text: # This file is part of QuTiP: Quantum Toolbox in Python.
  #
  #    Copyright (c) 2011 and later, Paul D. Nation and Robert J. Johansson.
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  ###############################################################################
  """
  Module contains functions for solving for the steady state density matrix of
  open quantum systems defined by a Liouvillian or Hamiltonian and a list of
  collapse operators.
  """
  
  __all__ = ['steadystate', 'steady', 'build_preconditioner',
             'pseudo_inverse']
  
  import functools
  import time
  import warnings
  
  from packaging.version import parse as _parse_version
  import numpy as np
  from numpy.linalg import svd
  import scipy
  import scipy.sparse as sp
  import scipy.linalg as la
  from scipy.sparse.linalg import (
      use_solver, splu, spilu, eigs, LinearOperator, gmres, lgmres, bicgstab,
  )
  from qutip.qobj import Qobj, issuper, isoper
  
  from qutip.superoperator import liouvillian, vec2mat, spre
  from qutip.sparse import sp_permute, sp_bandwidth, sp_profile
  from qutip.cy.spmath import zcsr_kron
  from qutip.graph import weighted_bipartite_matching
  from qutip import (mat2vec, tensor, identity, operator_to_vector)
  import qutip.settings as settings
  from qutip.cy.spconvert import dense2D_to_fastcsr_fmode
  
  import qutip.logging_utils
  logger = qutip.logging_utils.get_logger()
  logger.setLevel('DEBUG')
  
  # Load MKL spsolve if avaiable
  if settings.has_mkl:
      from qutip._mkl.spsolve import (mkl_splu, mkl_spsolve)
  
  
  def _eat_kwargs(function, names):
      """
      Return a wrapped version of `function` that simply removes any keyword
      arguments with one of the given names.
      """
      @functools.wraps(function)
      def out(*args, **kwargs):
          for name in names:
              if name in kwargs:
                  del kwargs[name]
          return function(*args, **kwargs)
      return out
  
  
  # From SciPy 1.4 onwards we need to pass the `callback_type='legacy'` argument
  # to gmres to maintain the same behaviour we used to have.  Since this should
  # be the default behaviour, we use that in the main code and just "eat" the
  # argument if passed to a lower version of SciPy that doesn't know about it.
  # Similarly, SciPy < 1.1 does not recognise the `atol` keyword.
  #
  # Respective checks can be removed when SciPy version requirements are raised.
  
  if _parse_version(scipy.__version__) < _parse_version("1.1"):
      gmres = _eat_kwargs(gmres, ['atol', 'callback_type'])
      lgmres = _eat_kwargs(lgmres, ['atol'])
      bicgstab = _eat_kwargs(bicgstab, ['atol'])
  elif _parse_version(scipy.__version__) < _parse_version("1.4"):
      gmres = _eat_kwargs(gmres, ['callback_type'])
  
  
  def _empty_info_dict():
      def_info = {'perm': [], 'solution_time': None,
                  'residual_norm': None,
                  'solver': None, 'method': None}
  
      return def_info
  
  
  def _default_steadystate_args():
      def_args = {'sparse': True, 'use_rcm': False,
                  'use_wbm': False, 'use_precond': False,
                  'all_states': False, 'M': None, 'x0': None, 'drop_tol': 1e-4,
                  'fill_factor': 100, 'diag_pivot_thresh': None, 'maxiter': 1000,
                  'permc_spec': 'COLAMD', 'ILU_MILU': 'smilu_2',
                  'restart': 20,
                  'max_iter_refine': 10,
                  'scaling_vectors': True,
                  'weighted_matching': True,
                  'return_info': False, 'info': _empty_info_dict(),
                  'verbose': False, 'solver': 'scipy', 'weight': None,
                  'tol': 1e-12, 'matol': 1e-15, 'mtol': None}
      return def_args
  
  
  def steadystate(A, c_op_list=[], method='direct', solver=None, **kwargs):
      """
      Calculates the steady state for quantum evolution subject to the supplied
      Hamiltonian or Liouvillian operator and (if given a Hamiltonian) a list of
      collapse operators.
  
      If the user passes a Hamiltonian then it, along with the list of collapse
      operators, will be converted into a Liouvillian operator in Lindblad form.
  
      Parameters
      ----------
      A : :obj:`~Qobj`
          A Hamiltonian or Liouvillian operator.
  
      c_op_list : list
          A list of collapse operators.
  
      solver : {'scipy', 'mkl'}, optional
          Selects the sparse solver to use.  Default is to auto-select based on
          the availability of the MKL library.
  
      method : str, default 'direct'
          The allowed methods are
  
          - 'direct'
          - 'eigen'
          - 'iterative-gmres'
          - 'iterative-lgmres'
          - 'iterative-bicgstab'
          - 'svd'
          - 'power'
          - 'power-gmres'
          - 'power-lgmres'
          - 'power-bicgstab'
  
          Method for solving the underlying linear equation. Direct LU solver
          'direct' (default), sparse eigenvalue problem 'eigen', iterative GMRES
          method 'iterative-gmres', iterative LGMRES method 'iterative-lgmres',
          iterative BICGSTAB method 'iterative-bicgstab', SVD 'svd' (dense), or
          inverse-power method 'power'. The iterative power methods
          'power-gmres', 'power-lgmres', 'power-bicgstab' use the same solvers as
          their direct counterparts.
  
      return_info : bool, default False
          Return a dictionary of solver-specific infomation about the solution
          and how it was obtained.
  
      sparse : bool, default True
          Solve for the steady state using sparse algorithms. If set to False,
          the underlying Liouvillian operator will be converted into a dense
          matrix. Use only for 'smaller' systems.
  
      use_rcm : bool, default False
          Use reverse Cuthill-Mckee reordering to minimize fill-in in the LU
          factorization of the Liouvillian.
  
      use_wbm : bool, default False
          Use Weighted Bipartite Matching reordering to make the Liouvillian
          diagonally dominant.  This is useful for iterative preconditioners
          only, and is set to ``True`` by default when finding a preconditioner.
  
      weight : float, optional
          Sets the size of the elements used for adding the unity trace condition
          to the linear solvers.  This is set to the average abs value of the
          Liouvillian elements if not specified by the user.
  
      max_iter_refine : int, default 10
          MKL ONLY. Max. number of iterative refinements to perform.
  
      scaling_vectors : bool
          MKL ONLY.  Scale matrix to unit norm columns and rows.
  
      weighted_matching : bool
          MKL ONLY.  Use weighted matching to better condition diagonal.
  
      x0 : ndarray, optional
          ITERATIVE ONLY. Initial guess for solution vector.
  
      maxiter : int, default 1000
          ITERATIVE ONLY. Maximum number of iterations to perform.
  
      tol : float, default 1e-12
          ITERATIVE ONLY. Tolerance used for terminating solver.
  
      mtol : float, optional
          ITERATIVE 'power' methods ONLY. Tolerance for lu solve method.  If None
          given then ``max(0.1*tol, 1e-15)`` is used.
  
      matol : float, default 1e-15
          ITERATIVE ONLY. Absolute tolerance for lu solve method.
  
      permc_spec : str, optional
          ITERATIVE ONLY. Column ordering used internally by superLU for the
          'direct' LU decomposition method. Options include 'COLAMD' (default)
          and 'NATURAL'. If using RCM then this is set to 'NATURAL' automatically
          unless explicitly specified.
  
      use_precond : bool, default False
          ITERATIVE ONLY. Use an incomplete sparse LU decomposition as a
          preconditioner for the 'iterative' GMRES and BICG solvers.  Speeds up
          convergence time by orders of magnitude in many cases.
  
      M : {sparse matrix, dense matrix, LinearOperator}, optional
          ITERATIVE ONLY. Preconditioner for A. The preconditioner should
          approximate the inverse of A. Effective preconditioning can
          dramatically improve the rate of convergence for iterative methods.
          If no preconditioner is given and ``use_precond = True``, then one
          is generated automatically.
  
      fill_factor : float, default 100
          ITERATIVE ONLY. Specifies the fill ratio upper bound (>=1) of the iLU
          preconditioner.  Lower values save memory at the cost of longer
          execution times and a possible singular factorization.
  
      drop_tol : float, default 1e-4
          ITERATIVE ONLY. Sets the threshold for the magnitude of preconditioner
          elements that should be dropped.  Can be reduced for a courser
          factorization at the cost of an increased number of iterations, and a
          possible singular factorization.
  
      diag_pivot_thresh : float, optional
          ITERATIVE ONLY. Sets the threshold between [0,1] for which diagonal
          elements are considered acceptable pivot points when using a
          preconditioner.  A value of zero forces the pivot to be the diagonal
          element.
  
      ILU_MILU : str, default 'smilu_2'
          ITERATIVE ONLY. Selects the incomplete LU decomposition method algoithm
          used in creating the preconditoner. Should only be used by advanced
          users.
  
      Returns
      -------
      dm : qobj
          Steady state density matrix.
      info : dict, optional
          Dictionary containing solver-specific information about the solution.
  
      Notes
      -----
      The SVD method works only for dense operators (i.e. small systems).
      """
      if solver is None:
          solver = 'scipy'
          if settings.has_mkl:
              if method in ['direct', 'power']:
                  solver = 'mkl'
      elif solver == 'mkl' and \
              (method not in ['direct', 'power']):
          raise Exception('MKL solver only for direct or power methods.')
  
      elif solver not in ['scipy', 'mkl']:
          raise Exception('Invalid solver kwarg.')
  
      ss_args = _default_steadystate_args()
      ss_args['method'] = method
      if solver is not None:
          ss_args['solver'] = solver
      ss_args['info']['solver'] = ss_args['solver']
      ss_args['info']['method'] = ss_args['method']
  
      for key in kwargs.keys():
          if key in ss_args.keys():
              ss_args[key] = kwargs[key]
          else:
              raise Exception(
                  "Invalid keyword argument '"+key+"' passed to steadystate.")
  
      # Set column perm to NATURAL if using RCM and not specified by user
      if ss_args['use_rcm'] and ('permc_spec' not in kwargs.keys()):
          ss_args['permc_spec'] = 'NATURAL'
  
      # Create & check Liouvillian
      A = _steadystate_setup(A, c_op_list)
  
      # Set weight parameter to avg abs val in L if not set explicitly
      if 'weight' not in kwargs.keys():
          ss_args['weight'] = np.mean(np.abs(A.data.data.max()))
          ss_args['info']['weight'] = ss_args['weight']
  
      if ss_args['method'] == 'direct':
          if (ss_args['solver'] == 'scipy' and ss_args['sparse']) \
                  or ss_args['solver'] == 'mkl':
              return _steadystate_direct_sparse(A, ss_args)
          else:
              return _steadystate_direct_dense(A, ss_args)
  
      elif ss_args['method'] == 'eigen':
          return _steadystate_eigen(A, ss_args)
  
      elif ss_args['method'] in ['iterative-gmres',
                                 'iterative-lgmres', 'iterative-bicgstab']:
          return _steadystate_iterative(A, ss_args)
  
      elif ss_args['method'] == 'svd':
          return _steadystate_svd_dense(A, ss_args)
  
      elif ss_args['method'] in ['power', 'power-gmres',
                                 'power-lgmres', 'power-bicgstab']:
          return _steadystate_power(A, ss_args)
  
      else:
          raise ValueError('Invalid method argument for steadystate.')
  
  
  def _steadystate_setup(A, c_op_list):
      """Build Liouvillian (if necessary) and check input.
      """
      if isoper(A):
          if len(c_op_list) > 0:
              return liouvillian(A, c_op_list)
  
          raise TypeError('Cannot calculate the steady state for a ' +
                          'non-dissipative system ' +
                          '(no collapse operators given)')
      elif issuper(A):
          return A
      else:
          raise TypeError('Solving for steady states requires ' +
                          'Liouvillian (super) operators')
  
  
  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 = sp.csgraph.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 steady(L, maxiter=10, tol=1e-12, itertol=1e-15, method='solve',
             use_precond=False):
      """
      Deprecated. See steadystate instead.
      """
      message = "steady has been deprecated, use steadystate instead"
      warnings.warn(message, DeprecationWarning)
      return steadystate(L, [], maxiter=maxiter, tol=tol,
                         use_precond=use_precond)
  
  
  def _steadystate_direct_sparse(L, ss_args):
      """
      Direct solver that uses scipy sparse matrices
      """
      if settings.debug:
          logger.debug('Starting direct LU solver.')
  
      dims = L.dims[0]
      n = int(np.sqrt(L.shape[0]))
      b = np.zeros(n ** 2, dtype=complex)
      b[0] = ss_args['weight']
  
      if ss_args['solver'] == 'mkl':
          has_mkl = 1
      else:
          has_mkl = 0
  
      ss_lu_liouv_list = _steadystate_LU_liouvillian(L, ss_args, has_mkl)
      L, perm, perm2, rev_perm, ss_args = ss_lu_liouv_list
      if np.any(perm):
          b = b[np.ix_(perm,)]
      if np.any(perm2):
          b = b[np.ix_(perm2,)]
  
      if ss_args['solver'] == 'scipy':
          ss_args['info']['permc_spec'] = ss_args['permc_spec']
          ss_args['info']['drop_tol'] = ss_args['drop_tol']
          ss_args['info']['diag_pivot_thresh'] = ss_args['diag_pivot_thresh']
          ss_args['info']['fill_factor'] = ss_args['fill_factor']
          ss_args['info']['ILU_MILU'] = ss_args['ILU_MILU']
  
      if not ss_args['solver'] == 'mkl':
          # Use superLU solver
          orig_nnz = L.nnz
          _direct_start = time.time()
          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)
          _direct_end = time.time()
          ss_args['info']['solution_time'] = _direct_end - _direct_start
          if (settings.debug or ss_args['return_info']):
              L_nnz = lu.L.nnz
              U_nnz = lu.U.nnz
              ss_args['info']['l_nnz'] = L_nnz
              ss_args['info']['u_nnz'] = U_nnz
              ss_args['info']['lu_fill_factor'] = (L_nnz + U_nnz)/L.nnz
              if settings.debug:
                  logger.debug('L NNZ: %i ; U NNZ: %i' % (L_nnz, U_nnz))
                  logger.debug('Fill factor: %f' % ((L_nnz + U_nnz)/orig_nnz))
  
      else:  # Use MKL solver
          if len(ss_args['info']['perm']) != 0:
              in_perm = np.arange(n**2, dtype=np.int32)
          else:
              in_perm = None
          _direct_start = time.time()
          v = mkl_spsolve(L, b, perm=in_perm, verbose=ss_args['verbose'],
                          max_iter_refine=ss_args['max_iter_refine'],
                          scaling_vectors=ss_args['scaling_vectors'],
                          weighted_matching=ss_args['weighted_matching'])
          _direct_end = time.time()
          ss_args['info']['solution_time'] = _direct_end-_direct_start
  
      if ss_args['return_info']:
          ss_args['info']['residual_norm'] = la.norm(b - L*v, np.inf)
          ss_args['info']['max_iter_refine'] = ss_args['max_iter_refine']
          ss_args['info']['scaling_vectors'] = ss_args['scaling_vectors']
          ss_args['info']['weighted_matching'] = ss_args['weighted_matching']
  
      if ss_args['use_rcm']:
          v = v[np.ix_(rev_perm,)]
  
      data = dense2D_to_fastcsr_fmode(vec2mat(v), n, n)
      data = 0.5 * (data + data.H)
      if ss_args['return_info']:
          return Qobj(data, dims=dims, isherm=True), ss_args['info']
      else:
          return Qobj(data, dims=dims, isherm=True)
  
  
  def _steadystate_direct_dense(L, ss_args):
      """
      Direct solver that uses numpy arrays. Suitable for small systems with few
      states.
      """
      if settings.debug:
          logger.debug('Starting direct dense solver.')
  
      dims = L.dims[0]
      n = int(np.sqrt(L.shape[0]))
      b = np.zeros(n ** 2)
      b[0] = ss_args['weight']
  
      L = L.full()
      L[0, :] = np.diag(ss_args['weight']*np.ones(n)).reshape((1, n ** 2))
      _dense_start = time.time()
      v = np.linalg.solve(L, b)
      _dense_end = time.time()
      ss_args['info']['solution_time'] = _dense_end-_dense_start
      if ss_args['return_info']:
          ss_args['info']['residual_norm'] = la.norm(b - L@v, np.inf)
      data = vec2mat(v)
      data = 0.5 * (data + data.conj().T)
  
      return Qobj(data, dims=dims, isherm=True)
  
  
  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 = sp.csgraph.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 _iterative_precondition(A, n, ss_args):
      """
      Internal function for preconditioning the steadystate problem for use
      with iterative solvers.
      """
      if settings.debug:
          logger.debug('Starting preconditioner.')
      _precond_start = time.time()
      try:
          P = spilu(A, permc_spec=ss_args['permc_spec'],
                    drop_tol=ss_args['drop_tol'],
                    diag_pivot_thresh=ss_args['diag_pivot_thresh'],
                    fill_factor=ss_args['fill_factor'],
                    options=dict(ILU_MILU=ss_args['ILU_MILU']))
  
          P_x = lambda x: P.solve(x)
          M = LinearOperator((n ** 2, n ** 2), matvec=P_x)
          _precond_end = time.time()
          ss_args['info']['permc_spec'] = ss_args['permc_spec']
          ss_args['info']['drop_tol'] = ss_args['drop_tol']
          ss_args['info']['diag_pivot_thresh'] = ss_args['diag_pivot_thresh']
          ss_args['info']['fill_factor'] = ss_args['fill_factor']
          ss_args['info']['ILU_MILU'] = ss_args['ILU_MILU']
          ss_args['info']['precond_time'] = _precond_end-_precond_start
  
          if settings.debug or ss_args['return_info']:
              if settings.debug:
                  logger.debug('Preconditioning succeeded.')
                  logger.debug('Precond. time: %f' %
                               (_precond_end - _precond_start))
              L_nnz = P.L.nnz
              U_nnz = P.U.nnz
              ss_args['info']['l_nnz'] = L_nnz
              ss_args['info']['u_nnz'] = U_nnz
              ss_args['info']['ilu_fill_factor'] = (L_nnz+U_nnz)/A.nnz
              e = np.ones(n ** 2, dtype=int)
              condest = la.norm(M*e, np.inf)
              ss_args['info']['ilu_condest'] = condest
              if settings.debug:
                  logger.debug('L NNZ: %i ; U NNZ: %i' % (L_nnz, U_nnz))
                  logger.debug('Fill factor: %f' % ((L_nnz+U_nnz)/A.nnz))
                  logger.debug('iLU condest: %f' % condest)
  
      except:
          raise Exception("Failed to build preconditioner. Try increasing " +
                          "fill_factor and/or drop_tol.")
  
      return M, ss_args
  
  
  def _steadystate_iterative(L, ss_args):
      """
      Iterative steady state solver using the GMRES, LGMRES, or BICGSTAB
      algorithm and a sparse incomplete LU preconditioner.
      """
      ss_iters = {'iter': 0}
  
      def _iter_count(r):
          ss_iters['iter'] += 1
          return
  
      if settings.debug:
          logger.debug('Starting %s solver.' % ss_args['method'])
  
      dims = L.dims[0]
      n = int(np.sqrt(L.shape[0]))
      b = np.zeros(n ** 2)
      b[0] = ss_args['weight']
  
      L, perm, perm2, rev_perm, ss_args = _steadystate_LU_liouvillian(L, ss_args)
      if np.any(perm):
          b = b[np.ix_(perm,)]
      if np.any(perm2):
          b = b[np.ix_(perm2,)]
  
      use_solver(assumeSortedIndices=True)
  
      if ss_args['M'] is None and ss_args['use_precond']:
          ss_args['M'], ss_args = _iterative_precondition(L, n, ss_args)
          if ss_args['M'] is None:
              warnings.warn("Preconditioning failed. Continuing without.",
                            UserWarning)
  
      # Select iterative solver type
      _iter_start = time.time()
      if ss_args['method'] == 'iterative-gmres':
          v, check = gmres(L, b, tol=ss_args['tol'], atol=ss_args['matol'],
                           M=ss_args['M'], x0=ss_args['x0'],
                           restart=ss_args['restart'],
                           maxiter=ss_args['maxiter'],
                           callback=_iter_count, callback_type='legacy')
      elif ss_args['method'] == 'iterative-lgmres':
          v, check = lgmres(L, b, tol=ss_args['tol'], atol=ss_args['matol'],
                            M=ss_args['M'], x0=ss_args['x0'],
                            maxiter=ss_args['maxiter'],
                            callback=_iter_count)
      elif ss_args['method'] == 'iterative-bicgstab':
          v, check = bicgstab(L, b, tol=ss_args['tol'],
                              atol=ss_args['matol'],
                              M=ss_args['M'], x0=ss_args['x0'],
                              maxiter=ss_args['maxiter'],
                              callback=_iter_count)
      else:
          raise Exception("Invalid iterative solver method.")
      _iter_end = time.time()
  
      ss_args['info']['iter_time'] = _iter_end - _iter_start
      if 'precond_time' in ss_args['info'].keys():
          ss_args['info']['solution_time'] = (ss_args['info']['iter_time'] +
                                              ss_args['info']['precond_time'])
      else:
          ss_args['info']['solution_time'] = ss_args['info']['iter_time']
      ss_args['info']['iterations'] = ss_iters['iter']
      if ss_args['return_info']:
          ss_args['info']['residual_norm'] = la.norm(b - L*v, np.inf)
  
      if settings.debug:
          logger.debug('Number of Iterations: %i' % ss_iters['iter'])
          logger.debug('Iteration. time: %f' % (_iter_end - _iter_start))
  
      if check > 0:
          raise Exception("Steadystate error: Did not reach tolerance after " +
                          str(ss_args['maxiter']) + " steps." +
                          "\nResidual norm: " +
                          str(ss_args['info']['residual_norm']))
  
      elif check < 0:
          raise Exception(
              "Steadystate error: 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)
      if ss_args['return_info']:
          return Qobj(data, dims=dims, isherm=True), ss_args['info']
      else:
          return Qobj(data, dims=dims, isherm=True)
  
  
  def _steadystate_svd_dense(L, ss_args):
      """
      Find the steady state(s) of an open quantum system by solving for the
      nullspace of the Liouvillian.
      """
      ss_args['info'].pop('weight', None)
      atol = 1e-12
      rtol = 1e-12
      if settings.debug:
          logger.debug('Starting SVD solver.')
      _svd_start = time.time()
      u, s, vh = svd(L.full(), full_matrices=False)
      tol = max(atol, rtol * s[0])
      nnz = (s >= tol).sum()
      ns = vh[nnz:].conj().T
      _svd_end = time.time()
      ss_args['info']['solution_time'] = _svd_end-_svd_start
      if ss_args['all_states']:
          rhoss_list = []
          for n in range(ns.shape[1]):
              rhoss = Qobj(vec2mat(ns[:, n]), dims=L.dims[0])
              rhoss_list.append(rhoss / rhoss.tr())
          if ss_args['return_info']:
              return rhoss_list, ss_args['info']
          else:
              if ss_args['return_info']:
                  return rhoss_list, ss_args['info']
              else:
                  return rhoss_list
      else:
          rhoss = Qobj(vec2mat(ns[:, 0]), dims=L.dims[0])
          return rhoss / rhoss.tr()
  
  
  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 = sp.csgraph.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 _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']
      mtol = ss_args['mtol']
      if mtol is None:
          mtol = max(0.1*tol, 1e-15)
      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 = L.shape[0]
      # Build Liouvillian
      if ss_args['solver'] == 'mkl' and ss_args['method'] == 'power':
          has_mkl = 1
      else:
          has_mkl = 0
      L, perm, perm2, rev_perm, ss_args = _steadystate_power_liouvillian(L,
                                                                         ss_args,
                                                                         has_mkl)
      orig_nnz = L.nnz
      # start with all ones as RHS
      v = np.ones(n, dtype=complex)
      if ss_args['use_rcm']:
          v = v[np.ix_(perm2,)]
  
      # Do preconditioning
      if ss_args['solver'] == 'scipy':
          if ss_args['M'] is None and ss_args['use_precond'] and \
                  ss_args['method'] in ['power-gmres',
                                        'power-lgmres',
                                        'power-bicgstab']:
              ss_args['M'], ss_args = _iterative_precondition(L, int(np.sqrt(n)),
                                                              ss_args)
              if ss_args['M'] is None:
                  warnings.warn("Preconditioning failed. Continuing without.",
                                UserWarning)
  
      ss_iters = {'iter': 0}
  
      def _iter_count(r):
          ss_iters['iter'] += 1
          return
  
      _power_start = time.time()
      # Get LU factors
      if ss_args['method'] == 'power':
          if ss_args['solver'] == 'mkl':
              lu = mkl_splu(L, max_iter_refine=ss_args['max_iter_refine'],
                            scaling_vectors=ss_args['scaling_vectors'],
                            weighted_matching=ss_args['weighted_matching'])
          else:
              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:
                  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):
          check = 0
          if ss_args['method'] == 'power':
              v = lu.solve(v)
          elif ss_args['method'] == 'power-gmres':
              v, check = gmres(L, v, tol=mtol, atol=ss_args['matol'],
                               M=ss_args['M'], x0=ss_args['x0'],
                               restart=ss_args['restart'],
                               maxiter=ss_args['maxiter'],
                               callback=_iter_count, callback_type='legacy')
          elif ss_args['method'] == 'power-lgmres':
              v, check = lgmres(L, v, tol=mtol, atol=ss_args['matol'],
                                M=ss_args['M'], x0=ss_args['x0'],
                                maxiter=ss_args['maxiter'],
                                callback=_iter_count)
          elif ss_args['method'] == 'power-bicgstab':
              v, check = bicgstab(L, v, tol=mtol, atol=ss_args['matol'],
                                  M=ss_args['M'], x0=ss_args['x0'],
                                  maxiter=ss_args['maxiter'],
                                  callback=_iter_count)
          else:
              raise Exception("Invalid iterative solver method.")
          if check > 0:
              raise Exception("{} failed to find solution in "
                              "{} iterations.".format(ss_args['method'],
                                                      check))
          if check < 0:
              raise Exception("Breakdown in {}".format(ss_args['method']))
          v = v / la.norm(v, np.inf)
          it += 1
      if ss_args['method'] == 'power' and ss_args['solver'] == 'mkl':
          lu.delete()
          if ss_args['return_info']:
              ss_args['info']['max_iter_refine'] = ss_args['max_iter_refine']
              ss_args['info']['scaling_vectors'] = ss_args['scaling_vectors']
              ss_args['info']['weighted_matching'] = ss_args['weighted_matching']
  
      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 = v[::rhoss.shape[0]+1]
          data = v / np.sum(trow)
      else:
          data = data / la.norm(v)
  
      data = dense2D_to_fastcsr_fmode(vec2mat(data),
                                      rhoss.shape[0],
                                      rhoss.shape[0])
      rhoss.data = 0.5 * (data + data.H)
      rhoss.isherm = True
      if ss_args['return_info']:
          return rhoss, ss_args['info']
      else:
          return rhoss
  
  
  def build_preconditioner(A, c_op_list=[], **kwargs):
      """Constructs a iLU preconditioner necessary for solving for
      the steady state density matrix using the iterative linear solvers
      in the 'steadystate' function.
  
      Parameters
      ----------
      A : qobj
          A Hamiltonian or Liouvillian operator.
  
      c_op_list : list
          A list of collapse operators.
  
      return_info : bool, optional, default = False
          Return a dictionary of solver-specific infomation about the
          solution and how it was obtained.
  
      use_rcm : bool, optional, default = False
          Use reverse Cuthill-Mckee reordering to minimize fill-in in the
          LU factorization of the Liouvillian.
  
      use_wbm : bool, optional, default = False
          Use Weighted Bipartite Matching reordering to make the Liouvillian
          diagonally dominant.  This is useful for iterative preconditioners
          only, and is set to ``True`` by default when finding a preconditioner.
  
      weight : float, optional
          Sets the size of the elements used for adding the unity trace condition
          to the linear solvers.  This is set to the average abs value of the
          Liouvillian elements if not specified by the user.
  
      method : str, default = 'iterative'
          Tells the preconditioner what type of Liouvillian to build for
          iLU factorization.  For direct iterative methods use 'iterative'.
          For power iterative methods use 'power'.
  
      permc_spec : str, optional, default='COLAMD'
          Column ordering used internally by superLU for the
          'direct' LU decomposition method. Options include 'COLAMD' and
          'NATURAL'. If using RCM then this is set to 'NATURAL' automatically
          unless explicitly specified.
  
      fill_factor : float, optional, default = 100
          Specifies the fill ratio upper bound (>=1) of the iLU
          preconditioner.  Lower values save memory at the cost of longer
          execution times and a possible singular factorization.
  
      drop_tol : float, optional, default = 1e-4
          Sets the threshold for the magnitude of preconditioner
          elements that should be dropped.  Can be reduced for a courser
          factorization at the cost of an increased number of iterations, and a
          possible singular factorization.
  
      diag_pivot_thresh : float, optional, default = None
          Sets the threshold between [0,1] for which diagonal
          elements are considered acceptable pivot points when using a
          preconditioner.  A value of zero forces the pivot to be the diagonal
          element.
  
      ILU_MILU : str, optional, default = 'smilu_2'
          Selects the incomplete LU decomposition method algoithm used in
          creating the preconditoner. Should only be used by advanced users.
  
      Returns
      -------
      lu : object
          Returns a SuperLU object representing iLU preconditioner.
  
      info : dict, optional
          Dictionary containing solver-specific information.
      """
      ss_args = _default_steadystate_args()
      ss_args['method'] = 'iterative'
      for key in kwargs.keys():
          if key in ss_args.keys():
              ss_args[key] = kwargs[key]
          else:
              raise Exception("Invalid keyword argument '" + key +
                              "' passed to steadystate.")
  
      # Set column perm to NATURAL if using RCM and not specified by user
      if ss_args['use_rcm'] and ('permc_spec' not in kwargs.keys()):
          ss_args['permc_spec'] = 'NATURAL'
  
      L = _steadystate_setup(A, c_op_list)
      # Set weight parameter to avg abs val in L if not set explicitly
      if 'weight' not in kwargs.keys():
          ss_args['weight'] = np.mean(np.abs(L.data.data.max()))
          ss_args['info']['weight'] = ss_args['weight']
  
      n = int(np.sqrt(L.shape[0]))
      if ss_args['method'] == 'iterative':
          ss_list = _steadystate_LU_liouvillian(L, ss_args)
          L, perm, perm2, rev_perm, ss_args = ss_list
      elif ss_args['method'] == 'power':
          ss_list = _steadystate_power_liouvillian(L, ss_args)
          L, perm, perm2, rev_perm, ss_args = ss_list
      else:
          raise Exception("Invalid preconditioning method.")
  
      M, ss_args = _iterative_precondition(L, n, ss_args)
  
      if ss_args['return_info']:
          return M, ss_args['info']
      else:
          return M
  
  
  def _pseudo_inverse_dense(L, rhoss, w=None, **pseudo_args):
      """
      Internal function for computing the pseudo inverse of an Liouvillian using
      dense matrix methods. See pseudo_inverse for details.
      """
      rho_vec = np.transpose(mat2vec(rhoss.full()))
  
      tr_mat = tensor([identity(n) for n in L.dims[0][0]])
      tr_vec = np.transpose(mat2vec(tr_mat.full()))
      N = np.prod(L.dims[0][0])
      I = np.identity(N * N)
      P = np.kron(np.transpose(rho_vec), tr_vec)
      Q = I - P
  
      if w is None:
          L = L
      else:
          L = 1.0j*w*spre(tr_mat)+L
  
      if pseudo_args['method'] == 'direct':
          try:
              LIQ = np.linalg.solve(L.full(), Q)
          except:
              LIQ = np.linalg.lstsq(L.full(), Q)[0]
  
          R = np.dot(Q, LIQ)
  
          return Qobj(R, dims=L.dims)
  
      elif pseudo_args['method'] == 'numpy':
          return Qobj(np.dot(Q, np.dot(np.linalg.pinv(L.full()), Q)),
                      dims=L.dims)
  
      elif pseudo_args['method'] == 'scipy':
          # return Qobj(la.pinv(L.full()), dims=L.dims)
          return Qobj(np.dot(Q, np.dot(la.pinv(L.full()), Q)),
                      dims=L.dims)
  
      elif pseudo_args['method'] == 'scipy2':
          # return Qobj(la.pinv2(L.full()), dims=L.dims)
          return Qobj(np.dot(Q, np.dot(la.pinv2(L.full()), Q)),
                      dims=L.dims)
  
      else:
          raise ValueError("Unsupported method '%s'. Use 'direct' or 'numpy'" %
                           method)
  
  
  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 = sp.csgraph.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 pseudo_inverse(L, rhoss=None, w=None, sparse=True,  **kwargs):
      """
      Compute the pseudo inverse for a Liouvillian superoperator, optionally
      given its steady state density matrix (which will be computed if not
      given).
  
      Returns
      -------
      L : Qobj
          A Liouvillian superoperator for which to compute the pseudo inverse.
  
  
      rhoss : Qobj
          A steadystate density matrix as Qobj instance, for the Liouvillian
          superoperator L.
  
      w : double
          frequency at which to evaluate pseudo-inverse.  Can be zero for dense
          systems and large sparse systems. Small sparse systems can fail for
          zero frequencies.
  
      sparse : bool
          Flag that indicate whether to use sparse or dense matrix methods when
          computing the pseudo inverse.
  
      method : string
          Name of method to use. For sparse=True, allowed values are 'spsolve',
          'splu' and 'spilu'. For sparse=False, allowed values are 'direct' and
          'numpy'.
  
      kwargs : dictionary
          Additional keyword arguments for setting parameters for solver methods.
  
      Returns
      -------
      R : Qobj
          Returns a Qobj instance representing the pseudo inverse of L.
  
      Note
      ----
      In general the inverse of a sparse matrix will be dense.  If you
      are applying the inverse to a density matrix then it is better to
      cast the problem as an Ax=b type problem where the explicit calculation
      of the inverse is not required. See page 67 of "Electrons in
      nanostructures" C. Flindt, PhD Thesis available online:
      https://orbit.dtu.dk/fedora/objects/orbit:82314/datastreams/
      file_4732600/content
  
      Note also that the definition of the pseudo-inverse herein is different
      from numpys pinv() alone, as it includes pre and post projection onto
      the subspace defined by the projector Q.
  
      """
      pseudo_args = _default_steadystate_args()
      for key in kwargs.keys():
          if key in pseudo_args.keys():
              pseudo_args[key] = kwargs[key]
          else:
              raise Exception(
                  "Invalid keyword argument '"+key+"' passed to pseudo_inverse.")
      if 'method' not in kwargs.keys():
          pseudo_args['method'] = 'splu'
  
      # Set column perm to NATURAL if using RCM and not specified by user
      if pseudo_args['use_rcm'] and ('permc_spec' not in kwargs.keys()):
          pseudo_args['permc_spec'] = 'NATURAL'
  
      if rhoss is None:
          rhoss = steadystate(L, **pseudo_args)
  
      if sparse:
          return _pseudo_inverse_sparse(L, rhoss, w=w, **pseudo_args)
      else:
          if pseudo_args['method'] != 'splu':
              pseudo_args['method'] = pseudo_args['method']
          else:
              pseudo_args['method'] = 'direct'
          return _pseudo_inverse_dense(L, rhoss, w=w, **pseudo_args)
  