def _wigner_clenshaw(rho, xvec, yvec, g=sqrt(2), sparse=False):
r"""
Using Clenshaw summation - numerically stable and efficient
iterative algorithm to evaluate polynomial series.
The Wigner function is calculated as
:math:`W = e^(-0.5*x^2)/pi * \sum_{L} c_L (2x)^L / \sqrt(L!)` where
:math:`c_L = \sum_n \rho_{n,L+n} LL_n^L` where
:math:`LL_n^L = (-1)^n \sqrt(L!n!/(L+n)!) LaguerreL[n,L,x]`
"""
M = np.prod(rho.shape[0])
X, Y = np.meshgrid(xvec, yvec)
#A = 0.5 * g * (X + 1.0j * Y)
A2 = g * (X + 1.0j * Y) #this is A2 = 2*A
B = np.abs(A2)
B *= B
w0 = (2 * rho.data[0, -1]) * np.ones_like(A2)
L = M - 1
#calculation of \sum_{L} c_L (2x)^L / \sqrt(L!)
#using Horner's method
if not sparse:
rho = rho.full() * (2 * np.ones((M, M)) - np.diag(np.ones(M)))
while L > 0:
L -= 1
#here c_L = _wig_laguerre_val(L, B, np.diag(rho, L))
w0 = _wig_laguerre_val(L, B, np.diag(rho,
L)) + w0 * A2 * (L + 1)**-0.5
else:
while L > 0:
L -= 1
diag = _csr_get_diag(rho.data.data, rho.data.indices,
rho.data.indptr, L)
if L != 0:
diag *= 2
#here c_L = _wig_laguerre_val(L, B, np.diag(rho, L))
w0 = _wig_laguerre_val(L, B, diag) + w0 * A2 * (L + 1)**-0.5
return w0.real * np.exp(-B * 0.5) * (g * g * 0.5 / pi)
def _wigner_clenshaw(rho, xvec, yvec, g=sqrt(2), sparse=False):
"""
Using Clenshaw summation - numerically stable and efficient
iterative algorithm to evaluate polynomial series.
The Wigner function is calculated as
:math:`W = e^(-0.5*x^2)/pi * \sum_{L} c_L (2x)^L / sqrt(L!)` where
:math:`c_L = \sum_n \\rho_{n,L+n} LL_n^L` where
:math:`LL_n^L = (-1)^n sqrt(L!n!/(L+n)!) LaguerreL[n,L,x]`
"""
M = np.prod(rho.shape[0])
X,Y = np.meshgrid(xvec, yvec)
#A = 0.5 * g * (X + 1.0j * Y)
A2 = g * (X + 1.0j * Y) #this is A2 = 2*A
B = np.abs(A2)
B *= B
w0 = (2*rho.data[0,-1])*np.ones_like(A2)
L = M-1
#calculation of \sum_{L} c_L (2x)^L / sqrt(L!)
#using Horner's method
if not sparse:
rho = rho.full() * (2*np.ones((M,M)) - np.diag(np.ones(M)))
while L > 0:
L -= 1
#here c_L = _wig_laguerre_val(L, B, np.diag(rho, L))
w0 = _wig_laguerre_val(L, B, np.diag(rho, L)) + w0 * A2 * (L+1)**-0.5
else:
while L > 0:
L -= 1
diag = _csr_get_diag(rho.data.data,rho.data.indices,
rho.data.indptr,L)
if L != 0:
diag *= 2
#here c_L = _wig_laguerre_val(L, B, np.diag(rho, L))
w0 = _wig_laguerre_val(L, B, diag) + w0 * A2 * (L+1)**-0.5
return w0.real * np.exp(-B*0.5) * (g*g*0.5 / pi)