# Source code for qutip.steadystate

    """
    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', 'steadystate_floquet',
               '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('qutip.steadystate')
    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

    def _rename_kwargs(function, names_pairs):
        """
        Return a wrapped version of `function` that rename any keyword
        arguments from the first value of the pair, to the second.
        """
        @functools.wraps(function)
        def out(*args, **kwargs):
            for old, new in names_pairs:
                if old in kwargs:
                    kwargs[new] = kwargs.pop(old)
            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'])

    # From SciPy 1.12, the `tol` keyword argument to iterative solvers was renamed
    # to `rtol`.
    if _parse_version(scipy.__version__) >= _parse_version("1.12"):
        gmres = _rename_kwargs(gmres, [('tol', 'rtol')])
        lgmres = _rename_kwargs(lgmres, [('tol', 'rtol')])
        bicgstab = _rename_kwargs(bicgstab, [('tol', 'rtol')])

    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': 0.1, '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:`~qutip.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 ValueError('MKL solver only for direct or power methods.')

        elif solver not in ['scipy', 'mkl']:
            raise ValueError('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 TypeError(
                    "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():
            # set the weight to the mean of the non-zero absoluate values in A:
            ss_args['weight'] = np.mean(np.abs(A.data.data))
            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()
            with warnings.catch_warnings():
                warnings.filterwarnings(
                    "ignore", "qutip graph functions are deprecated",
                    DeprecationWarning,
                )
                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(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']))

            M = LinearOperator((n ** 2, n ** 2), matvec=P.solve)
            _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 Exception:
            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'
        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()
            with warnings.catch_warnings():
                warnings.filterwarnings(
                    "ignore", "qutip graph functions are deprecated",
                    DeprecationWarning,
                )
                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 steadystate_floquet(H_0, c_ops, Op_t, w_d=1.0, n_it=3, sparse=False):
        """
        Calculates the effective steady state for a driven
         system with a time-dependent cosinusoidal term:

        .. math::

            \\mathcal{\\hat{H}}(t) = \\hat{H}_0 +
             \\mathcal{\\hat{O}} \\cos(\\omega_d t)

        Parameters
        ----------
        H_0 : :obj:`~qutip.Qobj`
            A Hamiltonian or Liouvillian operator.

        c_ops : list
            A list of collapse operators.

        Op_t : :obj:`~qutip.Qobj`
            The the interaction operator which is multiplied by the cosine

        w_d : float, default 1.0
            The frequency of the drive

        n_it : int, default 3
            The number of iterations for the solver

        sparse : bool, default False
            Solve for the steady state using sparse algorithms.
            Actually, dense seems to be faster.

        Returns
        -------
        dm : qobj
            Steady state density matrix.

        .. note::
            See: Sze Meng Tan,
            https://copilot.caltech.edu/documents/16743/qousersguide.pdf,
            Section (10.16)
        """
        if sparse:
            N = H_0.shape[0]

            L_0 = liouvillian(H_0, c_ops).data.tocsc()
            L_t = liouvillian(Op_t)
            L_p = (0.5 * L_t).data.tocsc()
            # L_p and L_m correspond to the positive and negative
            # frequency terms respectively.
            # They are independent in the model, so we keep both names.
            L_m = L_p
            L_p_array = L_p.todense()
            L_m_array = L_p_array

            Id = sp.eye(N ** 2, format="csc", dtype=np.complex128)
            S = T = sp.csc_matrix((N ** 2, N ** 2), dtype=np.complex128)

            for n_i in np.arange(n_it, 0, -1):
                L = sp.csc_matrix(L_0 - 1j * n_i * w_d * Id + L_m.dot(S))
                L.sort_indices()
                LU = splu(L)
                S = - LU.solve(L_p_array)

                L = sp.csc_matrix(L_0 + 1j * n_i * w_d * Id + L_p.dot(T))
                L.sort_indices()
                LU = splu(L)
                T = - LU.solve(L_m_array)

            M_subs = L_0 + L_m.dot(S) + L_p.dot(T)
        else:
            N = H_0.shape[0]

            L_0 = liouvillian(H_0, c_ops).full()
            L_t = liouvillian(Op_t)
            L_p = (0.5 * L_t).full()
            L_m = L_p

            Id = np.eye(N ** 2)
            S, T = np.zeros((N ** 2, N ** 2)), np.zeros((N ** 2, N ** 2))

            for n_i in np.arange(n_it, 0, -1):
                L = L_0 - 1j * n_i * w_d * Id + np.matmul(L_m, S)
                lu, piv = la.lu_factor(L)
                S = - la.lu_solve((lu, piv), L_p)

                L = L_0 + 1j * n_i * w_d * Id + np.matmul(L_p, T)
                lu, piv = la.lu_factor(L)
                T = - la.lu_solve((lu, piv), L_m)

            M_subs = L_0 + np.matmul(L_m, S) + np.matmul(L_p, T)

        return steadystate(Qobj(M_subs, type="super", dims=L_t.dims))

    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 TypeError("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 ValueError("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 Exception:
                LIQ = np.linalg.lstsq(L.full(), Q, rcond=None)[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', 'numpy', 'scipy' or"
                " 'scipy2'" % pseudo_args['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 pseudo_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']
                lu = sp.linalg.splu(A, permc_spec=pspec,
                                    diag_pivot_thresh=diag_p_thresh,
                                    options=dict(ILU_MILU=pseudo_args['ILU_MILU']))
                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'" % pseudo_args['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,
                       method='splu', **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 TypeError(
                    "Invalid keyword argument '"+key+"' passed to pseudo_inverse.")
        pseudo_args['method'] = method

        # 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)