def _update_after_truncate(self, n_last_trunc, old_l, n_first_trunc,
old_r):
if self.canonical_form == 'right':
self.r[0][0, 0] = 1
for n in xrange(1, self.N):
self.l[n] = m.simple_diag_matrix(old_l[n].diag[-self.D[n]:],
dtype=self.typ)
self.l[self.N][0, 0] = 1
for n in xrange(self.N - 1, n_last_trunc - 1, -1):
self.r[n] = m.eyemat(self.D[n], dtype=self.typ)
self.calc_r(n_high=n_last_trunc - 1)
else:
self.l[0][0, 0] = 1
for n in xrange(1, self.N):
self.r[n] = m.simple_diag_matrix(old_r[n].diag[-self.D[n]:],
dtype=self.typ)
self.r[0][0, 0] = 1
for n in xrange(1, n_first_trunc):
self.l[n] = m.eyemat(self.D[n], dtype=self.typ)
self.calc_l(n_low=n_first_trunc)
self.simple_renorm()
def _update_after_truncate(self, n_last_trunc, old_l, n_first_trunc, old_r):
if self.canonical_form == 'right':
self.r[0][0, 0] = 1
for n in xrange(1, self.N):
self.l[n] = m.simple_diag_matrix(old_l[n].diag[-self.D[n]:], dtype=self.typ)
self.l[self.N][0, 0] = 1
for n in xrange(self.N - 1, n_last_trunc - 1, - 1):
self.r[n] = m.eyemat(self.D[n], dtype=self.typ)
self.calc_r(n_high=n_last_trunc - 1)
else:
self.l[0][0, 0] = 1
for n in xrange(1, self.N):
self.r[n] = m.simple_diag_matrix(old_r[n].diag[-self.D[n]:], dtype=self.typ)
self.r[0][0, 0] = 1
for n in xrange(1, n_first_trunc):
self.l[n] = m.eyemat(self.D[n], dtype=self.typ)
self.calc_l(n_low=n_first_trunc)
self.simple_renorm()
def restore_LCF_r(A, r, Gi, sanity_checks=False):
if Gi is None:
x = r
else:
x = Gi.dot(r.dot(Gi.conj().T))
M = eps_r_noop(x, A, A)
ev, EV = la.eigh(M) #wraps lapack routines, which return eigenvalues in ascending order
if sanity_checks:
assert np.all(ev == np.sort(ev)), "unexpected eigenvalue ordering"
rm1 = mm.simple_diag_matrix(ev, dtype=A.dtype)
Gm1 = EV.conj().T
if Gi is None:
Gi = EV #for left uniform case
r = rm1 #for sanity check
for s in xrange(A.shape[0]):
A[s] = Gm1.dot(A[s].dot(Gi))
if sanity_checks:
rm1_ = eps_r_noop(r, A, A)
if not sp.allclose(rm1_, rm1, atol=1E-12, rtol=1E-12):
log.warning("Sanity Fail in restore_LCF_r!: r is bad!")
log.warning(la.norm(rm1_ - rm1))
Gm1_i = EV
return rm1, Gm1, Gm1_i
def restore_LCF_r(A, r, Gi, sanity_checks=False):
if Gi is None:
x = r
else:
x = Gi.dot(r.dot(Gi.conj().T))
M = eps_r_noop(x, A, A)
ev, EV = la.eigh(
M) #wraps lapack routines, which return eigenvalues in ascending order
if sanity_checks:
assert np.all(ev == np.sort(ev)), "unexpected eigenvalue ordering"
rm1 = mm.simple_diag_matrix(ev, dtype=A.dtype)
Gm1 = EV.conj().T
if Gi is None:
Gi = EV #for left uniform case
r = rm1 #for sanity check
for s in xrange(A.shape[0]):
A[s] = Gm1.dot(A[s].dot(Gi))
if sanity_checks:
rm1_ = eps_r_noop(r, A, A)
if not sp.allclose(rm1_, rm1, atol=1E-12, rtol=1E-12):
log.warning("Sanity Fail in restore_LCF_r!: r is bad!")
log.warning(la.norm(rm1_ - rm1))
Gm1_i = EV
return rm1, Gm1, Gm1_i
def calc_BB_2s(Y, Vlh, Vrh_p1, l_si_m1, r_si_p1, dD_max=16, sv_tol=1E-14):
try:
U, sv, Vh = la.svd(Y)
except la.LinAlgError:
return None, None, 0
dDn = min(sp.count_nonzero(sv > sv_tol), dD_max)
sv = mm.simple_diag_matrix(sv[:dDn])
ss = sv.sqrt()
Z1 = ss.dot_left(U[:, :dDn])
Z2 = ss.dot(Vh[:dDn, :])
BB12n = sp.zeros((Vlh.shape[0], l_si_m1.shape[0], dDn), dtype=Y.dtype)
for s in xrange(Vlh.shape[0]):
BB12n[s] = l_si_m1.dot(Vlh[s].conj().T).dot(Z1)
BB21np1 = sp.zeros((Vrh_p1.shape[0], dDn, Vrh_p1.shape[1]), dtype=Y.dtype)
try:
for s in xrange(Vrh_p1.shape[0]):
BB21np1[s] = r_si_p1.dot_left(Z2.dot(Vrh_p1[s].conj().T))
except AttributeError:
for s in xrange(Vrh_p1.shape[0]):
BB21np1[s] = Z2.dot(Vrh_p1[s].conj().T).dot(r_si_p1)
return BB12n, BB21np1, dDn
def calc_BB_2s(Y, Vlh, Vrh_p1, l_si_m1, r_si_p1, dD_max=16, sv_tol=1E-14):
try:
U, sv, Vh = la.svd(Y)
except la.LinAlgError:
return None, None, 0
dDn = min(sp.count_nonzero(sv > sv_tol), dD_max)
sv = mm.simple_diag_matrix(sv[:dDn])
ss = sv.sqrt()
Z1 = ss.dot_left(U[:, :dDn])
Z2 = ss.dot(Vh[:dDn, :])
BB12n = sp.zeros((Vlh.shape[0], l_si_m1.shape[0], dDn), dtype=Y.dtype)
for s in xrange(Vlh.shape[0]):
BB12n[s] = l_si_m1.dot(Vlh[s].conj().T).dot(Z1)
BB21np1 = sp.zeros((Vrh_p1.shape[0], dDn, Vrh_p1.shape[1]), dtype=Y.dtype)
try:
for s in xrange(Vrh_p1.shape[0]):
BB21np1[s] = r_si_p1.dot_left(Z2.dot(Vrh_p1[s].conj().T))
except AttributeError:
for s in xrange(Vrh_p1.shape[0]):
BB21np1[s] = Z2.dot(Vrh_p1[s].conj().T).dot(r_si_p1)
return BB12n, BB21np1, dDn
def restore_LCF_l(A, lm1, Gm1, sanity_checks=False, zero_tol=1E-15):
if Gm1 is None:
GhGm1 = lm1
else:
GhGm1 = Gm1.conj().T.dot(lm1.dot(Gm1))
M = eps_l_noop(GhGm1, A, A)
G, Gi, new_D = herm_fac_with_inv(M,
zero_tol=zero_tol,
return_rank=True,
sanity_checks=sanity_checks)
if Gm1 is None:
Gm1 = G
if sanity_checks:
if new_D == A.shape[2]:
eye = sp.eye(A.shape[2])
else:
eye = mm.simple_diag_matrix(np.append(np.zeros(A.shape[2] - new_D),
np.ones(new_D)),
dtype=A.dtype)
if not sp.allclose(G.dot(Gi), eye, atol=1E-13, rtol=1E-13):
log.warning("Sanity Fail in restore_LCF_l!: Bad GT!")
for s in xrange(A.shape[0]):
A[s] = Gm1.dot(A[s]).dot(Gi)
if new_D == A.shape[2]:
l = mm.eyemat(A.shape[2], A.dtype)
else:
l = mm.simple_diag_matrix(np.append(np.zeros(A.shape[2] - new_D),
np.ones(new_D)),
dtype=A.dtype)
if sanity_checks:
lm1_ = mm.eyemat(A.shape[1], A.dtype)
l_ = eps_l_noop(lm1_, A, A)
if not sp.allclose(l_, l.A, atol=1E-13, rtol=1E-13):
log.warning("Sanity Fail in restore_LCF_l!: l is bad")
log.warning(la.norm(l_ - l))
return l, G, Gi
def restore_LCF_l(A, lm1, Gm1, sanity_checks=False, zero_tol=1E-15):
if Gm1 is None:
GhGm1 = lm1
else:
GhGm1 = Gm1.conj().T.dot(lm1.dot(Gm1))
M = eps_l_noop(GhGm1, A, A)
G, Gi, new_D = herm_fac_with_inv(M, zero_tol=zero_tol, return_rank=True,
sanity_checks=sanity_checks)
if Gm1 is None:
Gm1 = G
if sanity_checks:
if new_D == A.shape[2]:
eye = sp.eye(A.shape[2])
else:
eye = mm.simple_diag_matrix(np.append(np.zeros(A.shape[2] - new_D),
np.ones(new_D)), dtype=A.dtype)
if not sp.allclose(G.dot(Gi), eye, atol=1E-13, rtol=1E-13):
log.warning("Sanity Fail in restore_LCF_l!: Bad GT!")
for s in xrange(A.shape[0]):
A[s] = Gm1.dot(A[s]).dot(Gi)
if new_D == A.shape[2]:
l = mm.eyemat(A.shape[2], A.dtype)
else:
l = mm.simple_diag_matrix(np.append(np.zeros(A.shape[2] - new_D),
np.ones(new_D)), dtype=A.dtype)
if sanity_checks:
lm1_ = mm.eyemat(A.shape[1], A.dtype)
l_ = eps_l_noop(lm1_, A, A)
if not sp.allclose(l_, l.A, atol=1E-13, rtol=1E-13):
log.warning("Sanity Fail in restore_LCF_l!: l is bad")
log.warning(la.norm(l_ - l))
return l, G, Gi
def truncate(self, newD, update=True):
assert newD < self.D, 'new bond-dimension must be smaller!'
tmp_A = self.A
tmp_l = self.l.diag
self._init_arrays(newD, self.q)
if self.symm_gauge:
self.l = m.simple_diag_matrix(tmp_l[:self.D], dtype=self.typ)
self.r = m.simple_diag_matrix(tmp_l[:self.D], dtype=self.typ)
self.A = tmp_A[:, :self.D, :self.D]
else:
self.l = m.simple_diag_matrix(tmp_l[-self.D:], dtype=self.typ)
self.r = m.eyemat(self.D, dtype=self.typ)
self.A = tmp_A[:, -self.D:, -self.D:]
self.l_before_CF = self.l.A
self.r_before_CF = self.r.A
if update:
self.update()
def restore_RCF_l(self):
G_nm1 = None
l_nm1 = self.l[0]
for n in xrange(self.N + 1):
if n == 0:
x = l_nm1
else:
x = mm.mmul(mm.H(G_nm1), l_nm1, G_nm1)
M = self.eps_l(n, x)
ev, EV = la.eigh(M)
self.l[n] = mm.simple_diag_matrix(ev, dtype=self.typ)
G_n_i = EV
if n == 0:
G_nm1 = mm.H(EV) #for left uniform case
l_nm1 = self.l[n] #for sanity check
self.u_gnd_l.r = mm.mmul(G_nm1, self.u_gnd_l.r, G_n_i) #since r is not eye
for s in xrange(self.q[n]):
self.A[n][s] = mm.mmul(G_nm1, self.A[n][s], G_n_i)
if self.sanity_checks:
l = self.eps_l(n, l_nm1)
if not sp.allclose(l, self.l[n], atol=1E-12, rtol=1E-12):
print "Sanity Fail in restore_RCF_l!: l_%u is bad" % n
print la.norm(l - self.l[n])
G_nm1 = mm.H(EV)
l_nm1 = self.l[n]
if self.sanity_checks:
if not sp.allclose(sp.dot(G_nm1, G_n_i), sp.eye(G_n_i.shape[0]),
atol=1E-12, rtol=1E-12):
print "Sanity Fail in restore_RCF_l!: Bad GT for l_%u" % n
#Now G_nm1 = G_N
G_nm1_i = mm.H(G_nm1)
for s in xrange(self.q[self.N + 1]):
self.A[self.N + 1][s] = mm.mmul(G_nm1, self.A[self.N + 1][s], G_nm1_i)
##This should not be necessary if G_N is really unitary
#self.r[self.N] = mm.mmul(G_nm1, self.r[self.N], mm.H(G_nm1))
#self.r[self.N + 1] = self.r[self.N]
self.u_gnd_r.l[:] = mm.mmul(mm.H(G_nm1_i), self.u_gnd_r.l, G_nm1_i)
self.S_hc = sp.zeros((self.N), dtype=sp.complex128)
for n in xrange(1, self.N + 1):
self.S_hc[n-1] = -sp.sum(self.l[n].diag * sp.log2(self.l[n].diag))
def restore_SCF(self):
X = la.cholesky(self.r, lower=True)
Y = la.cholesky(self.l, lower=False)
U, sv, Vh = la.svd(Y.dot(X))
#s contains the Schmidt coefficients,
lam = sv**2
self.S_hc = - np.sum(lam * sp.log2(lam))
S = m.simple_diag_matrix(sv, dtype=self.typ)
Srt = S.sqrt()
g = m.mmul(Srt, Vh, m.invtr(X, lower=True))
g_i = m.mmul(m.invtr(Y, lower=False), U, Srt)
for s in xrange(self.q):
self.A[s] = m.mmul(g, self.A[s], g_i)
if self.sanity_checks:
Sfull = np.asarray(S)
if not np.allclose(g.dot(g_i), np.eye(self.D)):
print "Sanity check failed! Restore_SCF, bad GT!"
l = m.mmul(m.H(g_i), self.l, g_i)
r = m.mmul(g, self.r, m.H(g))
if not np.allclose(Sfull, l):
print "Sanity check failed: Restorce_SCF, left failed!"
if not np.allclose(Sfull, r):
print "Sanity check failed: Restorce_SCF, right failed!"
l = self.eps_l(Sfull)
r = self.eps_r(Sfull)
if not np.allclose(Sfull, l, rtol=self.itr_rtol*self.check_fac,
atol=self.itr_atol*self.check_fac):
print "Sanity check failed: Restorce_SCF, left bad!"
if not np.allclose(Sfull, r, rtol=self.itr_rtol*self.check_fac,
atol=self.itr_atol*self.check_fac):
print "Sanity check failed: Restorce_SCF, right bad!"
self.l = S
self.r = S
def herm_fac_with_inv(A, lower=False, zero_tol=1E-15, return_rank=False,
calc_inv=True, force_evd=False,
sanity_checks=False, sc_data=''):
"""Factorizes a Hermitian matrix using either Cholesky or eigenvalue decomposition.
Decomposes a Hermitian A as A = X*X or, if lower == True, A = XX*.
Tries Cholesky first by default, then falls back to EVD if the matrix is
not positive-definite. If Cholesky decomposition is used, X is upper (or lower)
triangular. For the EVD decomposition, the inverse becomes a pseudo-inverse
and all eigenvalues below the zero-tolerance are set to zero.
Parameters
----------
A : ndarray
The Hermitian matrix to be factorized.
lower : bool
Refers to Cholesky factorization. If True, factorize as A = XX*, otherwise as A = X*X
zero_tol : float
Tolerance for detection of zeros in EVD case.
return_rank : bool
Whether to return the rank of A. The detected rank is affected by zero_tol.
calc_inv : bool
Whether to calculate (and return) the inverse of the factor.
force_evd : bool
Whether to force eigenvalue instead of Cholesky decomposition.
sanity_checks : bool
Whether to perform soem basic sanity checks.
"""
if not force_evd:
try:
x = la.cholesky(A, lower=lower)
if calc_inv:
xi = mm.invtr(x, lower=lower)
else:
xi = None
nonzeros = A.shape[0]
except sp.linalg.LinAlgError: #this usually means a is not pos. def.
force_evd = True
if force_evd:
ev, EV = la.eigh(A, turbo=True) #wraps lapack routines, which return eigenvalues in ascending order
if sanity_checks:
assert np.all(ev == np.sort(ev)), "Sanity fail in herm_fac_with_inv(): Unexpected eigenvalue ordering"
if ev.min() < -zero_tol:
log.warning("Sanity fail in herm_fac_with_inv(): Discarding negative eigenvalues! %s %s",
ev.min(), sc_data)
nonzeros = np.count_nonzero(ev > zero_tol)
ev_sq = sp.zeros_like(ev, dtype=A.dtype)
ev_sq[-nonzeros:] = sp.sqrt(ev[-nonzeros:])
ev_sq = mm.simple_diag_matrix(ev_sq, dtype=A.dtype)
if calc_inv:
#Replace almost-zero values with zero and perform a pseudo-inverse
ev_sq_i = sp.zeros_like(ev, dtype=A.dtype)
ev_sq_i[-nonzeros:] = 1. / ev_sq[-nonzeros:]
ev_sq_i = mm.simple_diag_matrix(ev_sq_i, dtype=A.dtype)
xi = None
if lower:
x = ev_sq.dot_left(EV)
if calc_inv:
xi = ev_sq_i.dot(EV.conj().T)
else:
x = ev_sq.dot(EV.conj().T)
if calc_inv:
xi = ev_sq_i.dot_left(EV)
if sanity_checks:
if not sp.allclose(A, A.conj().T, atol=1E-13, rtol=1E-13):
log.warning("Sanity fail in herm_fac_with_inv(): A is not Hermitian! %s %s",
la.norm(A - A.conj().T), sc_data)
eye = sp.zeros((A.shape[0]), dtype=A.dtype)
eye[-nonzeros:] = 1
eye = mm.simple_diag_matrix(eye)
if lower:
if calc_inv:
if not sp.allclose(xi.dot(x), eye, atol=1E-13, rtol=1E-13):
log.warning("Sanity fail in herm_fac_with_inv(): Bad left inverse! %s %s",
la.norm(xi.dot(x) - eye), sc_data)
if not sp.allclose(xi.dot(A).dot(xi.conj().T), eye, atol=1E-13, rtol=1E-13):
log.warning("Sanity fail in herm_fac_with_inv(): Bad A inverse! %s %s",
la.norm(xi.conj().T.dot(A).dot(xi) - eye), sc_data)
if not sp.allclose(x.dot(x.conj().T), A, atol=1E-13, rtol=1E-13):
log.warning("Sanity fail in herm_fac_with_inv(): Bad decomp! %s %s",
la.norm(x.dot(x.conj().T) - A), sc_data)
else:
if calc_inv:
if not sp.allclose(x.dot(xi), eye, atol=1E-13, rtol=1E-13):
log.warning("Sanity fail in herm_fac_with_inv(): Bad right inverse! %s %s",
la.norm(x.dot(xi) - eye), sc_data)
if not sp.allclose(xi.conj().T.dot(A).dot(xi), eye, atol=1E-13, rtol=1E-13):
log.warning("Sanity fail in herm_fac_with_inv(): Bad A inverse! %s %s",
la.norm(xi.conj().T.dot(A).dot(xi) - eye), sc_data)
if not sp.allclose(x.conj().T.dot(x), A, atol=1E-13, rtol=1E-13):
log.warning("Sanity fail in herm_fac_with_inv(): Bad decomp! %s %s",
la.norm(x.conj().T.dot(x) - A), sc_data)
if calc_inv:
if return_rank:
return x, xi, nonzeros
else:
return x, xi
else:
if return_rank:
return x, nonzeros
else:
return x
def restore_RCF_l(A, lm1, Gm1, sanity_checks=False):
"""Transforms a single A[n] to obtain diagonal l[n].
Applied after restore_RCF_r(), this completes the full canonical form
of sub-section 3.1, theorem 1 of arXiv:quant-ph/0608197v2.
This function must be called for each n in turn, starting at 1,
passing the gauge transformation matrix from the previous step
as an argument.
Finds a G[n] such that orthonormalization is fulfilled for n.
The diagonal entries of l[n] are sorted in
ascending order (for example l[n] = diag([0, 0, 0.1, 0.2, ...])).
Parameters
----------
A : ndarray
The parameter tensor for the nth site A[n].
lm1 : ndarray or object with array interface
The matrix l[n - 1].
Gm1 : ndarray
The gauge transform matrix for site n obtained in the previous step (for n - 1).
sanity_checks : bool (False)
Whether to perform additional sanity checks.
Returns
-------
l : ndarray or simple_diag_matrix
The new, diagonal matrix l[n].
G : ndarray
The gauge transformation matrix for site n.
G_i : ndarray
Inverse of G.
"""
if Gm1 is None:
x = lm1
else:
x = Gm1.conj().T.dot(lm1.dot(Gm1))
M = eps_l_noop(x, A, A)
ev, EV = la.eigh(
M) #wraps lapack routines, which return eigenvalues in ascending order
if sanity_checks:
assert np.all(ev == np.sort(ev)), "unexpected eigenvalue ordering"
l = mm.simple_diag_matrix(ev, dtype=A.dtype)
G_i = EV
if Gm1 is None:
Gm1 = EV.conj().T #for left uniform case
lm1 = l #for sanity check
for s in xrange(A.shape[0]):
A[s] = Gm1.dot(A[s].dot(G_i))
if sanity_checks:
l_ = eps_l_noop(lm1, A, A)
if not sp.allclose(l_, l, atol=1E-12, rtol=1E-12):
log.warning("Sanity Fail in restore_RCF_l!: l is bad!")
log.warning(la.norm(l_ - l))
G = EV.conj().T
return l, G, G_i
def restore_RCF_r(A,
r,
G_n_i,
zero_tol=1E-15,
sanity_checks=False,
sc_data=''):
"""Transforms a single A[n] to obtain r[n - 1] = eye(D).
Implements the condition for right-orthonormalization from sub-section
3.1, theorem 1 of arXiv:quant-ph/0608197v2.
This function must be called for each n in turn, starting at N + 1,
passing the gauge transformation matrix from the previous step
as an argument.
Finds a G[n-1] such that orthonormalization is fulfilled for n.
If rank-deficiency is encountered, the result fulfills the orthonormality
condition in the occupied subspace with the zeros at the top-left
(for example r = diag([0, 0, 1, 1, 1, 1, 1])).
Parameters
----------
A : ndarray
The parameter tensor for the nth site A[n].
r : ndarray or object with array interface
The matrix r[n].
G_n_i : ndarray
The inverse gauge transform matrix for site n obtained in the previous step (for n + 1).
sanity_checks : bool (False)
Whether to perform additional sanity checks.
zero_tol : float
Tolerance for detecting zeros.
Returns
-------
r_nm1 : ndarray or simple_diag_matrix or eyemat
The new matrix r[n - 1].
G_nm1 : ndarray
The gauge transformation matrix for the site n - 1.
G_n_m1_i : ndarray
The inverse gauge transformation matrix for the site n - 1.
"""
if G_n_i is None:
GGh_n_i = r
else:
GGh_n_i = G_n_i.dot(r.dot(G_n_i.conj().T))
M = eps_r_noop(GGh_n_i, A, A)
X, Xi, new_D = herm_fac_with_inv(M,
zero_tol=zero_tol,
return_rank=True,
sanity_checks=sanity_checks)
G_nm1 = Xi.conj().T
G_nm1_i = X.conj().T
if G_n_i is None:
G_n_i = G_nm1_i
if sanity_checks:
#GiG may not be equal to eye in the case of rank-deficiency,
#but the rest should lie in the null space of A.
GiG = G_nm1_i.dot(G_nm1)
As = np.sum(A, axis=0)
if not sp.allclose(
GiG.dot(As).dot(G_n_i), As.dot(G_n_i), atol=1E-13, rtol=1E-13):
log.warning("Sanity Fail in restore_RCF_r!: Bad GT! %s %s",
la.norm(GiG.dot(As).dot(G_n_i) - As.dot(G_n_i)),
sc_data)
for s in xrange(A.shape[0]):
A[s] = G_nm1.dot(A[s]).dot(G_n_i)
if new_D == A.shape[1]:
r_nm1 = mm.eyemat(A.shape[1], A.dtype)
else:
r_nm1 = sp.zeros((A.shape[1]), dtype=A.dtype)
r_nm1[-new_D:] = 1
r_nm1 = mm.simple_diag_matrix(r_nm1, dtype=A.dtype)
if sanity_checks:
r_nm1_ = G_nm1.dot(M).dot(G_nm1.conj().T)
if not sp.allclose(r_nm1_, r_nm1.A, atol=1E-13, rtol=1E-13):
log.warning(
"Sanity Fail in restore_RCF_r!: r != g old_r gH! %s %s",
la.norm(r_nm1_ - r_nm1), sc_data)
r_nm1_ = eps_r_noop(r, A, A)
if not sp.allclose(r_nm1_, r_nm1.A, atol=1E-13, rtol=1E-13):
log.warning("Sanity Fail in restore_RCF_r!: r is bad! %s %s",
la.norm(r_nm1_ - r_nm1), sc_data)
return r_nm1, G_nm1, G_nm1_i
def herm_fac_with_inv(A,
lower=False,
zero_tol=1E-15,
return_rank=False,
calc_inv=True,
force_evd=False,
sanity_checks=False,
sc_data=''):
"""Factorizes a Hermitian matrix using either Cholesky or eigenvalue decomposition.
Decomposes a Hermitian A as A = X*X or, if lower == True, A = XX*.
Tries Cholesky first by default, then falls back to EVD if the matrix is
not positive-definite. If Cholesky decomposition is used, X is upper (or lower)
triangular. For the EVD decomposition, the inverse becomes a pseudo-inverse
and all eigenvalues below the zero-tolerance are set to zero.
Parameters
----------
A : ndarray
The Hermitian matrix to be factorized.
lower : bool
Refers to Cholesky factorization. If True, factorize as A = XX*, otherwise as A = X*X
zero_tol : float
Tolerance for detection of zeros in EVD case.
return_rank : bool
Whether to return the rank of A. The detected rank is affected by zero_tol.
calc_inv : bool
Whether to calculate (and return) the inverse of the factor.
force_evd : bool
Whether to force eigenvalue instead of Cholesky decomposition.
sanity_checks : bool
Whether to perform soem basic sanity checks.
"""
if not force_evd:
try:
x = la.cholesky(A, lower=lower)
if calc_inv:
xi = mm.invtr(x, lower=lower)
else:
xi = None
nonzeros = A.shape[0]
except sp.linalg.LinAlgError: #this usually means a is not pos. def.
force_evd = True
if force_evd:
ev, EV = la.eigh(
A, turbo=True
) #wraps lapack routines, which return eigenvalues in ascending order
if sanity_checks:
assert np.all(
ev == np.sort(ev)
), "Sanity fail in herm_fac_with_inv(): Unexpected eigenvalue ordering"
if ev.min() < -zero_tol:
log.warning(
"Sanity fail in herm_fac_with_inv(): Discarding negative eigenvalues! %s %s",
ev.min(), sc_data)
nonzeros = np.count_nonzero(ev > zero_tol)
ev_sq = sp.zeros_like(ev, dtype=A.dtype)
ev_sq[-nonzeros:] = sp.sqrt(ev[-nonzeros:])
ev_sq = mm.simple_diag_matrix(ev_sq, dtype=A.dtype)
if calc_inv:
#Replace almost-zero values with zero and perform a pseudo-inverse
ev_sq_i = sp.zeros_like(ev, dtype=A.dtype)
ev_sq_i[-nonzeros:] = 1. / ev_sq[-nonzeros:]
ev_sq_i = mm.simple_diag_matrix(ev_sq_i, dtype=A.dtype)
xi = None
if lower:
x = ev_sq.dot_left(EV)
if calc_inv:
xi = ev_sq_i.dot(EV.conj().T)
else:
x = ev_sq.dot(EV.conj().T)
if calc_inv:
xi = ev_sq_i.dot_left(EV)
if sanity_checks:
if not sp.allclose(A, A.conj().T, atol=1E-13, rtol=1E-13):
log.warning(
"Sanity fail in herm_fac_with_inv(): A is not Hermitian! %s %s",
la.norm(A - A.conj().T), sc_data)
eye = sp.zeros((A.shape[0]), dtype=A.dtype)
eye[-nonzeros:] = 1
eye = mm.simple_diag_matrix(eye)
if lower:
if calc_inv:
if not sp.allclose(xi.dot(x), eye, atol=1E-13, rtol=1E-13):
log.warning(
"Sanity fail in herm_fac_with_inv(): Bad left inverse! %s %s",
la.norm(xi.dot(x) - eye), sc_data)
if not sp.allclose(
xi.dot(A).dot(
xi.conj().T), eye, atol=1E-13, rtol=1E-13):
log.warning(
"Sanity fail in herm_fac_with_inv(): Bad A inverse! %s %s",
la.norm(xi.conj().T.dot(A).dot(xi) - eye), sc_data)
if not sp.allclose(x.dot(x.conj().T), A, atol=1E-13, rtol=1E-13):
log.warning(
"Sanity fail in herm_fac_with_inv(): Bad decomp! %s %s",
la.norm(x.dot(x.conj().T) - A), sc_data)
else:
if calc_inv:
if not sp.allclose(x.dot(xi), eye, atol=1E-13, rtol=1E-13):
log.warning(
"Sanity fail in herm_fac_with_inv(): Bad right inverse! %s %s",
la.norm(x.dot(xi) - eye), sc_data)
if not sp.allclose(
xi.conj().T.dot(A).dot(xi), eye, atol=1E-13,
rtol=1E-13):
log.warning(
"Sanity fail in herm_fac_with_inv(): Bad A inverse! %s %s",
la.norm(xi.conj().T.dot(A).dot(xi) - eye), sc_data)
if not sp.allclose(x.conj().T.dot(x), A, atol=1E-13, rtol=1E-13):
log.warning(
"Sanity fail in herm_fac_with_inv(): Bad decomp! %s %s",
la.norm(x.conj().T.dot(x) - A), sc_data)
if calc_inv:
if return_rank:
return x, xi, nonzeros
else:
return x, xi
else:
if return_rank:
return x, nonzeros
else:
return x
def restore_SCF(self, ret_g=False, zero_tol=None):
"""Restores symmetric canonical form.
In this canonical form, self.l == self.r and are diagonal matrices
with the Schmidt coefficients corresponding to the half-chain
decomposition form the diagonal entries.
Parameters
----------
ret_g : bool
Whether to return the gauge-transformation matrices used.
Returns
-------
g, g_i : ndarray
Gauge transformation matrix g and its inverse g_i.
"""
if zero_tol is None:
zero_tol = self.zero_tol
X, Xi = tm.herm_fac_with_inv(self.r, lower=True, zero_tol=zero_tol)
Y, Yi = tm.herm_fac_with_inv(self.l, lower=False, zero_tol=zero_tol)
U, sv, Vh = la.svd(Y.dot(X))
#s contains the Schmidt coefficients,
lam = sv**2
self.S_hc = - np.sum(lam * sp.log2(lam))
S = m.simple_diag_matrix(sv, dtype=self.typ)
Srt = S.sqrt()
g = m.mmul(Srt, Vh, Xi)
g_i = m.mmul(Yi, U, Srt)
for s in xrange(self.q):
self.A[s] = m.mmul(g, self.A[s], g_i)
if self.sanity_checks:
Sfull = np.asarray(S)
if not np.allclose(g.dot(g_i), np.eye(self.D)):
log.warning("Sanity check failed! Restore_SCF, bad GT!")
l = m.mmul(m.H(g_i), self.l, g_i)
r = m.mmul(g, self.r, m.H(g))
if not np.allclose(Sfull, l):
log.warning("Sanity check failed: Restorce_SCF, left failed!")
if not np.allclose(Sfull, r):
log.warning("Sanity check failed: Restorce_SCF, right failed!")
l = tm.eps_l_noop(Sfull, self.A, self.A)
r = tm.eps_r_noop(Sfull, self.A, self.A)
if not np.allclose(Sfull, l, rtol=self.itr_rtol*self.check_fac,
atol=self.itr_atol*self.check_fac):
log.warning("Sanity check failed: Restorce_SCF, left bad!")
if not np.allclose(Sfull, r, rtol=self.itr_rtol*self.check_fac,
atol=self.itr_atol*self.check_fac):
log.warning("Sanity check failed: Restorce_SCF, right bad!")
self.l = S
self.r = S
if ret_g:
return g, g_i
else:
return
def herm_sqrt_inv(x,
zero_tol=1E-15,
sanity_checks=False,
return_rank=False,
sc_data=''):
if isinstance(x, mm.eyemat):
x_sqrt = x
x_sqrt_i = x
rank = x.shape[0]
else:
try:
ev = x.diag #simple_diag_matrix
EV = None
except AttributeError:
ev, EV = la.eigh(x)
zeros = ev <= zero_tol #throw away negative results too!
ev_sqrt = sp.sqrt(ev)
err = sp.seterr(divide='ignore', invalid='ignore')
try:
ev_sqrt_i = 1 / ev_sqrt
ev_sqrt[zeros] = 0
ev_sqrt_i[zeros] = 0
finally:
sp.seterr(divide=err['divide'], invalid=err['invalid'])
if EV is None:
x_sqrt = mm.simple_diag_matrix(ev_sqrt, dtype=x.dtype)
x_sqrt_i = mm.simple_diag_matrix(ev_sqrt_i, dtype=x.dtype)
else:
B = mm.mmul_diag(ev_sqrt, EV.conj().T)
x_sqrt = EV.dot(B)
B = mm.mmul_diag(ev_sqrt_i, EV.conj().T)
x_sqrt_i = EV.dot(B)
rank = x.shape[0] - np.count_nonzero(zeros)
if sanity_checks:
if ev.min() < -zero_tol:
log.warning(
"Sanity Fail in herm_sqrt_inv(): Throwing away negative eigenvalues! %s %s",
ev.min(), sc_data)
if not np.allclose(x_sqrt.dot(x_sqrt), x):
log.warning(
"Sanity Fail in herm_sqrt_inv(): x_sqrt is bad! %s %s",
la.norm(x_sqrt.dot(x_sqrt) - x), sc_data)
if EV is None:
nulls = sp.zeros(x.shape[0])
nulls[zeros] = 1
nulls = sp.diag(nulls)
else: #if we did an EVD then we use the eigenvectors
nulls = EV.copy()
nulls[:, sp.invert(zeros)] = 0
nulls = nulls.dot(nulls.conj().T)
eye = np.eye(x.shape[0])
if not np.allclose(x_sqrt.dot(x_sqrt_i), eye - nulls):
log.warning(
"Sanity Fail in herm_sqrt_inv(): x_sqrt_i is bad! %s %s",
la.norm(x_sqrt.dot(x_sqrt_i) - eye + nulls), sc_data)
if return_rank:
return x_sqrt, x_sqrt_i, rank
else:
return x_sqrt, x_sqrt_i
def restore_RCF(self, ret_g=False, zero_tol=None):
"""Restores right canonical form.
In this form, self.r = sp.eye(self.D) and self.l is diagonal, with
the squared Schmidt coefficients corresponding to the half-chain
decomposition as eigenvalues.
Parameters
----------
ret_g : bool
Whether to return the gauge-transformation matrices used.
Returns
-------
g, g_i : ndarray
Gauge transformation matrix g and its inverse g_i.
"""
if zero_tol is None:
zero_tol = self.zero_tol
#First get G such that r = eye
G, G_i, rank = tm.herm_fac_with_inv(self.r, lower=True, zero_tol=zero_tol,
return_rank=True)
self.l = m.mmul(m.H(G), self.l, G)
#Now bring l into diagonal form, trace = 1 (guaranteed by r = eye..?)
ev, EV = la.eigh(self.l)
G = G.dot(EV)
G_i = m.H(EV).dot(G_i)
for s in xrange(self.q):
self.A[s] = m.mmul(G_i, self.A[s], G)
#ev contains the squares of the Schmidt coefficients,
self.S_hc = - np.sum(ev * sp.log2(ev))
self.l = m.simple_diag_matrix(ev, dtype=self.typ)
r_old = self.r
if rank == self.D:
self.r = m.eyemat(self.D, self.typ)
else:
self.r = sp.zeros((self.D), dtype=self.typ)
self.r[-rank:] = 1
self.r = m.simple_diag_matrix(self.r, dtype=self.typ)
if self.sanity_checks:
r_ = m.mmul(G_i, r_old, m.H(G_i))
if not np.allclose(self.r, r_,
rtol=self.itr_rtol*self.check_fac,
atol=self.itr_atol*self.check_fac):
log.warning("Sanity check failed: RestoreRCF, bad r (bad GT).")
l = tm.eps_l_noop(self.l, self.A, self.A)
r = tm.eps_r_noop(self.r, self.A, self.A)
if not np.allclose(r, self.r,
rtol=self.itr_rtol*self.check_fac,
atol=self.itr_atol*self.check_fac):
log.warning("Sanity check failed: Restore_RCF, r not eigenvector! %s", la.norm(r - self.r))
if not np.allclose(l, self.l,
rtol=self.itr_rtol*self.check_fac,
atol=self.itr_atol*self.check_fac):
log.warning("Sanity check failed: Restore_RCF, l not eigenvector! %s", la.norm(l - self.l))
if ret_g:
return G, G_i
else:
return
def herm_sqrt_inv(x, zero_tol=1E-15, sanity_checks=False, return_rank=False, sc_data=''):
if isinstance(x, mm.eyemat):
x_sqrt = x
x_sqrt_i = x
rank = x.shape[0]
else:
try:
ev = x.diag #simple_diag_matrix
EV = None
except AttributeError:
ev, EV = la.eigh(x)
zeros = ev <= zero_tol #throw away negative results too!
ev_sqrt = sp.sqrt(ev)
err = sp.seterr(divide='ignore', invalid='ignore')
try:
ev_sqrt_i = 1 / ev_sqrt
ev_sqrt[zeros] = 0
ev_sqrt_i[zeros] = 0
finally:
sp.seterr(divide=err['divide'], invalid=err['invalid'])
if EV is None:
x_sqrt = mm.simple_diag_matrix(ev_sqrt, dtype=x.dtype)
x_sqrt_i = mm.simple_diag_matrix(ev_sqrt_i, dtype=x.dtype)
else:
B = mm.mmul_diag(ev_sqrt, EV.conj().T)
x_sqrt = EV.dot(B)
B = mm.mmul_diag(ev_sqrt_i, EV.conj().T)
x_sqrt_i = EV.dot(B)
rank = x.shape[0] - np.count_nonzero(zeros)
if sanity_checks:
if ev.min() < -zero_tol:
log.warning("Sanity Fail in herm_sqrt_inv(): Throwing away negative eigenvalues! %s %s",
ev.min(), sc_data)
if not np.allclose(x_sqrt.dot(x_sqrt), x):
log.warning("Sanity Fail in herm_sqrt_inv(): x_sqrt is bad! %s %s",
la.norm(x_sqrt.dot(x_sqrt) - x), sc_data)
if EV is None:
nulls = sp.zeros(x.shape[0])
nulls[zeros] = 1
nulls = sp.diag(nulls)
else: #if we did an EVD then we use the eigenvectors
nulls = EV.copy()
nulls[:, sp.invert(zeros)] = 0
nulls = nulls.dot(nulls.conj().T)
eye = np.eye(x.shape[0])
if not np.allclose(x_sqrt.dot(x_sqrt_i), eye - nulls):
log.warning("Sanity Fail in herm_sqrt_inv(): x_sqrt_i is bad! %s %s",
la.norm(x_sqrt.dot(x_sqrt_i) - eye + nulls), sc_data)
if return_rank:
return x_sqrt, x_sqrt_i, rank
else:
return x_sqrt, x_sqrt_i
def restore_RCF_l(A, lm1, Gm1, sanity_checks=False):
"""Transforms a single A[n] to obtain diagonal l[n].
Applied after restore_RCF_r(), this completes the full canonical form
of sub-section 3.1, theorem 1 of arXiv:quant-ph/0608197v2.
This function must be called for each n in turn, starting at 1,
passing the gauge transformation matrix from the previous step
as an argument.
Finds a G[n] such that orthonormalization is fulfilled for n.
The diagonal entries of l[n] are sorted in
ascending order (for example l[n] = diag([0, 0, 0.1, 0.2, ...])).
Parameters
----------
A : ndarray
The parameter tensor for the nth site A[n].
lm1 : ndarray or object with array interface
The matrix l[n - 1].
Gm1 : ndarray
The gauge transform matrix for site n obtained in the previous step (for n - 1).
sanity_checks : bool (False)
Whether to perform additional sanity checks.
Returns
-------
l : ndarray or simple_diag_matrix
The new, diagonal matrix l[n].
G : ndarray
The gauge transformation matrix for site n.
G_i : ndarray
Inverse of G.
"""
if Gm1 is None:
x = lm1
else:
x = Gm1.conj().T.dot(lm1.dot(Gm1))
M = eps_l_noop(x, A, A)
ev, EV = la.eigh(M) #wraps lapack routines, which return eigenvalues in ascending order
if sanity_checks:
assert np.all(ev == np.sort(ev)), "unexpected eigenvalue ordering"
l = mm.simple_diag_matrix(ev, dtype=A.dtype)
G_i = EV
if Gm1 is None:
Gm1 = EV.conj().T #for left uniform case
lm1 = l #for sanity check
for s in xrange(A.shape[0]):
A[s] = Gm1.dot(A[s].dot(G_i))
if sanity_checks:
l_ = eps_l_noop(lm1, A, A)
if not sp.allclose(l_, l, atol=1E-12, rtol=1E-12):
log.warning("Sanity Fail in restore_RCF_l!: l is bad!")
log.warning(la.norm(l_ - l))
G = EV.conj().T
return l, G, G_i
def restore_RCF(self, dbg=False):
if dbg:
self.calc_l()
self.calc_r()
print "BEFORE..."
h_before, h_left_before, h_right_before = self.restore_RCF_dbg()
print (h_left_before, h_before, h_right_before)
self.restore_RCF_r()
if dbg:
self.calc_l()
print "MIDDLE..."
h_mid, h_left_mid, h_right_mid = self.restore_RCF_dbg()
print (h_left_mid, h_mid, h_right_mid)
fac = 1 / self.l[0].trace().real
if dbg:
print "Scale l[0]: %g" % fac
self.l[0] *= fac
self.u_gnd_l.r *= 1/fac
self.restore_RCF_l()
if dbg:
print "Uni left:"
self.u_gnd_l.A = self.A[0]
self.u_gnd_l.l = self.l[0]
self.u_gnd_l.calc_lr() #Ensures largest ev of E=1
self.l[0] = self.u_gnd_l.l #No longer diagonal!
self.A[0] = self.u_gnd_l.A
if self.sanity_checks:
if not sp.allclose(self.l[0], sp.diag(sp.diag(self.l[0])), atol=1E-12, rtol=1E-12):
print "Sanity Fail in restore_RCF!: True l[0] not diagonal!"
self.l[0] = mm.simple_diag_matrix(sp.diag(self.l[0]))
fac = 1 / sp.trace(self.l[0]).real
if dbg:
print "Scale l[0]: %g" % fac
self.l[0] *= fac
self.u_gnd_l.r *= 1/fac
self.u_gnd_l.l = self.l[0]
if dbg:
print "Uni right:"
self.u_gnd_r.A = self.A[self.N + 1]
self.u_gnd_r.r = self.r[self.N]
self.u_gnd_r.calc_lr() #Ensures largest ev of E=1
self.r[self.N] = self.u_gnd_r.r
self.A[self.N + 1] = self.u_gnd_r.A
if self.sanity_checks:
if not sp.allclose(self.r[self.N], sp.eye(self.D[self.N]), atol=1E-12, rtol=1E-12):
print "Sanity Fail in restore_RCF!: True r[N] not eye!"
self.r[self.N] = mm.eyemat(self.D[self.N], dtype=self.typ)
self.u_gnd_r.r = self.r[self.N]
self.r[self.N + 1] = self.r[self.N]
self.l[self.N + 1][:] = self.eps_l(self.N + 1, self.l[self.N])
if self.sanity_checks:
l_n = self.l[0]
for n in xrange(0, self.N + 1):
l_n = self.eps_l(n, l_n)
if not sp.allclose(l_n, self.l[n], atol=1E-12, rtol=1E-12):
print "Sanity Fail in restore_RCF!: l_%u is bad" % n
r_nm1 = self.r[self.N + 1]
for n in reversed(xrange(1, self.N + 2)):
r_nm1 = self.eps_r(n, r_nm1)
if not sp.allclose(r_nm1, self.r[n - 1], atol=1E-12, rtol=1E-12):
print "Sanity Fail in restore_RCF!: r_%u is bad" % (n - 1)
if dbg:
print "AFTER..."
h_after, h_left_after, h_right_after = self.restore_RCF_dbg()
print (h_left_after, h_after, h_right_after)
print h_after - h_before
print (h_after.sum() - h_before.sum() + h_left_after - h_left_before
+ h_right_after - h_right_before)
def restore_CF(self, ret_g=False):
if self.symm_gauge:
self.restore_SCF()
else:
#First get G such that r = eye
G = la.cholesky(self.r, lower=True)
G_i = m.invtr(G, lower=True)
self.l = m.mmul(m.H(G), self.l, G)
#Now bring l into diagonal form, trace = 1 (guaranteed by r = eye..?)
ev, EV = la.eigh(self.l)
G = G.dot(EV)
G_i = m.H(EV).dot(G_i)
for s in xrange(self.q):
self.A[s] = m.mmul(G_i, self.A[s], G)
#ev contains the squares of the Schmidt coefficients,
self.S_hc = - np.sum(ev * sp.log2(ev))
self.l = m.simple_diag_matrix(ev, dtype=self.typ)
if self.sanity_checks:
M = np.zeros_like(self.r)
for s in xrange(self.q):
M += m.mmul(self.A[s], m.H(self.A[s]))
self.r = m.mmul(G_i, self.r, m.H(G_i))
if not np.allclose(M, self.r,
rtol=self.itr_rtol*self.check_fac,
atol=self.itr_atol*self.check_fac):
print "Sanity check failed: RestoreRCF, bad M."
print "Off by: " + str(la.norm(M - self.r))
if not np.allclose(self.r, np.eye(self.D),
rtol=self.itr_rtol*self.check_fac,
atol=self.itr_atol*self.check_fac):
print "Sanity check failed: r not identity."
print "Off by: " + str(la.norm(np.eye(self.D) - self.r))
l = self.eps_l(self.l)
r = self.eps_r(self.r)
if not np.allclose(r, self.r,
rtol=self.itr_rtol*self.check_fac,
atol=self.itr_atol*self.check_fac):
print "Sanity check failed: Restore_RCF, bad r!"
print "Off by: " + str(la.norm(r - self.r))
if not np.allclose(l, self.l,
rtol=self.itr_rtol*self.check_fac,
atol=self.itr_atol*self.check_fac):
print "Sanity check failed: Restore_RCF, bad l!"
print "Off by: " + str(la.norm(l - self.l))
self.r = m.eyemat(self.D, dtype=self.typ)
if ret_g:
return G, G_i
else:
return
def restore_RCF_r(A, r, G_n_i, zero_tol=1E-15, sanity_checks=False, sc_data=''):
"""Transforms a single A[n] to obtain r[n - 1] = eye(D).
Implements the condition for right-orthonormalization from sub-section
3.1, theorem 1 of arXiv:quant-ph/0608197v2.
This function must be called for each n in turn, starting at N + 1,
passing the gauge transformation matrix from the previous step
as an argument.
Finds a G[n-1] such that orthonormalization is fulfilled for n.
If rank-deficiency is encountered, the result fulfills the orthonormality
condition in the occupied subspace with the zeros at the top-left
(for example r = diag([0, 0, 1, 1, 1, 1, 1])).
Parameters
----------
A : ndarray
The parameter tensor for the nth site A[n].
r : ndarray or object with array interface
The matrix r[n].
G_n_i : ndarray
The inverse gauge transform matrix for site n obtained in the previous step (for n + 1).
sanity_checks : bool (False)
Whether to perform additional sanity checks.
zero_tol : float
Tolerance for detecting zeros.
Returns
-------
r_nm1 : ndarray or simple_diag_matrix or eyemat
The new matrix r[n - 1].
G_nm1 : ndarray
The gauge transformation matrix for the site n - 1.
G_n_m1_i : ndarray
The inverse gauge transformation matrix for the site n - 1.
"""
if G_n_i is None:
GGh_n_i = r
else:
GGh_n_i = G_n_i.dot(r.dot(G_n_i.conj().T))
M = eps_r_noop(GGh_n_i, A, A)
X, Xi, new_D = herm_fac_with_inv(M, zero_tol=zero_tol, return_rank=True,
sanity_checks=sanity_checks)
G_nm1 = Xi.conj().T
G_nm1_i = X.conj().T
if G_n_i is None:
G_n_i = G_nm1_i
if sanity_checks:
#GiG may not be equal to eye in the case of rank-deficiency,
#but the rest should lie in the null space of A.
GiG = G_nm1_i.dot(G_nm1)
As = np.sum(A, axis=0)
if not sp.allclose(GiG.dot(As).dot(G_n_i),
As.dot(G_n_i), atol=1E-13, rtol=1E-13):
log.warning("Sanity Fail in restore_RCF_r!: Bad GT! %s %s",
la.norm(GiG.dot(As).dot(G_n_i) - As.dot(G_n_i)), sc_data)
for s in xrange(A.shape[0]):
A[s] = G_nm1.dot(A[s]).dot(G_n_i)
if new_D == A.shape[1]:
r_nm1 = mm.eyemat(A.shape[1], A.dtype)
else:
r_nm1 = sp.zeros((A.shape[1]), dtype=A.dtype)
r_nm1[-new_D:] = 1
r_nm1 = mm.simple_diag_matrix(r_nm1, dtype=A.dtype)
if sanity_checks:
r_nm1_ = G_nm1.dot(M).dot(G_nm1.conj().T)
if not sp.allclose(r_nm1_, r_nm1.A, atol=1E-13, rtol=1E-13):
log.warning("Sanity Fail in restore_RCF_r!: r != g old_r gH! %s %s",
la.norm(r_nm1_ - r_nm1), sc_data)
r_nm1_ = eps_r_noop(r, A, A)
if not sp.allclose(r_nm1_, r_nm1.A, atol=1E-13, rtol=1E-13):
log.warning("Sanity Fail in restore_RCF_r!: r is bad! %s %s",
la.norm(r_nm1_ - r_nm1), sc_data)
return r_nm1, G_nm1, G_nm1_i
