def _qfunc_pure(psi, alpha_vec):
"""
Calculate the Q-function for a pure state.
"""
n = np.prod(psi.shape)
# Gengyan: maximun number to use factorial()
nmax = 170
if isinstance(psi, Qobj):
psi = psi.full().flatten()
else:
psi = psi.T
if n < nmax:
qvec = abs(
polyval(
fliplr([psi / sqrt(factorial(arange(n)))])[0],
conjugate(alpha_vec)))**2 * exp(-abs(alpha_vec)**2)
else:
# Gengyan: for m < nmax, use factorial()
qvec = polyval(
fliplr([psi[0:nmax] / sqrt(factorial(arange(nmax)))])[0],
conjugate(alpha_vec)) * exp(-abs(alpha_vec)**2 / 2)
# Gengyan: for m >= nmax, use Stirling's approximation
for m in range(nmax, n):
qvec += (conjugate(alpha_vec)/sqrt(m))**m*psi[m] * \
exp((m-abs(alpha_vec)**2)/2)*(2*pi*m)**(-0.25)
qvec = abs(qvec)**2
return np.real(qvec) / pi
def func(self, X, V):
k = self.C.TFdata.k
v1 = self.C.TFdata.v1
w1 = self.C.TFdata.w1
if k >=0:
J_coords = self.F.sysfunc.J_coords
w = sqrt(k)
q = v1 - (1j/w)*matrixmultiply(self.F.sysfunc.J_coords,v1)
p = w1 + (1j/w)*matrixmultiply(transpose(self.F.sysfunc.J_coords),w1)
p /= linalg.norm(p)
q /= linalg.norm(q)
p = reshape(p,(p.shape[0],))
q = reshape(q,(q.shape[0],))
direc = conjugate(1/matrixmultiply(transpose(conjugate(p)),q))
p = direc*p
l1 = firstlyapunov(X, self.F.sysfunc, w, J_coords=J_coords, p=p, q=q)
return array([l1])
else:
return array([1])
def to_minimize_fidelity(theta):
temp_z_gate = np.matmul(
sp.linalg.expm(-2 * np.pi * 1j * theta * self.H_zeeman),
U_ideal)
temp_m = np.matmul(sp.conjugate(sp.transpose(target)),
temp_z_gate)
return np.real(1. - (sp.trace(
np.matmul(temp_m, sp.conjugate(sp.transpose(temp_m)))) +
np.abs(sp.trace(temp_m))**2.) / 20.)
def process(self, X, V, C):
"""Tolerance for eigenvalues a possible problem when checking for neutral saddles."""
BifPoint.process(self, X, V, C)
J_coords = C.CorrFunc.jac(X, C.coords)
eigs, LV, RV = linalg.eig(J_coords,left=1,right=1)
# Check for neutral saddles
found = False
for i in range(len(eigs)):
if abs(imag(eigs[i])) < 1e-5:
for j in range(i+1,len(eigs)):
if C.verbosity >= 2:
if abs(eigs[i]) < 1e-5 and abs(eigs[j]) < 1e-5:
print 'Fold-Fold point found in Hopf!\n'
elif abs(imag(eigs[j])) < 1e-5 and abs(real(eigs[i]) + real(eigs[j])) < 1e-5:
print 'Neutral saddle found!\n'
elif abs(real(eigs[i])) < 1e-5:
for j in range(i+1, len(eigs)):
if abs(real(eigs[j])) < 1e-5 and abs(real(eigs[i]) - real(eigs[j])) < 1e-5:
found = True
w = abs(imag(eigs[i]))
if imag(eigs[i]) > 0:
p = conjugate(LV[:,j])/linalg.norm(LV[:,j])
q = RV[:,i]/linalg.norm(RV[:,i])
else:
p = conjugate(LV[:,i])/linalg.norm(LV[:,i])
q = RV[:,j]/linalg.norm(RV[:,j])
if not found:
del self.found[-1]
return False
direc = conjugate(1/matrixmultiply(conjugate(p),q))
p = direc*p
# Alternate way to compute 1st lyapunov coefficient (from Kuznetsov [4])
#print (1./(w*w))*real(1j*matrixmultiply(conjugate(p),b1)*matrixmultiply(conjugate(p),b3) + \
# w*matrixmultiply(conjugate(p),trilinearform(D,q,q,conjugate(q))))
self.found[-1].w = w
self.found[-1].l1 = firstlyapunov(X, C.CorrFunc, w, J_coords=J_coords, p=p, q=q, check=(C.verbosity==2))
self.found[-1].eigs = eigs
self.info(C, -1)
return True
def hg_conv(f, g): ub = tuple(array(f.shape) + array(g.shape) - 1); fh = rfftn(cpad(f,ub)); gh = rfftn(cpad(g,ub)); res = ifftshift(irfftn(fh * sp.conjugate(gh))); del fh, gh; return res;
def MakePulseDataRepLPC(pulse,spec,N,rep1,numtype = sp.complex128):
""" This will make data by assuming the data is an autoregressive process.
Inputs
spec - The properly weighted spectrum.
N - The size of the ar process used to model the filter.
pulse - The pulse shape.
rep1 - The number of repeats of the process.
Outputs
outdata - A numpy Array with the shape of the """
lp = len(pulse)
lenspec = len(spec)
r1 = scfft.ifft(scfft.ifftshift(spec))
rp1 = r1[:N]
rp2 = r1[1:N+1]
# Use Levinson recursion to find the coefs for the data
xr1 = sp.linalg.solve_toeplitz(rp1, rp2)
lpc = sp.r_[sp.ones(1), -xr1]
# The Gain term.
G = sp.sqrt(sp.sum(sp.conjugate(r1[:N+1])*lpc))
Gvec = sp.r_[G, sp.zeros(N)]
Npnt = (N+1)*3+lp
# Create the noise vector and normalize
xin = sp.random.randn(rep1, Npnt)+1j*sp.random.randn(rep1, Npnt)
xinsum = sp.tile(sp.sqrt(sp.mean(xin.real**2+xin.imag**2, axis=1))[:, sp.newaxis],(1, Npnt))
xin = xin/xinsum
outdata = sp.signal.lfilter(Gvec, lpc, xin, axis=1)
outpulse = sp.tile(pulse[sp.newaxis], (rep1, 1))
outdata = outpulse*outdata[:, 2*N:2*N+lp]
return outdata
def autocorrelation_1d(x):
"""Estimate the normalized autocorrelation function of a 1-D series
Args:
x: The series as a 1-D numpy array.
Returns:
array: The autocorrelation function of the time series.
Taken from: https://github.com/dfm/emcee/blob/master/emcee/autocorr.py
"""
x = scipy.atleast_1d(x)
if len(x.shape) != 1:
raise ValueError("invalid dimensions for 1D autocorrelation function")
def next_pow_two(n):
"""Returns the next power of two greater than or equal to `n`"""
i = 1
while i < n:
i = i << 1
return i
n = next_pow_two(len(x))
# Compute the FFT and then (from that) the auto-correlation function
f = fft.fft(x - scipy.mean(x), n=2 * n)
acf = fft.ifft(f * scipy.conjugate(f))[:len(x)].real
acf /= acf[0]
return acf
def task2(rx_sample_time, rx_signal, tx_envelope, pltfig=False):
rx_quad_demod = quadrature_demodulation(rx_signal, rx_sample_time)
rx_filtered = matched_filter(rx_quad_demod,
sp.conjugate(sp.flipud(tx_envelope)))
if pltfig:
plot_sqr_magn(rx_filtered, rx_sample_time)
return rx_filtered # workarround due to lack of time
def MakePulseDataRepLPC(pulse, spec, N, rep1, numtype=sp.complex128):
""" This will make data by assuming the data is an autoregressive process.
Inputs
spec - The properly weighted spectrum.
N - The size of the ar process used to model the filter.
pulse - The pulse shape.
rep1 - The number of repeats of the process.
Outputs
outdata - A numpy Array with the shape of the """
lp = len(pulse)
r1 = scfft.ifft(scfft.ifftshift(spec))
rp1 = r1[:N]
rp2 = r1[1:N + 1]
# Use Levinson recursion to find the coefs for the data
xr1 = sp.linalg.solve_toeplitz(rp1, rp2)
lpc = sp.r_[sp.ones(1), -xr1]
# The Gain term.
G = sp.sqrt(sp.sum(sp.conjugate(r1[:N + 1]) * lpc))
Gvec = sp.r_[G, sp.zeros(N)]
Npnt = (N + 1) * 3 + lp
# Create the noise vector and normalize
xin = sp.random.randn(rep1, Npnt) + 1j * sp.random.randn(rep1, Npnt)
xinsum = sp.tile(
sp.sqrt(sp.sum(xin.real**2 + xin.imag**2, axis=1))[:, sp.newaxis],
(1, Npnt))
xin = xin / xinsum / sp.sqrt(2.)
outdata = sp.signal.lfilter(Gvec, lpc, xin, axis=1)
outpulse = sp.tile(pulse[sp.newaxis], (rep1, 1))
outdata = outpulse * outdata[:, N:N + lp]
return outdata
def _cross_correlate(self, reference_image):
# FFT of the projection spectra
f_ref_xproj = fft(reference_image.proj_x)
f_ref_yproj = fft(reference_image.proj_y)
f_check_xproj = fft(self.proj_x)
f_check_yproj = fft(self.proj_y)
# cross correlate in and look for the maximium correlation
f_ref_xproj_conj = conjugate(f_ref_xproj)
f_ref_yproj_conj = conjugate(f_ref_yproj)
complex_sum_x = f_ref_xproj_conj * f_check_xproj
complex_sum_y = f_ref_yproj_conj * f_check_yproj
phi_ref_check_m_x = ifft(complex_sum_x)
phi_ref_check_m_y = ifft(complex_sum_y)
z_x = max(phi_ref_check_m_x)
z_pos_x = np.where(phi_ref_check_m_x == z_x)
z_y = max(phi_ref_check_m_y)
z_pos_y = np.where(phi_ref_check_m_y == z_y)
return z_pos_x, z_pos_y, phi_ref_check_m_x, phi_ref_check_m_y
def makeCovmat(lagsDatasum,lagsNoisesum,pulses_s,Nlags):
axvec=sp.roll(sp.arange(lagsDatasum.ndim),1)
# Get the covariance matrix
R=sp.transpose(lagsDatasum/sp.sqrt(2.*pulses_s),axes=axvec)
Rw=sp.transpose(lagsNoisesum/sp.sqrt(2.*pulses_s),axes=axvec)
l=sp.arange(Nlags)
T1,T2=sp.meshgrid(l,l)
R0=R[sp.zeros_like(T1)]
Rw0=Rw[sp.zeros_like(T1)]
Td=sp.absolute(T1-T2)
Tl = T1>T2
R12 =R[Td]
R12[Tl]=sp.conjugate(R12[Tl])
Rw12 =Rw[Td]
Rw12[Tl]=sp.conjugate(Rw12[Tl])
Ctt=R0*R12+R[T1]*sp.conjugate(R[T2])+Rw0*Rw12+Rw[T1]*sp.conjugate(Rw[T2])
avecback=sp.roll(sp.arange(Ctt.ndim),-2)
Cttout = sp.transpose(Ctt,avecback)
return Cttout
def makeCovmat(lagsDatasum,lagsNoisesum,pulses_s,Nlags):
axvec=sp.roll(sp.arange(lagsDatasum.ndim),1)
# Get the covariance matrix
R=sp.transpose(lagsDatasum/sp.sqrt(2.*pulses_s),axes=axvec)
Rw=sp.transpose(lagsNoisesum/sp.sqrt(2.*pulses_s),axes=axvec)
l=sp.arange(Nlags)
T1,T2=sp.meshgrid(l,l)
R0=R[sp.zeros_like(T1)]
Rw0=Rw[sp.zeros_like(T1)]
Td=sp.absolute(T1-T2)
Tl = T1>T2
R12 =R[Td]
R12[Tl]=sp.conjugate(R12[Tl])
Rw12 =Rw[Td]
Rw12[Tl]=sp.conjugate(Rw12[Tl])
Ctt=R0*R12+R[T1]*sp.conjugate(R[T2])+Rw0*Rw12+Rw[T1]*sp.conjugate(Rw[T2])
avecback=sp.roll(sp.arange(Ctt.ndim),-2)
Cttout = sp.transpose(Ctt,avecback)
return Cttout
def grad(self, A, B, q): #Compute relative transformation pr = norm(q.x) if(pr >= A.radius + B.radius ): return array([0., 0., 0., 0.]) #Load shape parameters fa = A.pft fb = B.pft da = A.pdft db = B.pdft ea = A.energy eb = B.energy #Estimate cutoff threshold cutoff = self.__get_cutoff(A, B) #Compute coordinate coefficients m = 2.j * pi / (2. * self.SHAPE_R + 1) * pr phi = atan2(q.x[1], q.x[0]) #Set up initial sums s_0 = real(fa[0][0] * fb[0][0]) s_x = 0. s_y = 0. s_ta = 0. s_tb = 0. for r in range(1, self.R): #Compute theta terms dtheta = 2. * pi / len(fa[r]) theta = arange(len(fa[r])) * dtheta #Construct multiplier / v mult = exp((m * r) * cos(theta + phi)) * r * dtheta u = pds.shift(conjugate(fb[r]), q.theta) * mult v = fa[r] * u #Check for early out s_0 += sum(real(v)) if(s_0 + min(ea[r], eb[r]) <= cutoff): return array([0.,0.,0.,0.]) #Sum up gradient vectors v = real(1.j * v) s_x -= sum(v * sin(theta + phi) ) s_y -= sum(v * cos(theta + phi) ) s_t += sum(real(da[r] * u)) if(s_0 <= cutoff): return array([0., 0., 0., 0.]) return array([s_x, s_y, s_ta, s_tb, s_0])
def _qfunc_pure(psi, alpha_mat):
"""
Calculate the Q-function for a pure state.
"""
n = np.prod(psi.shape)
if isinstance(psi, Qobj):
psi = psi.full().flatten()
else:
psi = psi.T
qmat = abs(polyval(fliplr([psi / sqrt(factorial(arange(n)))])[0],
conjugate(alpha_mat))) ** 2
return real(qmat) * exp(-abs(alpha_mat) ** 2) / pi
def levinsonSymmetricConstrainA1(A, y, ncoeffs=scipy.inf):
ncoeffs = min(ncoeffs, A.shape[1])
eps = A[0, 0]
ref = scipy.zeros(ncoeffs - 1)
x = scipy.zeros(ncoeffs)
x[0] = 1
for i in range(1, ncoeffs):
gamma = A[0, i] + scipy.dot(A[0, 1:i], x[1:i])
ref[i - 1] = -gamma / eps
eps *= (1 - ref[i - 1] * scipy.conjugate(ref[i - 1]))
temp = scipy.conjugate(scipy.flipud(x[:i + 1]))
x[:i + 1] += ref[i - 1] * temp
x[i] = ref[i - 1]
return (x, scipy.sqrt(eps))
def subspace_angles(A, B):
"""
Return the principle angles and vectors between two subspaces.
See Bjorck and Golub, "Numerical Methods for Computing Between Linear
Subspaces" _Mathematics_of_Computation_, 27(123), 1973, pp. 579-594
Or, for a more understandable exposition, Golub and Van Loan,
_Matrix_Computations_ (3rd ed.) pp. 603-604
"""
A, B = scipy.asarray(A), scipy.asarray(B)
if A.shape[0] != B.shape[0]:
raise ValueError('Input subspaces must live in the same dimensional '\
'space.')
# Get orthogonal bases for our two subspaces.
QA, QB = scipy.linalg.orth(A), scipy.linalg.orth(B)
M = scipy.matrixmultiply(scipy.transpose(scipy.conjugate(QA)), QB)
Y, C, Zh = scipy.linalg.svd(M)
U = scipy.matrixmultiply(QA, Y)
V = scipy.matrixmultiply(QB, scipy.transpose(scipy.conjugate(Zh)))
return scipy.arccos(C), U, V
def _qfunc_pure(psi, alpha_mat):
"""
Calculate the Q-function for a pure state.
"""
n = prod(psi.shape)
if isinstance(psi, Qobj):
psi = array(psi.trans().full())[0,:]
else:
psi = psi.T
qmat1 = abs(polyval(fliplr([psi/sqrt(factorial(arange(0, n)))])[0], conjugate(alpha_mat))) ** 2;
qmat1 = real(qmat1) * exp(-abs(alpha_mat)**2) / pi;
return qmat1
def levinsonSymmetricConstrainA1(A, y, ncoeffs = scipy.inf):
ncoeffs = min(ncoeffs, A.shape[1])
eps = A[0,0]
ref = scipy.zeros( ncoeffs - 1 )
x = scipy.zeros( ncoeffs )
x[0] = 1
for i in range(1, ncoeffs):
gamma = A[0, i] + scipy.dot(A[0, 1:i], x[1:i])
ref[i-1] = - gamma / eps
eps *= (1 - ref[i-1]*scipy.conjugate(ref[i-1]))
temp = scipy.conjugate(scipy.flipud(x[:i+1]))
x[:i+1] += ref[i-1] * temp
x[i] = ref[i-1]
return (x, scipy.sqrt(eps))
def _qfunc_pure(psi, alpha_mat):
"""
Calculate the Q-function for a pure state.
"""
n = prod(psi.shape)
if isinstance(psi, Qobj):
psi = array(psi.trans().full())[0, :]
else:
psi = psi.T
qmat1 = abs(
polyval(
fliplr([psi / sqrt(factorial(arange(0, n)))])[0],
conjugate(alpha_mat)))**2
qmat1 = real(qmat1) * exp(-abs(alpha_mat)**2) / pi
return qmat1
def RK4(t_tot, N, Ham, psi):
dt = t_tot / N #N is number of steps
t = sp.linspace(0, t_tot, N + 1)
y = [psi]
for i in range(len(t) - 1):
k1 = dt * TDSE_T(Ham, t[i], y[len(y) - 1])
k2 = dt * TDSE_T(Ham, t[i] + 0.5 * dt, y[len(y) - 1] + k1 / 2)
k3 = dt * TDSE_T(Ham, t[i] + 0.5 * dt, y[len(y) - 1] + k2 / 2)
k4 = dt * TDSE_T(Ham, t[i] + dt, y[len(y) - 1] + k3)
ynew = y[len(y) - 1] + (k1 + 2 * k2 + 2 * k3 + k4) / 6
y.append(ynew)
chi = []
for i in range(len(y)):
inneri = sp.matmul(sp.conjugate(psi), y[i])
inner = abs(inneri)**2
chi.append(inner)
return t, chi
def firstlyapunov(X, F, w, J_coords=None, V=None, W=None, p=None, q=None, check=False): if J_coords is None: J_coords = F.jac(X, F.coords) if p is None: alpha = bilinearform(transpose(J_coords),V[:,0],V[:,1]) - \ 1j*w*matrixmultiply(V[:,0],V[:,1]) beta = -1*bilinearform(transpose(J_coords),V[:,0],V[:,0]) + \ 1j*w*matrixmultiply(V[:,0],V[:,0]) q = alpha*V[:,0] + beta*V[:,1] alpha = bilinearform(J_coords,W[:,0],W[:,1]) + \ 1j*w*matrixmultiply(W[:,0],W[:,1]) beta = -1*bilinearform(J_coords,W[:,0],W[:,0]) - \ 1j*w*matrixmultiply(W[:,0],W[:,0]) p = alpha*W[:,0] + beta*W[:,1] p /= linalg.norm(p) q /= linalg.norm(q) direc = conjugate(1/matrixmultiply(conjugate(p),q)) p = direc*p if check: print 'Checking...' print ' |q| = %f' % linalg.norm(q) temp = matrixmultiply(conjugate(p),q) print ' |<p,q> - 1| = ', abs(temp-1) print ' |Aq - iwq| = %f' % linalg.norm(matrixmultiply(J_coords,q) - 1j*w*q) print ' |A*p + iwp| = %f\n' % linalg.norm(matrixmultiply(transpose(J_coords),p) + 1j*w*p) # Compute first lyapunov coefficient B = F.hess(X, F.coords, F.coords) D = hess3(F, X, F.coords) b1 = array([bilinearform(B[i,:,:], q, q) for i in range(B.shape[0])]) b2 = array([bilinearform(B[i,:,:], conjugate(q), linalg.solve(2*1j*w*eye(F.m) - J_coords, b1)) \ for i in range(B.shape[0])]) b3 = array([bilinearform(B[i,:,:], q, conjugate(q)) for i in range(B.shape[0])]) b4 = array([bilinearform(B[i,:,:], q, linalg.solve(J_coords, b3)) for i in range(B.shape[0])]) temp = array([trilinearform(D[i,:,:,:],q,q,conjugate(q)) for i in range(D.shape[0])]) + b2 - 2*b4 l1 = 0.5*real(matrixmultiply(conjugate(p), temp)) return l1
def f_osc(qw, E_X, a_2D, E_p = 25.0):
"""Formula for the oscilator strength per unit area [nm^-2] at k_par = 0.
Based on Eq. (2) Andreani and Bassani, PRB 41, 7536 (1990)
Keyword argumets:
qw -> quantum well object. Class defined in binding.py
E_X -> Exciton binding energy.
a_2D -> Exciton radial extension (aka lambda parameter)
E_p -> Kane parameter (dipole |S> |X> matrix element)
"""
E_tr = (qw.Elec.E - qw.Hole.E) + E_X
# Overlapp matrix element
I_cv = spi.trapz(sp.conjugate(qw.Elec.wf)*qw.Hole.wf, qw.grid)
F_QW_sq = 2.0/(sp.pi * a_2D**2)
return E_p/E_tr * I_cv.real**2 * F_QW_sq
def norm(self, component=None, summed=False):
r"""Calculate the :math:`L^2` norm :math:`\langle\Psi|\Psi\rangle` of the wavepacket :math:`\Psi`.
:param component: The index :math:`i` of the component :math:`\Phi_i` whose norm is calculated.
The default value is ``None`` which means to compute the norms of all :math:`N` components.
:type component: int or ``None``.
:param summed: Whether to sum up the norms :math:`\langle\Phi_i|\Phi_i\rangle` of the
individual components :math:`\Phi_i`.
:type summed: Boolean, default is ``False``.
:type summed: Boolean, default is ``False``.
:return: The norm of :math:`\Psi` or the norm of :math:`\Phi_i` or a list with the :math:`N`
norms of all components. Depending on the values of ``component`` and ``summed``.
"""
if component is not None:
result = norm(self._coefficients[component])
else:
result = [ norm(item) for item in self._coefficients ]
if summed is True:
result = reduce(lambda x,y: x+conjugate(y)*y, result, 0.0)
result = sqrt(result)
return result
def H(m, out=None):
"""Matrix conjugate transpose (adjoint).
This is just a shortcut for performing this operation on normal ndarrays.
Parameters
----------
m : ndarray
The input matrix.
out : ndarray
A matrix to hold the final result (dimensions must be correct). May be None.
May also be the same object as m.
Returns
-------
out : ndarray
The result.
"""
if out is None:
return m.T.conj()
else:
out = sp.conjugate(m.T, out)
return out
def H(m, out=None):
"""Matrix conjugate transpose (adjoint).
This is just a shortcut for performing this operation on normal ndarrays.
Parameters
----------
m : ndarray
The input matrix.
out : ndarray
A matrix to hold the final result (dimensions must be correct). May be None.
May also be the same object as m.
Returns
-------
out : ndarray
The result.
"""
if out is None:
return m.T.conj()
else:
out = sp.conjugate(m.T, out)
return out
def plv(x, y, identities):
"""Function for computing phase-locking values between x and y.
Output arguments:
=================
cplv : ndarray
Complex-valued phase-locking values.
"""
"""Change to amplitude 1, keep angle using Euler's formula."""
x = scipy.exp(1j * (asmatrix(scipy.angle(x))))
y = scipy.exp(1j * (asmatrix(scipy.angle(y))))
"""Get cPLV needed for flips and weighting."""
cplv = scipy.zeros(len(identities), dtype='complex')
for i, identity in enumerate(identities):
"""Compute cPLV only of parcel source pairs of sources that
belong to that parcel. One source belong to only one parcel."""
if (identities[i] >= 0):
cplv[i] = (scipy.sum((scipy.asarray(y[identity])) *
scipy.conjugate(scipy.asarray(x[i]))))
cplv /= np.shape(x)[1]
return cplv
def firstlyapunov(X,
F,
w,
J_coords=None,
V=None,
W=None,
p=None,
q=None,
check=False):
if J_coords is None:
J_coords = F.jac(X, F.coords)
if p is None:
alpha = bilinearform(transpose(J_coords),V[:,0],V[:,1]) - \
1j*w*matrixmultiply(V[:,0],V[:,1])
beta = -1*bilinearform(transpose(J_coords),V[:,0],V[:,0]) + \
1j*w*matrixmultiply(V[:,0],V[:,0])
q = alpha * V[:, 0] + beta * V[:, 1]
alpha = bilinearform(J_coords,W[:,0],W[:,1]) + \
1j*w*matrixmultiply(W[:,0],W[:,1])
beta = -1*bilinearform(J_coords,W[:,0],W[:,0]) - \
1j*w*matrixmultiply(W[:,0],W[:,0])
p = alpha * W[:, 0] + beta * W[:, 1]
p /= linalg.norm(p)
q /= linalg.norm(q)
direc = conjugate(1 / matrixmultiply(conjugate(p), q))
p = direc * p
if check:
print 'Checking...'
print ' |q| = %f' % linalg.norm(q)
temp = matrixmultiply(conjugate(p), q)
print ' |<p,q> - 1| = ', abs(temp - 1)
print ' |Aq - iwq| = %f' % linalg.norm(
matrixmultiply(J_coords, q) - 1j * w * q)
print ' |A*p + iwp| = %f\n' % linalg.norm(
matrixmultiply(transpose(J_coords), p) + 1j * w * p)
# Compute first lyapunov coefficient
B = F.hess(X, F.coords, F.coords)
D = hess3(F, X, F.coords)
b1 = array([bilinearform(B[i, :, :], q, q) for i in range(B.shape[0])])
b2 = array([bilinearform(B[i,:,:], conjugate(q), linalg.solve(2*1j*w*eye(F.m) - J_coords, b1)) \
for i in range(B.shape[0])])
b3 = array(
[bilinearform(B[i, :, :], q, conjugate(q)) for i in range(B.shape[0])])
b4 = array([
bilinearform(B[i, :, :], q, linalg.solve(J_coords, b3))
for i in range(B.shape[0])
])
temp = array([
trilinearform(D[i, :, :, :], q, q, conjugate(q))
for i in range(D.shape[0])
]) + b2 - 2 * b4
l1 = 0.5 * real(matrixmultiply(conjugate(p), temp))
return l1
def measure_shift(self, checkimage):
"""Generate a check image and measure its offset from the reference
This is done using the same settings as the reference image.
Parameters
----------
checkimage : str
Image filename to compare to reference image
Returns
-------
solution_n_x : float
The shift required, in X, to recentre the checkimage into the reference frame
solution_n_y : float
The shift required, in Y, to recentre the checkimage into the reference frame
Raises
------
None
"""
self.checkimage = checkimage
h = fits.open(self.checkimage)
self.check_image_section = h[self.image_ext].data[:, self.prescan_width:-self.overscan_width]
self.check_data = self.check_image_section[self.boarder:self.dimy - self.boarder,
self.boarder:self.dimx - self.boarder]
# adjust image if requested - same as reference
if self.subtract_bkg:
self.check_bkgmap = self.__generate_bkg_map(self.check_data, self.ntiles,
self.tilesizex, self.tilesizey)
self.check_data = self.check_data - self.check_bkgmap
if self.normalise:
self.check_data = self.check_data / self.texp
self.check_xproj = np.sum(self.check_data, axis=0)
self.check_yproj = np.sum(self.check_data, axis=1)
# FFT of the projection spectra
f_ref_xproj_n = fft(self.ref_xproj)
f_ref_yproj_n = fft(self.ref_yproj)
f_check_xproj_n = fft(self.check_xproj)
f_check_yproj_n = fft(self.check_yproj)
# cross correlate in and look for the maximium correlation
f_ref_xproj_conj_n = conjugate(f_ref_xproj_n)
f_ref_yproj_conj_n = conjugate(f_ref_yproj_n)
complex_sum_x_n = f_ref_xproj_conj_n * f_check_xproj_n
complex_sum_y_n = f_ref_yproj_conj_n * f_check_yproj_n
phi_ref_check_m_x_n = ifft(complex_sum_x_n)
phi_ref_check_m_y_n = ifft(complex_sum_y_n)
z_x_n = max(phi_ref_check_m_x_n)
z_pos_x_n = np.where(phi_ref_check_m_x_n == z_x_n)
z_y_n = max(phi_ref_check_m_y_n)
z_pos_y_n = np.where(phi_ref_check_m_y_n == z_y_n)
# turn the location of the maximum into shift in pixels
# quadratically interpolate over the 3 pixels surrounding
# the peak in the CCF. This gives sub pixel resolution.
tst_y = np.empty(3)
tst_x = np.empty(3)
# X
if z_pos_x_n[0][0] <= len(phi_ref_check_m_x_n) / 2:
lra_x = [z_pos_x_n[0][0] - 1, z_pos_x_n[0][0], z_pos_x_n[0][0] + 1]
tst_x[0] = phi_ref_check_m_x_n[lra_x[0]].real
tst_x[1] = phi_ref_check_m_x_n[lra_x[1]].real
tst_x[2] = phi_ref_check_m_x_n[lra_x[2]].real
coeffs_n_x = polyfit(lra_x, tst_x, 2)
self.solution_n_x = -(-coeffs_n_x[1] / (2 * coeffs_n_x[0]))
elif z_pos_x_n[0][0] > len(phi_ref_check_m_x_n) / 2 and z_pos_x_n[0][0] != len(phi_ref_check_m_x_n) - 1:
lra_x = [z_pos_x_n[0][0] - 1, z_pos_x_n[0][0], z_pos_x_n[0][0] + 1]
tst_x[0] = phi_ref_check_m_x_n[lra_x[0]].real
tst_x[1] = phi_ref_check_m_x_n[lra_x[1]].real
tst_x[2] = phi_ref_check_m_x_n[lra_x[2]].real
coeffs_n_x = polyfit(lra_x, tst_x, 2)
self.solution_n_x = len(phi_ref_check_m_x_n) + (coeffs_n_x[1] / (2 * coeffs_n_x[0]))
elif z_pos_x_n[0][0] == len(phi_ref_check_m_x_n) - 1:
lra_x = [-1, 0, 1]
tst_x[0] = phi_ref_check_m_x_n[-2].real
tst_x[1] = phi_ref_check_m_x_n[-1].real
tst_x[2] = phi_ref_check_m_x_n[0].real
coeffs_n_x = polyfit(lra_x, tst_x, 2)
self.solution_n_x = 1 + (coeffs_n_x[1] / (2 * coeffs_n_x[0]))
else: #if z_pos_x_n[0][0] == 0:
lra_x = [1, 0, -1]
tst_x[0] = phi_ref_check_m_x_n[-1].real
tst_x[1] = phi_ref_check_m_x_n[0].real
tst_x[2] = phi_ref_check_m_x_n[1].real
coeffs_n_x = polyfit(lra_x, tst_x, 2)
self.solution_n_x = -coeffs_n_x[1] / (2 * coeffs_n_x[0])
print("X: {0:.2f}".format(self.solution_n_x))
# Y
if z_pos_y_n[0][0] <= len(phi_ref_check_m_y_n)/2:
lra_y = [z_pos_y_n[0][0] - 1, z_pos_y_n[0][0], z_pos_y_n[0][0] + 1]
tst_y[0] = phi_ref_check_m_y_n[lra_y[0]].real
tst_y[1] = phi_ref_check_m_y_n[lra_y[1]].real
tst_y[2] = phi_ref_check_m_y_n[lra_y[2]].real
coeffs_n_y = polyfit(lra_y, tst_y, 2)
self.solution_n_y = -(-coeffs_n_y[1]/(2*coeffs_n_y[0]))
if z_pos_y_n[0][0] > len(phi_ref_check_m_y_n) / 2 and z_pos_y_n[0][0] != len(phi_ref_check_m_y_n) - 1:
lra_y = [z_pos_y_n[0][0]-1, z_pos_y_n[0][0], z_pos_y_n[0][0]+1]
tst_y[0] = phi_ref_check_m_y_n[lra_y[0]].real
tst_y[1] = phi_ref_check_m_y_n[lra_y[1]].real
tst_y[2] = phi_ref_check_m_y_n[lra_y[2]].real
coeffs_n_y = polyfit(lra_y, tst_y, 2)
self.solution_n_y = len(phi_ref_check_m_y_n) + (coeffs_n_y[1] / (2 * coeffs_n_y[0]))
if z_pos_y_n[0][0] == len(phi_ref_check_m_y_n) - 1:
lra_y = [-1, 0, 1]
tst_y[0] = phi_ref_check_m_y_n[-2].real
tst_y[1] = phi_ref_check_m_y_n[-1].real
tst_y[2] = phi_ref_check_m_y_n[0].real
coeffs_n_y = polyfit(lra_y, tst_y, 2)
self.solution_n_y = 1 + (coeffs_n_y[1] / (2 * coeffs_n_y[0]))
else: #if z_pos_y_n[0][0] == 0:
lra_y = [1, 0, -1]
tst_y[0] = phi_ref_check_m_y_n[-1].real
tst_y[1] = phi_ref_check_m_y_n[0].real
tst_y[2] = phi_ref_check_m_y_n[1].real
coeffs_n_y = polyfit(lra_y, tst_y, 2)
self.solution_n_y = -coeffs_n_y[1] / (2 * coeffs_n_y[0])
print("Y: {0:.2f}".format(self.solution_n_y))
return self.solution_n_x, self.solution_n_y
def evaluate_basis_at(self, grid, component, prefactor=False):
r"""Evaluate the basis functions :math:`\phi_k` recursively at the given nodes :math:`\gamma`.
:param grid: The grid :math:`\Gamma` containing the nodes :math:`\gamma`.
:type grid: A class having a :py:meth:`get_nodes(...)` method.
:param component: The index :math:`i` of a single component :math:`\Phi_i` to evaluate.
:param prefactor: Whether to include a factor of :math:`\frac{1}{\sqrt{\det(Q)}}`.
:type prefactor: Boolean, default is ``False``.
:return: A two-dimensional ndarray :math:`H` of shape :math:`(|\mathfrak{K}_i|, |\Gamma|)` where
the entry :math:`H[\mu(k), i]` is the value of :math:`\phi_k(\gamma_i)`.
"""
D = self._dimension
bas = self._basis_shapes[component]
bs = self._basis_sizes[component]
# The grid
grid = self._grid_wrap(grid)
nodes = grid.get_nodes()
nn = grid.get_number_nodes(overall=True)
# Allocate the storage array
phi = zeros((bs, nn), dtype=complexfloating)
# Precompute some constants
Pi = self.get_parameters(component=component)
q, p, Q, P, S = Pi
Qinv = inv(Q)
Qbar = conjugate(Q)
QQ = dot(Qinv, Qbar)
# Compute the ground state phi_0 via direct evaluation
mu0 = bas[tuple(D * [0])]
phi[mu0, :] = self._evaluate_phi0(component, nodes, prefactor=False)
# Compute all higher order states phi_k via recursion
for d in range(D):
# Iterator for all valid index vectors k
indices = bas.get_node_iterator(mode="chain", direction=d)
for k in indices:
# Current index vector
ki = vstack(k)
# Access predecessors
phim = zeros((D, nn), dtype=complexfloating)
for j, kpj in bas.get_neighbours(k, selection="backward"):
mukpj = bas[kpj]
phim[j, :] = phi[mukpj, :]
# Compute 3-term recursion
p1 = (nodes - q) * phi[bas[k], :]
p2 = sqrt(ki) * phim
t1 = sqrt(2.0 / self._eps**2) * dot(Qinv[d, :], p1)
t2 = dot(QQ[d, :], p2)
# Find multi-index where to store the result
kped = bas.get_neighbours(k, selection="forward", direction=d)
# Did we find this k?
if len(kped) > 0:
kped = kped[0]
# Store computed value
phi[bas[kped[1]], :] = (t1 - t2) / sqrt(ki[d] + 1.0)
if prefactor is True:
phi = phi / self._get_sqrt(component)(det(Q))
return phi
#!/home/paulk/software/bin/python from __future__ import division from sys import argv,exit,stderr import scipy import scipy.linalg as SLA # generate random values for X and Y X = scipy.transpose(scipy.mat(scipy.random.random((5,5)))) Y = scipy.transpose(scipy.mat(scipy.random.random(5))) print "X = \n",X print print "Y = \n",Y print print "-----------------------------------------" # use least squares to get B, the regression parameters A = SLA.lstsq(X,Y) print "scipy.lstsq: A = \n",A[0] print print "hermetian: A = \n",SLA.inv(scipy.conjugate(scipy.transpose(X))*X)*scipy.conjugate(scipy.transpose(X))*Y print print "non-hermetian: A = \n",SLA.inv(scipy.transpose(X)*X)*scipy.transpose(X)*Y print
def slim_recursion(self, grid, component, prefactor=False):
r"""Evaluate the Hagedorn wavepacket :math:`\Psi` at the given nodes :math:`\gamma`.
This routine is a slim version compared to the full basis evaluation. At every moment
we store only the data we really need to compute the next step until we hit the highest
order basis functions.
:param grid: The grid :math:`\Gamma` containing the nodes :math:`\gamma`.
:type grid: A class having a :py:meth:`get_nodes(...)` method.
:param component: The index :math:`i` of a single component :math:`\Phi_i` to evaluate.
:param prefactor: Whether to include a factor of :math:`\frac{1}{\sqrt{\det(Q)}}`.
:type prefactor: Boolean, default is ``False``.
:return: A list of arrays or a single array containing the values of the :math:`\Phi_i`
at the nodes :math:`\gamma`.
Note that this function does not include the global phase :math:`\exp(\frac{i S}{\varepsilon^2})`.
"""
D = self._dimension
# Precompute some constants
Pi = self.get_parameters(component=component)
q, p, Q, P, S = Pi
Qinv = inv(Q)
Qbar = conjugate(Q)
QQ = dot(Qinv, Qbar)
# The basis shape
bas = self._basis_shapes[component]
Z = tuple(D * [0])
# Book keeping
todo = []
newtodo = [Z]
olddelete = []
delete = []
tmp = {}
# The grid nodes
grid = self._grid_wrap(grid)
nn = grid.get_number_nodes(overall=True)
nodes = grid.get_nodes()
# Evaluate phi0
tmp[Z] = self._evaluate_phi0(component, nodes, prefactor=False)
psi = self._coefficients[component][bas[Z], 0] * tmp[Z]
# Iterate for higher order states
while len(newtodo) != 0:
# Delete results that never will be used again
for d in olddelete:
del tmp[d]
# Exchange queues
todo = newtodo
newtodo = []
olddelete = delete
delete = []
# Compute new results
for k in todo:
# Center stencil at node k
ki = vstack(k)
# Access predecessors
phim = zeros((D, nn), dtype=complexfloating)
for j, kpj in bas.get_neighbours(k, selection="backward"):
phim[j, :] = tmp[kpj]
# Compute the neighbours
for d, n in bas.get_neighbours(k, selection="forward"):
if n not in tmp.keys():
# Compute 3-term recursion
p1 = (nodes - q) * tmp[k]
p2 = sqrt(ki) * phim
t1 = sqrt(2.0 / self._eps**2) * dot(Qinv[d, :], p1)
t2 = dot(QQ[d, :], p2)
# Store computed value
tmp[n] = (t1 - t2) / sqrt(ki[d] + 1.0)
# And update the result
psi = psi + self._coefficients[component][bas[n],
0] * tmp[n]
newtodo.append(n)
delete.append(k)
if prefactor is True:
psi = psi / self._get_sqrt(component)(det(Q))
return psi
parcelTimeSeries = scipy.exp(1j *
(scipy.asmatrix(scipy.angle(parcelTimeSeries))))
########## Get cPLV needed for flips and weighting
cPLVArray = 1j * scipy.zeros(len(sourceIdentities),
dtype=float) # Initialize as zeros (complex).
for i, identity in enumerate(
sourceIdentities
): # Compute cPLV only of parcel source pairs of sources that belong to that parcel. One source belong to only one parcel.
if sourceIdentities[
i] >= 0: # Don't compute negative values. These should be sources not belonging to any parcel.
cPLVArray[i] = scipy.sum(
(scipy.asarray(parcelTimeSeries[identity])) *
scipy.conjugate(scipy.asarray(sourceTimeSeries[i])))
cPLVArray /= timeOutput # Normalize by samples. For debugging. Output doesn't change even if you don't do this.
########## Get weights and flip. This could be the output.
weights = scipy.zeros(len(sourceIdentities)) # Initialize as zeros
for i, cPLV in enumerate(cPLVArray):
weights[i] = scipy.real(cPLV)**2 * scipy.sign(
scipy.real(cPLV)) # Sign is the flip; weight (real part)^2
########## Create weighted inverse operator and normalize the norm of weighted inv op to match original inv op's norm.
weightedInvOp = np.dot(
scipy.eye(weights.shape[0]) * weights, inverseOperator
) # Multiply sensor dimension in inverseOperator by weight. This one would be the un-normalized operator.
def absq(x):
return x*sp.conjugate(x)
print "%.3f " % (time.clock() - t_solving), "secondes passed"
print "V_port = "
print V_port
print "I_port = "
print I_port
print "==" * 30
# In[]
t_cal_gain = time.clock()
direct = calGain()
D0 = np.max(direct[1][2]) / scipy.real(
scipy.sum(scipy.conjugate(V_port) * I_port)) * 2 * pi4
print "gain obtained"
print "%.3f " % (time.clock() - t_cal_gain), "secondes passed"
print "direct = ", D0
print "gain = ", 10. * np.log10(D0), 'dBi'
print "==" * 30
from mpl_toolkits.mplot3d import Axes3D
import matplotlib.pyplot as plt
import numpy as np
ths, phs, aug = direct[1]
ts, ps = np.meshgrid(ths, phs)
def evaluate_basis_at(self, grid, component, prefactor=False):
r"""Evaluate the basis functions :math:`\phi_k` recursively at the given nodes :math:`\gamma`.
:param grid: The grid :math:`\Gamma` containing the nodes :math:`\gamma`.
:type grid: A class having a :py:meth:`get_nodes(...)` method.
:param component: The index :math:`i` of a single component :math:`\Phi_i` to evaluate.
:param prefactor: Whether to include a factor of :math:`\frac{1}{\sqrt{\det(Q)}}`.
:type prefactor: Boolean, default is ``False``.
:return: A two-dimensional ndarray :math:`H` of shape :math:`(|\mathfrak{K}_i|, |\Gamma|)` where
the entry :math:`H[\mu(k), i]` is the value of :math:`\phi_k(\gamma_i)`.
"""
D = self._dimension
bas = self._basis_shapes[component]
bs = self._basis_sizes[component]
# The grid
grid = self._grid_wrap(grid)
nodes = grid.get_nodes()
nn = grid.get_number_nodes(overall=True)
# Allocate the storage array
phi = zeros((bs, nn), dtype=complexfloating)
# Precompute some constants
Pi = self.get_parameters(component=component)
q, p, Q, P, S = Pi
Qinv = inv(Q)
Qbar = conjugate(Q)
QQ = dot(Qinv, Qbar)
# Compute the ground state phi_0 via direct evaluation
mu0 = bas[tuple(D * [0])]
phi[mu0, :] = self._evaluate_phi0(component, nodes, prefactor=False)
# Compute all higher order states phi_k via recursion
for d in range(D):
# Iterator for all valid index vectors k
indices = bas.get_node_iterator(mode="chain", direction=d)
for k in indices:
# Current index vector
ki = vstack(k)
# Access predecessors
phim = zeros((D, nn), dtype=complexfloating)
for j, kpj in bas.get_neighbours(k, selection="backward"):
mukpj = bas[kpj]
phim[j, :] = phi[mukpj, :]
# Compute 3-term recursion
p1 = (nodes - q) * phi[bas[k], :]
p2 = sqrt(ki) * phim
t1 = sqrt(2.0 / self._eps**2) * dot(Qinv[d, :], p1)
t2 = dot(QQ[d, :], p2)
# Find multi-index where to store the result
kped = bas.get_neighbours(k, selection="forward", direction=d)
# Did we find this k?
if len(kped) > 0:
kped = kped[0]
# Store computed value
phi[bas[kped[1]], :] = (t1 - t2) / sqrt(ki[d] + 1.0)
if prefactor is True:
phi = phi / self._get_sqrt(component)(det(Q))
return phi
def slim_recursion(self, grid, component, prefactor=False):
r"""Evaluate the Hagedorn wavepacket :math:`\Psi` at the given nodes :math:`\gamma`.
This routine is a slim version compared to the full basis evaluation. At every moment
we store only the data we really need to compute the next step until we hit the highest
order basis functions.
:param grid: The grid :math:`\Gamma` containing the nodes :math:`\gamma`.
:type grid: A class having a :py:meth:`get_nodes(...)` method.
:param component: The index :math:`i` of a single component :math:`\Phi_i` to evaluate.
:param prefactor: Whether to include a factor of :math:`\frac{1}{\sqrt{\det(Q)}}`.
:type prefactor: Boolean, default is ``False``.
:return: A list of arrays or a single array containing the values of the :math:`\Phi_i`
at the nodes :math:`\gamma`.
Note that this function does not include the global phase :math:`\exp(\frac{i S}{\varepsilon^2})`.
"""
D = self._dimension
# Precompute some constants
Pi = self.get_parameters(component=component)
q, p, Q, P, S = Pi
Qinv = inv(Q)
Qbar = conjugate(Q)
QQ = dot(Qinv, Qbar)
# The basis shape
bas = self._basis_shapes[component]
Z = tuple(D * [0])
# Book keeping
todo = []
newtodo = [Z]
olddelete = []
delete = []
tmp = {}
# The grid nodes
grid = self._grid_wrap(grid)
nn = grid.get_number_nodes(overall=True)
nodes = grid.get_nodes()
# Evaluate phi0
tmp[Z] = self._evaluate_phi0(component, nodes, prefactor=False)
psi = self._coefficients[component][bas[Z], 0] * tmp[Z]
# Iterate for higher order states
while len(newtodo) != 0:
# Delete results that never will be used again
for d in olddelete:
del tmp[d]
# Exchange queues
todo = newtodo
newtodo = []
olddelete = delete
delete = []
# Compute new results
for k in todo:
# Center stencil at node k
ki = vstack(k)
# Access predecessors
phim = zeros((D, nn), dtype=complexfloating)
for j, kpj in bas.get_neighbours(k, selection="backward"):
phim[j, :] = tmp[kpj]
# Compute the neighbours
for d, n in bas.get_neighbours(k, selection="forward"):
if n not in tmp.keys():
# Compute 3-term recursion
p1 = (nodes - q) * tmp[k]
p2 = sqrt(ki) * phim
t1 = sqrt(2.0 / self._eps**2) * dot(Qinv[d, :], p1)
t2 = dot(QQ[d, :], p2)
# Store computed value
tmp[n] = (t1 - t2) / sqrt(ki[d] + 1.0)
# And update the result
psi = psi + self._coefficients[component][bas[n], 0] * tmp[n]
newtodo.append(n)
delete.append(k)
if prefactor is True:
psi = psi / self._get_sqrt(component)(det(Q))
return psi
def genRadiationPattern2(ths_phs):
try:
# print ths_phs
RPs = [0 for _ in ths_phs]
# 定义一个辅助空间保存各个单元的因子，其作用是利用向量乘积的办法计算远处的电场
rr = 100.0
# 对方向角度中的集合进行迭代
brs = np.zeros([len(ths_phs), 3])
for iid, th_ph in enumerate(ths_phs):
th0, ph0 = th_ph # 俯仰角和方位角
# 根据俯仰角和方位角计算三个向量
try:
th0, ph0 = th_ph # 俯仰角和方位角
# 根据俯仰角和方位角计算三个向量
if th0 >= 0 and th0 <= scipy.pi:
br = np.array([
scipy.sin(th0) * scipy.cos(ph0),
scipy.sin(th0) * scipy.sin(ph0),
scipy.cos(th0)
]) * rr # 1.j*k向量
elif th0 < 0 and th0 >= -scipy.pi:
th0 = -th0
ph0 = ph0 + scipy.pi
br = np.array([
scipy.sin(th0) * scipy.cos(ph0),
scipy.sin(th0) * scipy.sin(ph0),
scipy.cos(th0)
]) * rr # 1.j*k向量
elif th0 > scipy.pi and th0 <= scipy.pi * 2.0:
th0 = th0 - scipy.pi
ph0 = ph0 + scipy.pi
br = np.array([
scipy.sin(th0) * scipy.cos(ph0),
scipy.sin(th0) * scipy.sin(ph0),
scipy.cos(th0)
]) * rr # 1.j*k向量
except Exception as e:
print e
raise
brs[iid] = br
EEs = scipy.array([calE_total(br_cell) for br_cell in brs])
HHs = scipy.array([calH_total(br_cell) for br_cell in brs])
WWs = scipy.array([
scipy.real(scipy.cross(EE, scipy.conjugate(HH)))
for EE, HH in zip(EEs, HHs)
]) * 0.5
RPs = scipy.array(
[rr * np.dot(WW, br_cell) for WW, br_cell in zip(WWs, brs)])
# for br in brs:
# # 计算电场
# EE = calE_total(br)
# # 计算磁场
# HH = calH_total(br)
# # 计算坡印廷矢量
# WW = scipy.real(scipy.cross(EE,scipy.conjugate(HH)))*0.5
# # 计算功率密度
# UU = rr*np.dot(WW,br)
# # 保存，并返回
# RPs[iid] = UU
return RPs
except Exception as e:
print e
print ths_phs
print np.shape(brs)
print br
raise
def integrand(x):
return sp.conjugate( f(x) ) * f(x) * weight(x);
