def _steadystate_eigen(L, ss_args):
"""
Internal function for solving the steady state problem by
finding the eigenvector corresponding to the zero eigenvalue
of the Liouvillian using ARPACK.
"""
if settings.debug:
print('Starting Eigen solver...')
dims = L.dims[0]
shape = prod(dims[0])
L = L.data.tocsc()
if ss_args['use_rcm']:
if settings.debug:
old_band = sp_bandwidth(L)[0]
print('Original bandwidth:', old_band)
perm = reverse_cuthill_mckee(L)
rev_perm = np.argsort(perm)
L = sp_permute(L, perm, perm, 'csc')
if settings.debug:
rcm_band = sp_bandwidth(L)[0]
print('RCM bandwidth:', rcm_band)
print('Bandwidth reduction factor:', round(old_band/rcm_band, 1))
eigval, eigvec = eigs(L, k=1, sigma=1e-15, tol=ss_args['tol'],
which='LM', maxiter=ss_args['maxiter'])
if ss_args['use_rcm']:
eigvec = eigvec[np.ix_(rev_perm,)]
data = vec2mat(eigvec)
data = 0.5 * (data + data.conj().T)
out = Qobj(data, dims=dims, isherm=True)
return out/out.tr()
def _steadystate_LU_liouvillian(L, ss_args, has_mkl=0):
"""Creates modified Liouvillian for LU based SS methods.
"""
perm = None
perm2 = None
rev_perm = None
n = int(np.sqrt(L.shape[0]))
form = 'csr'
if has_mkl:
L = L.data + sp.csr_matrix(
(ss_args['weight']*np.ones(n), (np.zeros(n), [nn * (n + 1)
for nn in range(n)])),
shape=(n ** 2, n ** 2))
else:
form = 'csc'
L = L.data.tocsc() + sp.csc_matrix(
(ss_args['weight']*np.ones(n), (np.zeros(n), [nn * (n + 1)
for nn in range(n)])),
shape=(n ** 2, n ** 2))
if settings.debug:
old_band = sp_bandwidth(L)[0]
old_pro = sp_profile(L)[0]
logger.debug('Orig. NNZ: %i' % L.nnz)
if ss_args['use_rcm']:
logger.debug('Original bandwidth: %i' % old_band)
if ss_args['use_wbm']:
if settings.debug:
logger.debug('Calculating Weighted Bipartite Matching ordering...')
_wbm_start = time.time()
perm = weighted_bipartite_matching(L)
_wbm_end = time.time()
L = sp_permute(L, perm, [], form)
ss_args['info']['perm'].append('wbm')
ss_args['info']['wbm_time'] = _wbm_end-_wbm_start
if settings.debug:
wbm_band = sp_bandwidth(L)[0]
logger.debug('WBM bandwidth: %i' % wbm_band)
if ss_args['use_rcm']:
if settings.debug:
logger.debug('Calculating Reverse Cuthill-Mckee ordering...')
_rcm_start = time.time()
perm2 = reverse_cuthill_mckee(L)
_rcm_end = time.time()
rev_perm = np.argsort(perm2)
L = sp_permute(L, perm2, perm2, form)
ss_args['info']['perm'].append('rcm')
ss_args['info']['rcm_time'] = _rcm_end-_rcm_start
if settings.debug:
rcm_band = sp_bandwidth(L)[0]
rcm_pro = sp_profile(L)[0]
logger.debug('RCM bandwidth: %i' % rcm_band)
logger.debug('Bandwidth reduction factor: %f' %
(old_band/rcm_band))
logger.debug('Profile reduction factor: %f' %
(old_pro/rcm_pro))
L.sort_indices()
return L, perm, perm2, rev_perm, ss_args
def test_sp_bandwidth():
"Sparse: Bandwidth"
# Bandwidth test 1
A = create(25)+destroy(25)+qeye(25)
band = sp_bandwidth(A.data)
assert_equal(band[0], 3)
assert_equal(band[1] == band[2] == 1, 1)
# Bandwidth test 2
A = np.array([[1, 0, 0, 0, 1, 0, 0, 0],
[0, 1, 1, 0, 0, 1, 0, 1],
[0, 1, 1, 0, 1, 0, 0, 0],
[0, 0, 0, 1, 0, 0, 1, 0],
[1, 0, 1, 0, 1, 0, 0, 0],
[0, 1, 0, 0, 0, 1, 0, 1],
[0, 0, 0, 1, 0, 0, 1, 0],
[0, 1, 0, 0, 0, 1, 0, 1]], dtype=np.int32)
A = sp.csr_matrix(A)
out1 = sp_bandwidth(A)
assert_equal(out1[0], 13)
assert_equal(out1[1] == out1[2] == 6, 1)
# Bandwidth test 3
perm = reverse_cuthill_mckee(A)
B = sp_permute(A, perm, perm)
out2 = sp_bandwidth(B)
assert_equal(out2[0], 5)
assert_equal(out2[1] == out2[2] == 2, 1)
# Asymmetric bandwidth test
A = destroy(25)+qeye(25)
out1 = sp_bandwidth(A.data)
assert_equal(out1[0], 2)
assert_equal(out1[1], 0)
assert_equal(out1[2], 1)
def test_sp_bandwidth():
"Sparse: Bandwidth"
# Bandwidth test 1
A = create(25) + destroy(25) + qeye(25)
band = sp_bandwidth(A.data)
assert_equal(band[0], 3)
assert_equal(band[1] == band[2] == 1, 1)
# Bandwidth test 2
A = np.array([[1, 0, 0, 0, 1, 0, 0, 0], [0, 1, 1, 0, 0, 1, 0, 1],
[0, 1, 1, 0, 1, 0, 0, 0], [0, 0, 0, 1, 0, 0, 1, 0],
[1, 0, 1, 0, 1, 0, 0, 0], [0, 1, 0, 0, 0, 1, 0, 1],
[0, 0, 0, 1, 0, 0, 1, 0], [0, 1, 0, 0, 0, 1, 0, 1]],
dtype=np.int32)
A = sp.csr_matrix(A)
out1 = sp_bandwidth(A)
assert_equal(out1[0], 13)
assert_equal(out1[1] == out1[2] == 6, 1)
# Bandwidth test 3
perm = reverse_cuthill_mckee(A)
B = sp_permute(A, perm, perm)
out2 = sp_bandwidth(B)
assert_equal(out2[0], 5)
assert_equal(out2[1] == out2[2] == 2, 1)
# Asymmetric bandwidth test
A = destroy(25) + qeye(25)
out1 = sp_bandwidth(A.data)
assert_equal(out1[0], 2)
assert_equal(out1[1], 0)
assert_equal(out1[2], 1)
def _steadystate_power_liouvillian(L, ss_args, has_mkl=0):
"""Creates modified Liouvillian for power based SS methods.
"""
perm = None
perm2 = None
rev_perm = None
n = L.shape[0]
if ss_args['solver'] == 'mkl':
L = L.data - (1e-15) * sp.eye(n, n, format='csr')
kind = 'csr'
else:
L = L.data.tocsc() - (1e-15) * sp.eye(n, n, format='csc')
kind = 'csc'
orig_nnz = L.nnz
if settings.debug:
old_band = sp_bandwidth(L)[0]
old_pro = sp_profile(L)[0]
logger.debug('Original bandwidth: %i' % old_band)
logger.debug('Original profile: %i' % old_pro)
if ss_args['use_wbm']:
if settings.debug:
logger.debug('Calculating Weighted Bipartite Matching ordering...')
_wbm_start = time.time()
perm = weighted_bipartite_matching(L)
_wbm_end = time.time()
L = sp_permute(L, perm, [], kind)
ss_args['info']['perm'].append('wbm')
ss_args['info']['wbm_time'] = _wbm_end-_wbm_start
if settings.debug:
wbm_band = sp_bandwidth(L)[0]
wbm_pro = sp_profile(L)[0]
logger.debug('WBM bandwidth: %i' % wbm_band)
logger.debug('WBM profile: %i' % wbm_pro)
if ss_args['use_rcm']:
if settings.debug:
logger.debug('Calculating Reverse Cuthill-Mckee ordering...')
ss_args['info']['perm'].append('rcm')
_rcm_start = time.time()
perm2 = reverse_cuthill_mckee(L)
_rcm_end = time.time()
ss_args['info']['rcm_time'] = _rcm_end-_rcm_start
rev_perm = np.argsort(perm2)
L = sp_permute(L, perm2, perm2, kind)
if settings.debug:
new_band = sp_bandwidth(L)[0]
new_pro = sp_profile(L)[0]
logger.debug('RCM bandwidth: %i' % new_band)
logger.debug('Bandwidth reduction factor: %f'
% (old_band/new_band))
logger.debug('RCM profile: %i' % new_pro)
logger.debug('Profile reduction factor: %f'
% (old_pro/new_pro))
L.sort_indices()
return L, perm, perm2, rev_perm, ss_args
def _pseudo_inverse_sparse(L,
rhoss,
method='splu',
use_umfpack=False,
use_rcm=False):
"""
Internal function for computing the pseudo inverse of an Liouvillian using
sparse matrix methods. See pseudo_inverse for details.
"""
N = np.prod(L.dims[0][0])
rhoss_vec = operator_to_vector(rhoss)
tr_op = tensor([identity(n) for n in L.dims[0][0]])
tr_op_vec = operator_to_vector(tr_op)
P = sp.kron(rhoss_vec.data, tr_op_vec.data.T, format='csc')
I = sp.eye(N * N, N * N, format='csc')
Q = I - P
if use_rcm:
perm = reverse_cuthill_mckee(L.data)
A = sp_permute(L.data, perm, perm, 'csc').tocsc()
Q = sp_permute(Q, perm, perm, 'csc')
permc_spec = 'NATURAL'
else:
A = L.data.tocsc()
A.sort_indices()
permc_spec = 'COLAMD'
if method == 'spsolve':
sp.linalg.use_solver(assumeSortedIndices=True, useUmfpack=use_umfpack)
LIQ = sp.linalg.spsolve(A, Q)
elif method == 'splu':
lu = sp.linalg.splu(A, permc_spec=permc_spec)
LIQ = lu.solve(Q.toarray())
elif method == 'spilu':
lu = sp.linalg.spilu(A,
permc_spec=permc_spec,
fill_factor=10,
drop_tol=1e-8)
LIQ = lu.solve(Q.toarray())
else:
raise ValueError("unsupported method '%s'" % method)
R = sp.csc_matrix(Q * LIQ)
if use_rcm:
rev_perm = np.argsort(perm)
R = sp_permute(R, rev_perm, rev_perm, 'csc')
return Qobj(R, dims=L.dims)
def _steadystate_power_liouvillian(L, ss_args, has_mkl=0):
"""Creates modified Liouvillian for power based SS methods.
"""
perm = None
perm2 = None
rev_perm = None
n = L.shape[0]
if ss_args['solver'] == 'mkl':
L = L.data - (1e-15) * sp.eye(n, n, format='csr')
kind = 'csr'
else:
L = L.data.tocsc() - (1e-15) * sp.eye(n, n, format='csc')
kind = 'csc'
orig_nnz = L.nnz
if settings.debug:
old_band = sp_bandwidth(L)[0]
old_pro = sp_profile(L)[0]
logger.debug('Original bandwidth: %i' % old_band)
logger.debug('Original profile: %i' % old_pro)
if ss_args['use_wbm']:
if settings.debug:
logger.debug('Calculating Weighted Bipartite Matching ordering...')
_wbm_start = time.time()
perm = weighted_bipartite_matching(L)
_wbm_end = time.time()
L = sp_permute(L, perm, [], kind)
ss_args['info']['perm'].append('wbm')
ss_args['info']['wbm_time'] = _wbm_end - _wbm_start
if settings.debug:
wbm_band = sp_bandwidth(L)[0]
wbm_pro = sp_profile(L)[0]
logger.debug('WBM bandwidth: %i' % wbm_band)
logger.debug('WBM profile: %i' % wbm_pro)
if ss_args['use_rcm']:
if settings.debug:
logger.debug('Calculating Reverse Cuthill-Mckee ordering...')
ss_args['info']['perm'].append('rcm')
_rcm_start = time.time()
perm2 = reverse_cuthill_mckee(L)
_rcm_end = time.time()
ss_args['info']['rcm_time'] = _rcm_end - _rcm_start
rev_perm = np.argsort(perm2)
L = sp_permute(L, perm2, perm2, kind)
if settings.debug:
new_band = sp_bandwidth(L)[0]
new_pro = sp_profile(L)[0]
logger.debug('RCM bandwidth: %i' % new_band)
logger.debug('Bandwidth reduction factor: %f' %
(old_band / new_band))
logger.debug('RCM profile: %i' % new_pro)
logger.debug('Profile reduction factor: %f' % (old_pro / new_pro))
L.sort_indices()
return L, perm, perm2, rev_perm, ss_args
def _pseudo_inverse_sparse(L, rhoss, method='splu', **pseudo_args):
"""
Internal function for computing the pseudo inverse of an Liouvillian using
sparse matrix methods. See pseudo_inverse for details.
"""
N = np.prod(L.dims[0][0])
rhoss_vec = operator_to_vector(rhoss)
tr_op = tensor([identity(n) for n in L.dims[0][0]])
tr_op_vec = operator_to_vector(tr_op)
P = sp.kron(rhoss_vec.data, tr_op_vec.data.T, format='csc')
I = sp.eye(N * N, N * N, format='csc')
Q = I - P
if pseudo_args['use_rcm']:
perm = reverse_cuthill_mckee(L.data)
A = sp_permute(L.data, perm, perm, 'csc').tocsc()
Q = sp_permute(Q, perm, perm, 'csc')
else:
A = L.data.tocsc()
A.sort_indices()
if method == 'spsolve':
sp.linalg.use_solver(assumeSortedIndices=True, useUmfpack=use_umfpack)
LIQ = sp.linalg.spsolve(A, Q)
elif method == 'splu':
lu = sp.linalg.splu(A,
permc_spec=pseudo_args['permc_spec'],
diag_pivot_thresh=pseudo_args['diag_pivot_thresh'],
options=dict(ILU_MILU=pseudo_args['ILU_MILU']))
LIQ = lu.solve(Q.toarray())
elif method == 'spilu':
lu = sp.linalg.spilu(A,
permc_spec=pseudo_args['permc_spec'],
fill_factor=pseudo_args['fill_factor'],
drop_tol=pseudo_args['drop_tol'])
LIQ = lu.solve(Q.toarray())
else:
raise ValueError("unsupported method '%s'" % method)
R = sp.csc_matrix(Q * LIQ)
if pseudo_args['use_rcm']:
rev_perm = np.argsort(perm)
R = sp_permute(R, rev_perm, rev_perm, 'csc')
return Qobj(R, dims=L.dims)
def _pseudo_inverse_sparse(L, rhoss, method='splu', **pseudo_args):
"""
Internal function for computing the pseudo inverse of an Liouvillian using
sparse matrix methods. See pseudo_inverse for details.
"""
N = np.prod(L.dims[0][0])
rhoss_vec = operator_to_vector(rhoss)
tr_op = tensor([identity(n) for n in L.dims[0][0]])
tr_op_vec = operator_to_vector(tr_op)
P = sp.kron(rhoss_vec.data, tr_op_vec.data.T, format='csr')
I = sp.eye(N*N, N*N, format='csr')
Q = I - P
if pseudo_args['use_rcm']:
perm = reverse_cuthill_mckee(L.data)
A = sp_permute(L.data, perm, perm, 'csr')
Q = sp_permute(Q, perm, perm, 'csr')
else:
if not settings.has_mkl:
A = L.data.tocsc()
A.sort_indices()
if method == 'splu':
if settings.has_mkl:
LIQ = mkl_spsolve(A,Q.toarray())
else:
lu = sp.linalg.splu(A, permc_spec=pseudo_args['permc_spec'],
diag_pivot_thresh=pseudo_args['diag_pivot_thresh'],
options=dict(ILU_MILU=pseudo_args['ILU_MILU']))
LIQ = lu.solve(Q.toarray())
elif method == 'spilu':
lu = sp.linalg.spilu(A, permc_spec=pseudo_args['permc_spec'],
fill_factor=pseudo_args['fill_factor'],
drop_tol=pseudo_args['drop_tol'])
LIQ = lu.solve(Q.toarray())
else:
raise ValueError("unsupported method '%s'" % method)
R = sp.csr_matrix(Q * LIQ)
if pseudo_args['use_rcm']:
rev_perm = np.argsort(perm)
R = sp_permute(R, rev_perm, rev_perm, 'csr')
return Qobj(R, dims=L.dims)
def _pseudo_inverse_sparse(L, rhoss, method='splu', use_umfpack=False,
use_rcm=False):
"""
Internal function for computing the pseudo inverse of an Liouvillian using
sparse matrix methods. See pseudo_inverse for details.
"""
N = np.prod(L.dims[0][0])
rhoss_vec = operator_to_vector(rhoss)
tr_op = tensor([identity(n) for n in L.dims[0][0]])
tr_op_vec = operator_to_vector(tr_op)
P = sp.kron(rhoss_vec.data, tr_op_vec.data.T, format='csc')
I = sp.eye(N*N, N*N, format='csc')
Q = I - P
if use_rcm:
perm = reverse_cuthill_mckee(L.data)
A = sp_permute(L.data, perm, perm, 'csc').tocsc()
Q = sp_permute(Q, perm, perm, 'csc')
permc_spec = 'NATURAL'
else:
A = L.data.tocsc()
A.sort_indices()
permc_spec = 'COLAMD'
if method == 'spsolve':
sp.linalg.use_solver(assumeSortedIndices=True, useUmfpack=use_umfpack)
LIQ = sp.linalg.spsolve(A, Q)
elif method == 'splu':
lu = sp.linalg.splu(A, permc_spec=permc_spec)
LIQ = lu.solve(Q.toarray())
elif method == 'spilu':
lu = sp.linalg.spilu(A, permc_spec=permc_spec,
fill_factor=10, drop_tol=1e-8)
LIQ = lu.solve(Q.toarray())
else:
raise ValueError("unsupported method '%s'" % method)
R = sp.csc_matrix(Q * LIQ)
if use_rcm:
rev_perm = np.argsort(perm)
R = sp_permute(R, rev_perm, rev_perm, 'csc')
return Qobj(R, dims=L.dims)
def _steadystate_LU_liouvillian(L, ss_args):
"""Creates modified Liouvillian for LU based SS methods.
"""
perm = None
perm2 = None
rev_perm = None
dims = L.dims[0]
n = prod(L.dims[0][0])
L = L.data.tocsc() + sp.csc_matrix(
(ss_args['weight']*np.ones(n), (np.zeros(n), [nn * (n + 1)
for nn in range(n)])),
shape=(n ** 2, n ** 2))
if settings.debug:
old_band = sp_bandwidth(L)[0]
print('NNZ:', L.nnz)
if ss_args['use_rcm']:
print('Original bandwidth:', old_band)
if ss_args['use_wbm']:
if settings.debug:
print('Calculating Weighted Bipartite Matching ordering...')
_wbm_start = time.time()
perm = weighted_bipartite_matching(L)
_wbm_end = time.time()
L = sp_permute(L, perm, [], 'csc')
ss_args['info']['perm'].append('wbm')
ss_args['info']['wbm_time'] = _wbm_end-_wbm_start
if settings.debug:
wbm_band = sp_bandwidth(L)[0]
print('WBM bandwidth:', wbm_band)
if ss_args['use_rcm']:
if settings.debug:
print('Calculating Reverse Cuthill-Mckee ordering...')
_rcm_start = time.time()
perm2 = reverse_cuthill_mckee(L)
_rcm_end = time.time()
rev_perm = np.argsort(perm2)
L = sp_permute(L, perm2, perm2, 'csc')
ss_args['info']['perm'].append('rcm')
ss_args['info']['rcm_time'] = _rcm_end-_rcm_start
if settings.debug:
rcm_band = sp_bandwidth(L)[0]
print('RCM bandwidth:', rcm_band)
print('Bandwidth reduction factor:', round(
old_band/rcm_band, 1))
L.sort_indices()
return L, perm, perm2, rev_perm, ss_args
def _steadystate_eigen(L, ss_args):
"""
Internal function for solving the steady state problem by
finding the eigenvector corresponding to the zero eigenvalue
of the Liouvillian using ARPACK.
"""
ss_args['info'].pop('weight', None)
if settings.debug:
logger.debug('Starting Eigen solver.')
dims = L.dims[0]
L = L.data.tocsc()
if ss_args['use_rcm']:
ss_args['info']['perm'].append('rcm')
if settings.debug:
old_band = sp_bandwidth(L)[0]
logger.debug('Original bandwidth: %i' % old_band)
perm = reverse_cuthill_mckee(L)
rev_perm = np.argsort(perm)
L = sp_permute(L, perm, perm, 'csc')
if settings.debug:
rcm_band = sp_bandwidth(L)[0]
logger.debug('RCM bandwidth: %i' % rcm_band)
logger.debug('Bandwidth reduction factor: %f' %
(old_band / rcm_band))
_eigen_start = time.time()
eigval, eigvec = eigs(L,
k=1,
sigma=1e-15,
tol=ss_args['tol'],
which='LM',
maxiter=ss_args['maxiter'])
_eigen_end = time.time()
ss_args['info']['solution_time'] = _eigen_end - _eigen_start
if ss_args['return_info']:
ss_args['info']['residual_norm'] = la.norm(L * eigvec)
if ss_args['use_rcm']:
eigvec = eigvec[np.ix_(rev_perm, )]
data = vec2mat(eigvec)
data = 0.5 * (data + data.conj().T)
out = Qobj(data, dims=dims, isherm=True)
if ss_args['return_info']:
return out / out.tr(), ss_args['info']
else:
return out / out.tr()
def _steadystate_LU_liouvillian(L, ss_args):
"""Creates modified Liouvillian for LU based SS methods.
"""
perm = None
perm2 = None
rev_perm = None
dims = L.dims[0]
n = prod(L.dims[0][0])
L = L.data.tocsc() + sp.csc_matrix(
(ss_args['weight'] * np.ones(n),
(np.zeros(n), [nn * (n + 1) for nn in range(n)])),
shape=(n**2, n**2))
if settings.debug:
old_band = sp_bandwidth(L)[0]
print('NNZ:', L.nnz)
if ss_args['use_rcm']:
print('Original bandwidth:', old_band)
if ss_args['use_wbm']:
if settings.debug:
print('Calculating Weighted Bipartite Matching ordering...')
_wbm_start = time.time()
perm = weighted_bipartite_matching(L)
_wbm_end = time.time()
L = sp_permute(L, perm, [], 'csc')
ss_args['info']['perm'].append('wbm')
ss_args['info']['wbm_time'] = _wbm_end - _wbm_start
if settings.debug:
wbm_band = sp_bandwidth(L)[0]
print('WBM bandwidth:', wbm_band)
if ss_args['use_rcm']:
if settings.debug:
print('Calculating Reverse Cuthill-Mckee ordering...')
_rcm_start = time.time()
perm2 = reverse_cuthill_mckee(L)
_rcm_end = time.time()
rev_perm = np.argsort(perm2)
L = sp_permute(L, perm2, perm2, 'csc')
ss_args['info']['perm'].append('rcm')
ss_args['info']['rcm_time'] = _rcm_end - _rcm_start
if settings.debug:
rcm_band = sp_bandwidth(L)[0]
print('RCM bandwidth:', rcm_band)
print('Bandwidth reduction factor:', round(old_band / rcm_band, 1))
L.sort_indices()
return L, perm, perm2, rev_perm, ss_args
def _steadystate_eigen(L, ss_args):
"""
Internal function for solving the steady state problem by
finding the eigenvector corresponding to the zero eigenvalue
of the Liouvillian using ARPACK.
"""
ss_args['info'].pop('weight', None)
if settings.debug:
logger.debug('Starting Eigen solver.')
dims = L.dims[0]
L = L.data.tocsc()
if ss_args['use_rcm']:
ss_args['info']['perm'].append('rcm')
if settings.debug:
old_band = sp_bandwidth(L)[0]
logger.debug('Original bandwidth: %i' % old_band)
perm = reverse_cuthill_mckee(L)
rev_perm = np.argsort(perm)
L = sp_permute(L, perm, perm, 'csc')
if settings.debug:
rcm_band = sp_bandwidth(L)[0]
logger.debug('RCM bandwidth: %i' % rcm_band)
logger.debug('Bandwidth reduction factor: %f' %
(old_band/rcm_band))
_eigen_start = time.time()
eigval, eigvec = eigs(L, k=1, sigma=1e-15, tol=ss_args['tol'],
which='LM', maxiter=ss_args['maxiter'])
_eigen_end = time.time()
ss_args['info']['solution_time'] = _eigen_end - _eigen_start
if ss_args['return_info']:
ss_args['info']['residual_norm'] = la.norm(L*eigvec, np.inf)
if ss_args['use_rcm']:
eigvec = eigvec[np.ix_(rev_perm,)]
_temp = vec2mat(eigvec)
data = dense2D_to_fastcsr_fmode(_temp, _temp.shape[0], _temp.shape[1])
data = 0.5 * (data + data.H)
out = Qobj(data, dims=dims, isherm=True)
if ss_args['return_info']:
return out/out.tr(), ss_args['info']
else:
return out/out.tr()
def _steadystate_power(L, ss_args):
"""
Inverse power method for steady state solving.
"""
ss_args['info'].pop('weight', None)
if settings.debug:
logger.debug('Starting iterative inverse-power method solver.')
tol = ss_args['tol']
maxiter = ss_args['maxiter']
use_solver(assumeSortedIndices=True)
rhoss = Qobj()
sflag = issuper(L)
if sflag:
rhoss.dims = L.dims[0]
else:
rhoss.dims = [L.dims[0], 1]
n = prod(rhoss.shape)
L = L.data.tocsc() - (1e-15) * sp.eye(n, n, format='csc')
L.sort_indices()
orig_nnz = L.nnz
# start with all ones as RHS
v = np.ones(n, dtype=complex)
if ss_args['use_rcm']:
if settings.debug:
old_band = sp_bandwidth(L)[0]
logger.debug('Original bandwidth: %i' % old_band)
perm = reverse_cuthill_mckee(L)
rev_perm = np.argsort(perm)
L = sp_permute(L, perm, perm, 'csc')
v = v[np.ix_(perm, )]
if settings.debug:
new_band = sp_bandwidth(L)[0]
logger.debug('RCM bandwidth: %i' % new_band)
logger.debug('Bandwidth reduction factor: %f' %
round(old_band / new_band, 2))
_power_start = time.time()
# Get LU factors
lu = splu(L,
permc_spec=ss_args['permc_spec'],
diag_pivot_thresh=ss_args['diag_pivot_thresh'],
options=dict(ILU_MILU=ss_args['ILU_MILU']))
if settings.debug and _scipy_check:
L_nnz = lu.L.nnz
U_nnz = lu.U.nnz
logger.debug('L NNZ: %i ; U NNZ: %i' % (L_nnz, U_nnz))
logger.debug('Fill factor: %f' % ((L_nnz + U_nnz) / orig_nnz))
it = 0
while (la.norm(L * v, np.inf) > tol) and (it < maxiter):
v = lu.solve(v)
v = v / la.norm(v, np.inf)
it += 1
if it >= maxiter:
raise Exception('Failed to find steady state after ' + str(maxiter) +
' iterations')
_power_end = time.time()
ss_args['info']['solution_time'] = _power_end - _power_start
ss_args['info']['iterations'] = it
if ss_args['return_info']:
ss_args['info']['residual_norm'] = la.norm(L * v, np.inf)
if settings.debug:
logger.debug('Number of iterations: %i' % it)
if ss_args['use_rcm']:
v = v[np.ix_(rev_perm, )]
# normalise according to type of problem
if sflag:
trow = sp.eye(rhoss.shape[0], rhoss.shape[0], format='coo')
trow = sp_reshape(trow, (1, n))
data = v / sum(trow.dot(v))
else:
data = data / la.norm(v)
data = sp.csr_matrix(vec2mat(data))
rhoss.data = 0.5 * (data + data.conj().T)
rhoss.isherm = True
if ss_args['return_info']:
return rhoss, ss_args['info']
else:
return rhoss
def _steadystate_iterative(L, ss_args):
"""
Iterative steady state solver using the GMRES or LGMRES algorithm
and a sparse incomplete LU preconditioner.
"""
if settings.debug:
print('Starting '+ss_args['method']+' solver...')
dims = L.dims[0]
n = prod(L.dims[0][0])
b = np.zeros(n ** 2)
b[0] = ss_args['weight']
L = L.data.tocsc() + sp.csc_matrix(
(ss_args['weight']*np.ones(n), (np.zeros(n), [nn * (n + 1)
for nn in range(n)])),
shape=(n ** 2, n ** 2))
if ss_args['use_wbm']:
perm = weighted_bipartite_matching(L)
L = sp_permute(L, perm, [], 'csc')
b = b[np.ix_(perm,)]
if ss_args['use_rcm']:
if settings.debug:
old_band = sp_bandwidth(L)[0]
print('Original bandwidth ', old_band)
perm2 = reverse_cuthill_mckee(L)
rev_perm = np.argsort(perm2)
L = sp_permute(L, perm2, perm2, 'csc')
b = b[np.ix_(perm2,)]
if settings.debug:
rcm_band = sp_bandwidth(L)[0]
print('RCM bandwidth ', rcm_band)
print('Bandwidth reduction factor:', round(old_band/rcm_band, 1))
L.sort_indices()
if ss_args['M'] is None and ss_args['use_precond']:
ss_args['M'] = _iterative_precondition(L, n, ss_args)
# Select iterative solver type
if ss_args['method'] == 'iterative-gmres':
v, check = gmres(L, b, tol=ss_args['tol'], M=ss_args['M'],
maxiter=ss_args['maxiter'])
elif ss_args['method'] == 'iterative-lgmres':
v, check = lgmres(L, b, tol=ss_args['tol'], M=ss_args['M'],
maxiter=ss_args['maxiter'])
else:
raise Exception("Invalid iterative solver method.")
if check > 0:
raise Exception("Steadystate solver did not reach tolerance after " +
str(check) + " steps.")
elif check < 0:
raise Exception(
"Steadystate solver failed with fatal error: " + str(check) + ".")
if ss_args['use_rcm']:
v = v[np.ix_(rev_perm,)]
data = vec2mat(v)
data = 0.5 * (data + data.conj().T)
return Qobj(data, dims=dims, isherm=True)
def _pseudo_inverse_sparse(L, rhoss, w=None, **pseudo_args):
"""
Internal function for computing the pseudo inverse of an Liouvillian using
sparse matrix methods. See pseudo_inverse for details.
"""
N = np.prod(L.dims[0][0])
rhoss_vec = operator_to_vector(rhoss)
tr_op = tensor([identity(n) for n in L.dims[0][0]])
tr_op_vec = operator_to_vector(tr_op)
P = zcsr_kron(rhoss_vec.data, tr_op_vec.data.T)
I = sp.eye(N * N, N * N, format='csr')
Q = I - P
if w is None:
L = 1.0j * (1e-15) * spre(tr_op) + L
else:
if w != 0.0:
L = 1.0j * w * spre(tr_op) + L
else:
L = 1.0j * (1e-15) * spre(tr_op) + L
if pseudo_args['use_rcm']:
perm = reverse_cuthill_mckee(L.data)
A = sp_permute(L.data, perm, perm)
Q = sp_permute(Q, perm, perm)
else:
if ss_args['solver'] == 'scipy':
A = L.data.tocsc()
A.sort_indices()
if pseudo_args['method'] == 'splu':
if settings.has_mkl:
A = L.data.tocsr()
A.sort_indices()
LIQ = mkl_spsolve(A, Q.toarray())
else:
pspec = pseudo_args['permc_spec']
diag_p_thresh = pseudo_args['diag_pivot_thresh']
pseudo_args = pseudo_args['ILU_MILU']
lu = sp.linalg.splu(A,
permc_spec=pspec,
diag_pivot_thresh=diag_p_thresh,
options=dict(ILU_MILU=pseudo_args))
LIQ = lu.solve(Q.toarray())
elif pseudo_args['method'] == 'spilu':
lu = sp.linalg.spilu(A,
permc_spec=pseudo_args['permc_spec'],
fill_factor=pseudo_args['fill_factor'],
drop_tol=pseudo_args['drop_tol'])
LIQ = lu.solve(Q.toarray())
else:
raise ValueError("unsupported method '%s'" % method)
R = sp.csr_matrix(Q * LIQ)
if pseudo_args['use_rcm']:
rev_perm = np.argsort(perm)
R = sp_permute(R, rev_perm, rev_perm, 'csr')
return Qobj(R, dims=L.dims)
def _steadystate_power(L, ss_args):
"""
Inverse power method for steady state solving.
"""
ss_args['info'].pop('weight', None)
if settings.debug:
print('Starting iterative inverse-power method solver...')
tol = ss_args['tol']
maxiter = ss_args['maxiter']
use_solver(assumeSortedIndices=True)
rhoss = Qobj()
sflag = issuper(L)
if sflag:
rhoss.dims = L.dims[0]
else:
rhoss.dims = [L.dims[0], 1]
n = prod(rhoss.shape)
L = L.data.tocsc() - (1e-15) * sp.eye(n, n, format='csc')
L.sort_indices()
orig_nnz = L.nnz
# start with all ones as RHS
v = np.ones(n,dtype=complex)
if ss_args['use_rcm']:
if settings.debug:
old_band = sp_bandwidth(L)[0]
print('Original bandwidth:', old_band)
perm = reverse_cuthill_mckee(L)
rev_perm = np.argsort(perm)
L = sp_permute(L, perm, perm, 'csc')
v = v[np.ix_(perm,)]
if settings.debug:
new_band = sp_bandwidth(L)[0]
print('RCM bandwidth:', new_band)
print('Bandwidth reduction factor:', round(old_band/new_band, 2))
# Get LU factors
lu = splu(L, permc_spec=ss_args['permc_spec'],
diag_pivot_thresh=ss_args['diag_pivot_thresh'],
options=dict(ILU_MILU=ss_args['ILU_MILU']))
if settings.debug and _scipy_check:
L_nnz = lu.L.nnz
U_nnz = lu.U.nnz
print('L NNZ:', L_nnz, ';', 'U NNZ:', U_nnz)
print('Fill factor:', (L_nnz+U_nnz)/orig_nnz)
_power_start = time.time()
it = 0
while (la.norm(L * v, np.inf) > tol) and (it < maxiter):
v = lu.solve(v)
v = v / la.norm(v, np.inf)
it += 1
if it >= maxiter:
raise Exception('Failed to find steady state after ' +
str(maxiter) + ' iterations')
_power_end = time.time()
ss_args['info']['solution_time'] = _power_end-_power_start
ss_args['info']['iterations'] = it
if settings.debug:
print('Number of iterations:', it)
if ss_args['use_rcm']:
v = v[np.ix_(rev_perm,)]
# normalise according to type of problem
if sflag:
trow = sp.eye(rhoss.shape[0], rhoss.shape[0], format='coo')
trow = sp_reshape(trow, (1, n))
data = v / sum(trow.dot(v))
else:
data = data / la.norm(v)
data = sp.csr_matrix(vec2mat(data))
rhoss.data = 0.5 * (data + data.conj().T)
rhoss.isherm = True
if ss_args['return_info']:
return rhoss, ss_args['info']
else:
return rhoss
def _steadystate_direct_sparse(L, ss_args):
"""
Direct solver that uses scipy sparse matrices
"""
if settings.debug:
print('Starting direct LU solver...')
dims = L.dims[0]
n = prod(L.dims[0][0])
b = np.zeros(n ** 2, dtype=complex)
b[0] = ss_args['weight']
L = L.data.tocsc() + sp.csc_matrix(
(ss_args['weight']*np.ones(n), (np.zeros(n), [nn * (n + 1)
for nn in range(n)])),
shape=(n ** 2, n ** 2))
L.sort_indices()
use_solver(assumeSortedIndices=True, useUmfpack=ss_args['use_umfpack'])
if not ss_args['use_umfpack']:
# Use superLU solver
orig_nnz = L.nnz
if settings.debug:
old_band = sp_bandwidth(L)[0]
print('Original NNZ:', orig_nnz)
if ss_args['use_rcm']:
print('Original bandwidth:', old_band)
if ss_args['use_wbm']:
perm = weighted_bipartite_matching(L)
L = sp_permute(L, perm, [], 'csc')
b = b[np.ix_(perm,)]
if ss_args['use_rcm']:
perm2 = reverse_cuthill_mckee(L)
rev_perm = np.argsort(perm2)
L = sp_permute(L, perm2, perm2, 'csc')
b = b[np.ix_(perm2,)]
if settings.debug:
rcm_band = sp_bandwidth(L)[0]
print('RCM bandwidth:', rcm_band)
print('Bandwidth reduction factor:', round(
old_band/rcm_band, 1))
lu = splu(L, permc_spec=ss_args['permc_spec'],
diag_pivot_thresh=ss_args['diag_pivot_thresh'],
options=dict(ILU_MILU=ss_args['ILU_MILU']))
v = lu.solve(b)
if settings.debug and _scipy_check:
L_nnz = lu.L.nnz
U_nnz = lu.U.nnz
print('L NNZ:', L_nnz, ';', 'U NNZ:', U_nnz)
print('Fill factor:', (L_nnz+U_nnz)/orig_nnz)
else:
# Use umfpack solver
v = spsolve(L, b)
if (not ss_args['use_umfpack']) and ss_args['use_rcm']:
v = v[np.ix_(rev_perm,)]
data = vec2mat(v)
data = 0.5 * (data + data.conj().T)
return Qobj(data, dims=dims, isherm=True)
