * mu[3]) + exp(2 * mu[2])) * zeta[0] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[0] ** 3 - R / zeta[0] ** 2 + complex(0, 12) * xm * x[2] / zeta[0] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[0]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[2] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[0] - (-x[0] + complex(0, -1) * x[1]) / zeta[0] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[0]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[2] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[2] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[0] - x[2] / zeta[0] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[2] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[0] + complex(0, 2) * pi - 2 * mu[2])) / K ** 2 / (-RMBAR / zeta[0] ** 3 + 2 * R * x[2] / zeta[0] ** 2 + RPBAR / zeta[0] - x[2] * (SM ** 2 + SP ** 2))) * (exp(-2 * mu[1]) - exp(-2 * mu[0])) + (complex(0, -1) * (-exp(2 * mu[0]) + exp(2 * mu[3])) * zeta[2] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[2] ** 3 - R / zeta[2] ** 2 + complex(0, 12) * xm * x[2] / zeta[2] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[2]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[0] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[2] - (-x[0] + complex(0, -1) * x[1]) / zeta[2] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[2]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[2] - x[2] / zeta[2] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[0] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[2] + complex(0, -2) * pi - 2 * mu[0])) / K ** 2 / (-RMBAR / zeta[2] ** 3 + 2 * R * x[2] / zeta[2] ** 2 + RPBAR / zeta[2] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (exp(2 * mu[0]) - exp(2 * mu[2])) * zeta[3] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[3] ** 3 - R / zeta[3] ** 2 + complex(0, 12) * xm * x[2] / zeta[3] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[3]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[0] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[3] - (-x[0] + complex(0, -1) * x[1]) / zeta[3] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[3]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[0] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[0] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[3] - x[2] / zeta[3] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[0] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[3] + complex(0, -2) * pi - 2 * mu[0])) / K ** 2 / (-RMBAR / zeta[3] ** 3 + 2 * R * x[2] / zeta[3] ** 2 + RPBAR / zeta[3] - x[2] * (SM ** 2 + SP ** 2))) * (exp(-2 * mu[2]) - exp(-2 * mu[1])) + (complex(0, -1) * (-exp(2 * mu[0]) + exp(2 * mu[3])) * zeta[2] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[2] ** 3 - R / zeta[2] ** 2 + complex(0, 12) * xm * x[2] / zeta[2] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[2]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[1] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[2] - (-x[0] + complex(0, -1) * x[1]) / zeta[2] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[1] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[1] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[2]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[1] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[1] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[2] - x[2] / zeta[2] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[1] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[2] + complex(0, -2) * pi - 2 * mu[1])) / K ** 2 / (-RMBAR / zeta[2] ** 3 + 2 * R * x[2] / zeta[2] ** 2 + RPBAR / zeta[2] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (exp(2 * mu[0]) - exp(2 * mu[2])) * zeta[3] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[3] ** 3 - R / zeta[3] ** 2 + complex(0, 12) * xm * x[2] / zeta[3] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[3]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[1] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[3] - (-x[0] + complex(0, -1) * x[1]) / zeta[3] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[1] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[1] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[3]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[1] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[1] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[3] - x[2] / zeta[3] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[1] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[3] + complex(0, -2) * pi - 2 * mu[1])) / K ** 2 / (-RMBAR / zeta[3] ** 3 + 2 * R * x[2] / zeta[3] ** 2 + RPBAR / zeta[3] - x[2] * (SM ** 2 + SP ** 2)) + complex(0, -1) * (-exp(2 * mu[3]) + exp(2 * mu[2])) * zeta[0] * SP ** 2 * (complex(0, 4) * x[2] * xp / zeta[0] ** 3 - R / zeta[0] ** 2 + complex(0, 12) * xm * x[2] / zeta[0] + SM ** 2) * ((-x[2] + complex(0, 1) * (-x[0] + complex(0, -1) * x[1]) / zeta[0]) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, -1) * x[1]) ** 2) * zeta[1] - (x[0] + complex(0, -1) * x[1]) * x[2]) + (complex(0, -1) * x[2] / zeta[0] - (-x[0] + complex(0, -1) * x[1]) / zeta[0] ** 2) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[1] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[1] + K ** 2 * (-2 * k ** 2 + 1) / 8 - 0.3e1 / 0.2e1 * x[2] ** 2) + (-x[0] + complex(0, 1) * x[1] + complex(0, 1) * x[2] / zeta[0]) * (-(K ** 2 / 8 - (x[0] + complex(0, 1) * x[1]) ** 2 / 2) / zeta[1] ** 2 + complex(0, -2) * (x[0] + complex(0, 1) * x[1]) * x[2] / zeta[1] + K ** 2 * (-2 * k ** 2 + 1) / 8 + (x[0] + complex(0, -1) * x[1]) * (x[0] + complex(0, 1) * x[1]) - x[2] ** 2 / 2) + (complex(0, 1) * (complex(0, 1) * x[1] - x[0]) / zeta[0] - x[2] / zeta[0] ** 2) * (complex(0, 0.1e1 / 0.4e1) * (K ** 2 - 4 * (x[0] + complex(0, 1) * x[1]) ** 2) / zeta[1] - (x[0] + complex(0, 1) * x[1]) * x[2])) * (1 - exp(2 * mu[0] - 2 * mu[1])) / K ** 2 / (-RMBAR / zeta[0] ** 3 + 2 * R * x[2] / zeta[0] ** 2 + RPBAR / zeta[0] - x[2] * (SM ** 2 + SP ** 2))) * (-exp(-2 * mu[2]) + exp(-2 * mu[0]))]])
def test_timing(k, x1, x2, x3):
t0 = time.time()
zeta = calc_zeta(k ,x1, x2, x3)
eta = calc_eta(k, x1, x2, x3)
abel = calc_abel(k, zeta, eta)
mu = calc_mu(k, x1, x2, x3, zeta, abel)
x=[x1,x2,x3]
t1 = time.time()
K = complex64(ellipk(k**2))
xp = x[0]+complex(0,1)*x[1]
xm = x[0]-complex(0,1)*x[1]
DM = dmus(zeta, x, k)
DZ = dzetas(zeta, x,k)
DDM = ddmus(zeta, x, k)
DDZ = ddzetas(zeta, x,k)
t2 = time.time()
A = ddphis111(zeta, mu, DM, DZ, DDM, DDZ, [x1, x2, x3], k)
t3= time.time()
B = ddphis222(zeta, mu, DM, DZ, DDM, DDZ, [x1, x2, x3], k)
t4= time.time()
C = ddgrams211(zeta, mu, DM, DZ, DDM, DDZ, [x1, x2, x3], k)
t5= time.time()
return A, B, C
def z_Zolotarev(N, x, m):
r"""
Function to evaluate the Zolotarev polynomial (eq 1, [McNamara93]_).
:param N: Order of the Zolotarev polynomial
:param x: The argument at which one would like to evaluate the Zolotarev polynomial
:param m: m is the elliptic parameter (not the modulus k and not the nome q)
:rtype: Returns a float, the value of Zolotarev polynomial at x
"""
M = -ellipk(m) / N
x3 = ellipfun('sn', u= -M, m=m)
xbar = x3 * mp.sqrt((x ** 2 - 1) / (x ** 2 - x3 ** 2)) # rearranged eq 21, [Levy70]_
u = ellipf(mp.asin(xbar), m) # rearranged eq 20, [Levy70]_, asn(x) = F(asin(x)|m)
f = mp.cosh((N / 2) * mp.log(z_eta(M + u, m) / z_eta(M - u, m)))
if (f.imag / f.real > 1e-10):
print "imaginary part of the Zolotarev function is not negligible!"
print "f_imaginary = ", f.imag
else:
if (x > 0): # no idea why I am doing this ... anyhow, it seems working
f = -f.real
else:
f = f.real
return f
def order_roots(roots, segment, sign, theta):
"""
Determine the roots along a trajectory segment.
This is done by comparing the direction of the segment
in the complex plane with that determined by the
evolution equation using the roots in either order.
The right one is chosen by similarity with the direction
of the segment.
e1 and e2 are the colliding roots, e3 is the far one,
they are all expected to be evaluated at u1.
"""
e1_0, e2_0, e3_0 = roots
u0, u1 = segment
distances = map(abs, [e1_0 - e2_0, e2_0 - e3_0, e3_0 - e1_0])
# the pair e1, e2
pair = min(enumerate(distances), key=itemgetter(1))[0]
if pair == 0:
twins = [e1_0, e2_0]
e3 = e3_0
elif pair == 1:
twins = [e3_0, e2_0]
e3 = e1_0
elif pair == 2:
twins = [e3_0, e1_0]
e3 = e2_0
if abs(twins[0] - twins[1]) / abs(twins[0] - e3) < 0.3:
### First possibility ###
f1, f2 = twins
eta_u1 = (sign) * 4.0 * ((e3 - f1) ** (-0.5)) * \
mp.ellipk( ((f2 - f1) / (e3 - f1)) )
phase_1 = phase( \
cmath.exp(1j * (theta + cmath.pi - phase(eta_u1))) / \
(u1 - u0) \
)
### Second possibility ###
f1, f2 = twins[::-1]
eta_u1 = (sign) * 4.0 * ((e3 - f1) ** (-0.5)) * \
mp.ellipk( ((f2 - f1) / (e3 - f1)) )
phase_2 = phase( \
cmath.exp(1j * (theta + cmath.pi - phase(eta_u1))) / \
(u1 - u0) \
)
if abs(phase_1) > PRIMARY_BENDING_TOLERANCE and \
abs(phase_2) > PRIMARY_BENDING_TOLERANCE:
# print "Exceeding primary bending tolerance"
# print "phase_1=%s" % phase_1
# print "phase_2=%s" % phase_2
return 0
else:
if abs(phase_1) < abs(phase_2):
e1, e2 = twins
else:
e1, e2 = twins[::-1]
eta_u1 = (sign) * 4.0 * ((e3 - e1) ** (-0.5)) * \
mp.ellipk( ((e2 - e1) / (e3 - e1)) )
return [[e1, e2, e3], complex(eta_u1)]
else:
return 0
