def rotation_from_projection(proj_nw, fixed, ortho=True):
"""Determine rotation and coefficient matrices from projections
proj_nw = <psi_n|p_w>
psi_n: eigenstates
p_w: localized function
Nb (n) = Number of bands
Nw (w) = Number of wannier functions
M (f) = Number of fixed states
L (l) = Number of extra degrees of freedom
U (u) = Number of non-fixed states
"""
Nb, Nw = proj_nw.shape
M = fixed
L = Nw - M
U_ww = np.empty((Nw, Nw), dtype=proj_nw.dtype)
U_ww[:M] = proj_nw[:M]
if L > 0:
proj_uw = proj_nw[M:]
eig_w, C_ww = np.linalg.eigh(np.dot(dag(proj_uw), proj_uw))
C_ul = np.dot(proj_uw, C_ww[:, np.argsort(-eig_w.real)[:L]])
# eig_u, C_uu = np.linalg.eigh(np.dot(proj_uw, dag(proj_uw)))
# C_ul = C_uu[:, np.argsort(-eig_u.real)[:L]]
U_ww[M:] = np.dot(dag(C_ul), proj_uw)
else:
C_ul = np.empty((Nb - M, 0))
normalize(C_ul)
if ortho:
lowdin(U_ww)
else:
normalize(U_ww)
return U_ww, C_ul
def rotation_from_projection(proj_nw, fixed, ortho=True):
"""Determine rotation and coefficient matrices from projections
proj_nw = <psi_n|p_w>
psi_n: eigenstates
p_w: localized function
Nb (n) = Number of bands
Nw (w) = Number of wannier functions
M (f) = Number of fixed states
L (l) = Number of extra degrees of freedom
U (u) = Number of non-fixed states
"""
Nb, Nw = proj_nw.shape
M = fixed
L = Nw - M
U_ww = np.empty((Nw, Nw), dtype=proj_nw.dtype)
U_ww[:M] = proj_nw[:M]
if L > 0:
proj_uw = proj_nw[M:]
eig_w, C_ww = np.linalg.eigh(np.dot(dag(proj_uw), proj_uw))
C_ul = np.dot(proj_uw, C_ww[:, np.argsort(-eig_w.real)[:L]])
# eig_u, C_uu = np.linalg.eigh(np.dot(proj_uw, dag(proj_uw)))
# C_ul = C_uu[:, np.argsort(-eig_u.real)[:L]]
U_ww[M:] = np.dot(dag(C_ul), proj_uw)
else:
C_ul = np.empty((Nb - M, 0))
normalize(C_ul)
if ortho:
lowdin(U_ww)
else:
normalize(U_ww)
return U_ww, C_ul
def normalization_error(matrix):
old_matrix = matrix.copy()
normalize(matrix)
return np.abs(matrix - old_matrix).max()