# Buggy code
def lobpcg(
    A,
    X,
    B=None,
    M=None,
    Y=None,
    tol=None,
    maxiter=20,
    largest=True,
    verbosityLevel=0,
    retLambdaHistory=False,
    retResidualNormsHistory=False,
):
    """Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG)

    LOBPCG is a preconditioned eigensolver for large symmetric positive
    definite (SPD) generalized eigenproblems.

    Parameters
    ----------
    A : {sparse matrix, dense matrix, LinearOperator}
        The symmetric linear operator of the problem, usually a
        sparse matrix.  Often called the "stiffness matrix".
    X : array_like
        Initial approximation to the k eigenvectors. If A has
        shape=(n,n) then X should have shape shape=(n,k).
    B : {dense matrix, sparse matrix, LinearOperator}, optional
        the right hand side operator in a generalized eigenproblem.
        by default, B = Identity
        often called the "mass matrix"
    M : {dense matrix, sparse matrix, LinearOperator}, optional
        preconditioner to A; by default M = Identity
        M should approximate the inverse of A
    Y : array_like, optional
        n-by-sizeY matrix of constraints, sizeY < n
        The iterations will be performed in the B-orthogonal complement
        of the column-space of Y. Y must be full rank.
    tol : scalar, optional
        Solver tolerance (stopping criterion)
        by default: tol=n*sqrt(eps)
    maxiter : integer, optional
        maximum number of iterations
        by default: maxiter=min(n,20)
    largest : bool, optional
        when True, solve for the largest eigenvalues, otherwise the smallest
    verbosityLevel : integer, optional
        controls solver output.  default: verbosityLevel = 0.
    retLambdaHistory : boolean, optional
        whether to return eigenvalue history
    retResidualNormsHistory : boolean, optional
        whether to return history of residual norms

    Returns
    -------
    w : array
        Array of k eigenvalues
    v : array
        An array of k eigenvectors.  V has the same shape as X.
    lambdas : list of arrays, optional
        The eigenvalue history, if `retLambdaHistory` is True.
    rnorms : list of arrays, optional
        The history of residual norms, if `retResidualNormsHistory` is True.

    Examples
    --------

    Solve A x = lambda B x with constraints and preconditioning.

    >>> from scipy.sparse import spdiags, issparse
    >>> from scipy.sparse.linalg import lobpcg, LinearOperator
    >>> n = 100
    >>> vals = [np.arange(n, dtype=np.float64) + 1]
    >>> A = spdiags(vals, 0, n, n)
    >>> A.toarray()
    array([[  1.,   0.,   0., ...,   0.,   0.,   0.],
           [  0.,   2.,   0., ...,   0.,   0.,   0.],
           [  0.,   0.,   3., ...,   0.,   0.,   0.],
           ...,
           [  0.,   0.,   0., ...,  98.,   0.,   0.],
           [  0.,   0.,   0., ...,   0.,  99.,   0.],
           [  0.,   0.,   0., ...,   0.,   0., 100.]])

    Constraints.

    >>> Y = np.eye(n, 3)

    Initial guess for eigenvectors, should have linearly independent
    columns. Column dimension = number of requested eigenvalues.

    >>> X = np.random.rand(n, 3)

    Preconditioner -- inverse of A (as an abstract linear operator).

    >>> invA = spdiags([1./vals[0]], 0, n, n)
    >>> def precond( x ):
    ...     return invA  * x
    >>> M = LinearOperator(matvec=precond, shape=(n, n), dtype=float)

    Here, ``invA`` could of course have been used directly as a preconditioner.
    Let us then solve the problem:

    >>> eigs, vecs = lobpcg(A, X, Y=Y, M=M, largest=False)
    >>> eigs
    array([4., 5., 6.])

    Note that the vectors passed in Y are the eigenvectors of the 3 smallest
    eigenvalues. The results returned are orthogonal to those.

    Notes
    -----
    If both retLambdaHistory and retResidualNormsHistory are True,
    the return tuple has the following format
    (lambda, V, lambda history, residual norms history).

    In the following ``n`` denotes the matrix size and ``m`` the number
    of required eigenvalues (smallest or largest).

    The LOBPCG code internally solves eigenproblems of the size 3``m`` on every
    iteration by calling the "standard" dense eigensolver, so if ``m`` is not
    small enough compared to ``n``, it does not make sense to call the LOBPCG
    code, but rather one should use the "standard" eigensolver,
    e.g. numpy or scipy function in this case.
    If one calls the LOBPCG algorithm for 5``m``>``n``,
    it will most likely break internally, so the code tries to call
    the standard function instead.

    It is not that n should be large for the LOBPCG to work, but rather the
    ratio ``n``/``m`` should be large. It you call LOBPCG with ``m``=1
    and ``n``=10, it works though ``n`` is small. The method is intended
    for extremely large ``n``/``m``, see e.g., reference [28] in
    https://arxiv.org/abs/0705.2626

    The convergence speed depends basically on two factors:

    1. How well relatively separated the seeking eigenvalues are from the rest
       of the eigenvalues. One can try to vary ``m`` to make this better.

    2. How well conditioned the problem is. This can be changed by using proper
       preconditioning. For example, a rod vibration test problem (under tests
       directory) is ill-conditioned for large ``n``, so convergence will be
       slow, unless efficient preconditioning is used. For this specific
       problem, a good simple preconditioner function would be a linear solve
       for A, which is easy to code since A is tridiagonal.

    *Acknowledgements*

    lobpcg.py code was written by Robert Cimrman.
    Many thanks belong to Andrew Knyazev, the author of the algorithm,
    for lots of advice and support.

    References
    ----------
    .. [1] A. V. Knyazev (2001),
           Toward the Optimal Preconditioned Eigensolver: Locally Optimal
           Block Preconditioned Conjugate Gradient Method.
           SIAM Journal on Scientific Computing 23, no. 2,
           pp. 517-541. http://dx.doi.org/10.1137/S1064827500366124

    .. [2] A. V. Knyazev, I. Lashuk, M. E. Argentati, and E. Ovchinnikov
           (2007), Block Locally Optimal Preconditioned Eigenvalue Xolvers
           (BLOPEX) in hypre and PETSc. https://arxiv.org/abs/0705.2626

    .. [3] A. V. Knyazev's C and MATLAB implementations:
           https://bitbucket.org/joseroman/blopex
    """
    blockVectorX = X
    blockVectorY = Y
    residualTolerance = tol
    maxIterations = maxiter

    if blockVectorY is not None:
        sizeY = blockVectorY.shape[1]
    else:
        sizeY = 0

    # Block size.
    if len(blockVectorX.shape) != 2:
        raise ValueError("expected rank-2 array for argument X")

    n, sizeX = blockVectorX.shape

    if verbosityLevel:
        aux = "Solving "
        if B is None:
            aux += "standard"
        else:
            aux += "generalized"
        aux += " eigenvalue problem with"
        if M is None:
            aux += "out"
        aux += " preconditioning\n\n"
        aux += "matrix size %d\n" % n
        aux += "block size %d\n\n" % sizeX
        if blockVectorY is None:
            aux += "No constraints\n\n"
        else:
            if sizeY > 1:
                aux += "%d constraints\n\n" % sizeY
            else:
                aux += "%d constraint\n\n" % sizeY
        print(aux)

    A = _makeOperator(A, (n, n))
    B = _makeOperator(B, (n, n))
    M = _makeOperator(M, (n, n))

    if (n - sizeY) < (5 * sizeX):
        # warn('The problem size is small compared to the block size.' \
        #        ' Using dense eigensolver instead of LOBPCG.')

        sizeX = min(sizeX, n)

        if blockVectorY is not None:
            raise NotImplementedError(
                "The dense eigensolver does not support constraints."
            )

        # Define the closed range of indices of eigenvalues to return.
        if largest:
            eigvals = (n - sizeX, n - 1)
        else:
            eigvals = (0, sizeX - 1)

        A_dense = A(np.eye(n, dtype=A.dtype))
        B_dense = None if B is None else B(np.eye(n, dtype=B.dtype))

        vals, vecs = eigh(A_dense, B_dense, eigvals=eigvals, check_finite=False)
        if largest:
            # Reverse order to be compatible with eigs() in 'LM' mode.
            vals = vals[::-1]
            vecs = vecs[:, ::-1]

        return vals, vecs

    if (residualTolerance is None) or (residualTolerance <= 0.0):
        residualTolerance = np.sqrt(1e-15) * n

    # Apply constraints to X.
    if blockVectorY is not None:
        if B is not None:
            blockVectorBY = B(blockVectorY)
        else:
            blockVectorBY = blockVectorY

        # gramYBY is a dense array.
        gramYBY = np.dot(blockVectorY.T.conj(), blockVectorBY)
        try:
            # gramYBY is a Cholesky factor from now on...
            gramYBY = cho_factor(gramYBY)
        except LinAlgError:
            raise ValueError("cannot handle linearly dependent constraints")

        _applyConstraints(blockVectorX, gramYBY, blockVectorBY, blockVectorY)

    ##
    # B-orthonormalize X.
    blockVectorX, blockVectorBX = _b_orthonormalize(B, blockVectorX)

    ##
    # Compute the initial Ritz vectors: solve the eigenproblem.
    blockVectorAX = A(blockVectorX)
    gramXAX = np.dot(blockVectorX.T.conj(), blockVectorAX)

    _lambda, eigBlockVector = eigh(gramXAX, check_finite=False)
    ii = _get_indx(_lambda, sizeX, largest)
    _lambda = _lambda[ii]

    eigBlockVector = np.asarray(eigBlockVector[:, ii])
    blockVectorX = np.dot(blockVectorX, eigBlockVector)
    blockVectorAX = np.dot(blockVectorAX, eigBlockVector)
    if B is not None:
        blockVectorBX = np.dot(blockVectorBX, eigBlockVector)

    ##
    # Active index set.
    activeMask = np.ones((sizeX,), dtype=bool)

    lambdaHistory = [_lambda]
    residualNormsHistory = []

    previousBlockSize = sizeX
    ident = np.eye(sizeX, dtype=A.dtype)
    ident0 = np.eye(sizeX, dtype=A.dtype)

    ##
    # Main iteration loop.

    blockVectorP = None  # set during iteration
    blockVectorAP = None
    blockVectorBP = None

    iterationNumber = -1
    while iterationNumber < maxIterations:
        iterationNumber += 1
        if verbosityLevel > 0:
            print("iteration %d" % iterationNumber)

        if B is not None:
            aux = blockVectorBX * _lambda[np.newaxis, :]

        else:
            aux = blockVectorX * _lambda[np.newaxis, :]

        blockVectorR = blockVectorAX - aux

        aux = np.sum(blockVectorR.conjugate() * blockVectorR, 0)
        residualNorms = np.sqrt(aux)

        residualNormsHistory.append(residualNorms)

        ii = np.where(residualNorms > residualTolerance, True, False)
        activeMask = activeMask & ii
        if verbosityLevel > 2:
            print(activeMask)

        currentBlockSize = activeMask.sum()
        if currentBlockSize != previousBlockSize:
            previousBlockSize = currentBlockSize
            ident = np.eye(currentBlockSize, dtype=A.dtype)

        if currentBlockSize == 0:
            break

        if verbosityLevel > 0:
            print("current block size:", currentBlockSize)
            print("eigenvalue:", _lambda)
            print("residual norms:", residualNorms)
        if verbosityLevel > 10:
            print(eigBlockVector)

        activeBlockVectorR = _as2d(blockVectorR[:, activeMask])

        if iterationNumber > 0:
            activeBlockVectorP = _as2d(blockVectorP[:, activeMask])
            activeBlockVectorAP = _as2d(blockVectorAP[:, activeMask])
            if B is not None:
                activeBlockVectorBP = _as2d(blockVectorBP[:, activeMask])

        if M is not None:
            # Apply preconditioner T to the active residuals.
            activeBlockVectorR = M(activeBlockVectorR)

        ##
        # Apply constraints to the preconditioned residuals.
        if blockVectorY is not None:
            _applyConstraints(activeBlockVectorR, gramYBY, blockVectorBY, blockVectorY)

        ##
        # B-orthonormalize the preconditioned residuals.

        aux = _b_orthonormalize(B, activeBlockVectorR)
        activeBlockVectorR, activeBlockVectorBR = aux

        activeBlockVectorAR = A(activeBlockVectorR)

        if iterationNumber > 0:
            if B is not None:
                aux = _b_orthonormalize(
                    B, activeBlockVectorP, activeBlockVectorBP, retInvR=True
                )
                activeBlockVectorP, activeBlockVectorBP, invR = aux
                activeBlockVectorAP = np.dot(activeBlockVectorAP, invR)

            else:
                aux = _b_orthonormalize(B, activeBlockVectorP, retInvR=True)
                activeBlockVectorP, _, invR = aux
                activeBlockVectorAP = np.dot(activeBlockVectorAP, invR)

        ##
        # Perform the Rayleigh Ritz Procedure:
        # Compute symmetric Gram matrices:

        if B is not None:
            xaw = np.dot(blockVectorX.T.conj(), activeBlockVectorAR)
            waw = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorAR)
            xbw = np.dot(blockVectorX.T.conj(), activeBlockVectorBR)

            if iterationNumber > 0:
                xap = np.dot(blockVectorX.T.conj(), activeBlockVectorAP)
                wap = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorAP)
                pap = np.dot(activeBlockVectorP.T.conj(), activeBlockVectorAP)
                xbp = np.dot(blockVectorX.T.conj(), activeBlockVectorBP)
                wbp = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorBP)

                gramA = bmat(
                    [
                        [np.diag(_lambda), xaw, xap],
                        [xaw.T.conj(), waw, wap],
                        [xap.T.conj(), wap.T.conj(), pap],
                    ]
                )

                gramB = bmat(
                    [
                        [ident0, xbw, xbp],
                        [xbw.T.conj(), ident, wbp],
                        [xbp.T.conj(), wbp.T.conj(), ident],
                    ]
                )
            else:
                gramA = bmat([[np.diag(_lambda), xaw], [xaw.T.conj(), waw]])
                gramB = bmat([[ident0, xbw], [xbw.T.conj(), ident]])

        else:
            xaw = np.dot(blockVectorX.T.conj(), activeBlockVectorAR)
            waw = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorAR)
            xbw = np.dot(blockVectorX.T.conj(), activeBlockVectorR)

            if iterationNumber > 0:
                xap = np.dot(blockVectorX.T.conj(), activeBlockVectorAP)
                wap = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorAP)
                pap = np.dot(activeBlockVectorP.T.conj(), activeBlockVectorAP)
                xbp = np.dot(blockVectorX.T.conj(), activeBlockVectorP)
                wbp = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorP)

                gramA = bmat(
                    [
                        [np.diag(_lambda), xaw, xap],
                        [xaw.T.conj(), waw, wap],
                        [xap.T.conj(), wap.T.conj(), pap],
                    ]
                )

                gramB = bmat(
                    [
                        [ident0, xbw, xbp],
                        [xbw.T.conj(), ident, wbp],
                        [xbp.T.conj(), wbp.T.conj(), ident],
                    ]
                )
            else:
                gramA = bmat([[np.diag(_lambda), xaw], [xaw.T.conj(), waw]])
                gramB = bmat([[ident0, xbw], [xbw.T.conj(), ident]])

        if verbosityLevel > 0:
            _report_nonhermitian(gramA, 3, -1, "gramA")
            _report_nonhermitian(gramB, 3, -1, "gramB")

        if verbosityLevel > 10:
            _save(gramA, "gramA")
            _save(gramB, "gramB")

        # Solve the generalized eigenvalue problem.
        _lambda, eigBlockVector = eigh(gramA, gramB, check_finite=False)
        ii = _get_indx(_lambda, sizeX, largest)

        if verbosityLevel > 10:
            print(ii)
            print(_lambda)

        _lambda = _lambda[ii]
        eigBlockVector = eigBlockVector[:, ii]

        lambdaHistory.append(_lambda)

        if verbosityLevel > 10:
            print("lambda:", _lambda)
        #         # Normalize eigenvectors!
        #         aux = np.sum( eigBlockVector.conjugate() * eigBlockVector, 0 )
        #         eigVecNorms = np.sqrt( aux )
        #         eigBlockVector = eigBlockVector / eigVecNorms[np.newaxis, :]
        #         eigBlockVector, aux = _b_orthonormalize( B, eigBlockVector )

        if verbosityLevel > 10:
            print(eigBlockVector)

        # Compute Ritz vectors.
        if B is not None:
            if iterationNumber > 0:
                eigBlockVectorX = eigBlockVector[:sizeX]
                eigBlockVectorR = eigBlockVector[sizeX : sizeX + currentBlockSize]
                eigBlockVectorP = eigBlockVector[sizeX + currentBlockSize :]

                pp = np.dot(activeBlockVectorR, eigBlockVectorR)
                pp += np.dot(activeBlockVectorP, eigBlockVectorP)

                app = np.dot(activeBlockVectorAR, eigBlockVectorR)
                app += np.dot(activeBlockVectorAP, eigBlockVectorP)

                bpp = np.dot(activeBlockVectorBR, eigBlockVectorR)
                bpp += np.dot(activeBlockVectorBP, eigBlockVectorP)
            else:
                eigBlockVectorX = eigBlockVector[:sizeX]
                eigBlockVectorR = eigBlockVector[sizeX:]

                pp = np.dot(activeBlockVectorR, eigBlockVectorR)
                app = np.dot(activeBlockVectorAR, eigBlockVectorR)
                bpp = np.dot(activeBlockVectorBR, eigBlockVectorR)

            if verbosityLevel > 10:
                print(pp)
                print(app)
                print(bpp)

            blockVectorX = np.dot(blockVectorX, eigBlockVectorX) + pp
            blockVectorAX = np.dot(blockVectorAX, eigBlockVectorX) + app
            blockVectorBX = np.dot(blockVectorBX, eigBlockVectorX) + bpp

            blockVectorP, blockVectorAP, blockVectorBP = pp, app, bpp

        else:
            if iterationNumber > 0:
                eigBlockVectorX = eigBlockVector[:sizeX]
                eigBlockVectorR = eigBlockVector[sizeX : sizeX + currentBlockSize]
                eigBlockVectorP = eigBlockVector[sizeX + currentBlockSize :]

                pp = np.dot(activeBlockVectorR, eigBlockVectorR)
                pp += np.dot(activeBlockVectorP, eigBlockVectorP)

                app = np.dot(activeBlockVectorAR, eigBlockVectorR)
                app += np.dot(activeBlockVectorAP, eigBlockVectorP)
            else:
                eigBlockVectorX = eigBlockVector[:sizeX]
                eigBlockVectorR = eigBlockVector[sizeX:]

                pp = np.dot(activeBlockVectorR, eigBlockVectorR)
                app = np.dot(activeBlockVectorAR, eigBlockVectorR)

            if verbosityLevel > 10:
                print(pp)
                print(app)

            blockVectorX = np.dot(blockVectorX, eigBlockVectorX) + pp
            blockVectorAX = np.dot(blockVectorAX, eigBlockVectorX) + app

            blockVectorP, blockVectorAP = pp, app

    if B is not None:
        aux = blockVectorBX * _lambda[np.newaxis, :]

    else:
        aux = blockVectorX * _lambda[np.newaxis, :]

    blockVectorR = blockVectorAX - aux

    aux = np.sum(blockVectorR.conjugate() * blockVectorR, 0)
    residualNorms = np.sqrt(aux)

    if verbosityLevel > 0:
        print("final eigenvalue:", _lambda)
        print("final residual norms:", residualNorms)

    if retLambdaHistory:
        if retResidualNormsHistory:
            return _lambda, blockVectorX, lambdaHistory, residualNormsHistory
        else:
            return _lambda, blockVectorX, lambdaHistory
    else:
        if retResidualNormsHistory:
            return _lambda, blockVectorX, residualNormsHistory
        else:
            return _lambda, blockVectorX


# Traceback (numpy.linalg.LinAlgError)
Traceback (most recent call last):
File "./test_lobpcg.py", line 47, in <module>
test_lobpcg(read_guess=True)
File "./test_lobpcg.py", line 39, in test_lobpcg
λ, v = lobpcg(A, X, largest=False, M=P, maxiter=100, tol=tol)
File "/home/mfh/.julia/python_virtualenv/lib/python3.7/site-packages/scipy/sparse/linalg/eigen/lobpcg/lobpcg.py", line 476, in lobpcg
aux = _b_orthonormalize(B, activeBlockVectorP, retInvR=True)
File "/home/mfh/.julia/python_virtualenv/lib/python3.7/site-packages/scipy/sparse/linalg/eigen/lobpcg/lobpcg.py", line 93, in _b_orthonormalize
gramVBV = cholesky(gramVBV)
File "/home/mfh/.julia/python_virtualenv/lib/python3.7/site-packages/scipy/linalg/decomp_cholesky.py", line 91, in cholesky
check_finite=check_finite)
File "/home/mfh/.julia/python_virtualenv/lib/python3.7/site-packages/scipy/linalg/decomp_cholesky.py", line 40, in _cholesky
"definite" % info)
numpy.linalg.LinAlgError: 9-th leading minor of the array is not positive definite

# Fixed code
def lobpcg(
    A,
    X,
    B=None,
    M=None,
    Y=None,
    tol=None,
    maxiter=20,
    largest=True,
    verbosityLevel=0,
    retLambdaHistory=False,
    retResidualNormsHistory=False,
):
    """Locally Optimal Block Preconditioned Conjugate Gradient Method (LOBPCG)

    LOBPCG is a preconditioned eigensolver for large symmetric positive
    definite (SPD) generalized eigenproblems.

    Parameters
    ----------
    A : {sparse matrix, dense matrix, LinearOperator}
        The symmetric linear operator of the problem, usually a
        sparse matrix.  Often called the "stiffness matrix".
    X : ndarray, float32 or float64
        Initial approximation to the ``k`` eigenvectors (non-sparse). If `A`
        has ``shape=(n,n)`` then `X` should have shape ``shape=(n,k)``.
    B : {dense matrix, sparse matrix, LinearOperator}, optional
        The right hand side operator in a generalized eigenproblem.
        By default, ``B = Identity``.  Often called the "mass matrix".
    M : {dense matrix, sparse matrix, LinearOperator}, optional
        Preconditioner to `A`; by default ``M = Identity``.
        `M` should approximate the inverse of `A`.
    Y : ndarray, float32 or float64, optional
        n-by-sizeY matrix of constraints (non-sparse), sizeY < n
        The iterations will be performed in the B-orthogonal complement
        of the column-space of Y. Y must be full rank.
    tol : scalar, optional
        Solver tolerance (stopping criterion).
        The default is ``tol=n*sqrt(eps)``.
    maxiter : int, optional
        Maximum number of iterations.  The default is ``maxiter=min(n, 20)``.
    largest : bool, optional
        When True, solve for the largest eigenvalues, otherwise the smallest.
    verbosityLevel : int, optional
        Controls solver output.  The default is ``verbosityLevel=0``.
    retLambdaHistory : bool, optional
        Whether to return eigenvalue history.  Default is False.
    retResidualNormsHistory : bool, optional
        Whether to return history of residual norms.  Default is False.

    Returns
    -------
    w : ndarray
        Array of ``k`` eigenvalues
    v : ndarray
        An array of ``k`` eigenvectors.  `v` has the same shape as `X`.
    lambdas : list of ndarray, optional
        The eigenvalue history, if `retLambdaHistory` is True.
    rnorms : list of ndarray, optional
        The history of residual norms, if `retResidualNormsHistory` is True.

    Notes
    -----
    If both ``retLambdaHistory`` and ``retResidualNormsHistory`` are True,
    the return tuple has the following format
    ``(lambda, V, lambda history, residual norms history)``.

    In the following ``n`` denotes the matrix size and ``m`` the number
    of required eigenvalues (smallest or largest).

    The LOBPCG code internally solves eigenproblems of the size ``3m`` on every
    iteration by calling the "standard" dense eigensolver, so if ``m`` is not
    small enough compared to ``n``, it does not make sense to call the LOBPCG
    code, but rather one should use the "standard" eigensolver, e.g. numpy or
    scipy function in this case.
    If one calls the LOBPCG algorithm for ``5m > n``, it will most likely break
    internally, so the code tries to call the standard function instead.

    It is not that ``n`` should be large for the LOBPCG to work, but rather the
    ratio ``n / m`` should be large. It you call LOBPCG with ``m=1``
    and ``n=10``, it works though ``n`` is small. The method is intended
    for extremely large ``n / m``, see e.g., reference [28] in
    https://arxiv.org/abs/0705.2626

    The convergence speed depends basically on two factors:

    1. How well relatively separated the seeking eigenvalues are from the rest
       of the eigenvalues. One can try to vary ``m`` to make this better.

    2. How well conditioned the problem is. This can be changed by using proper
       preconditioning. For example, a rod vibration test problem (under tests
       directory) is ill-conditioned for large ``n``, so convergence will be
       slow, unless efficient preconditioning is used. For this specific
       problem, a good simple preconditioner function would be a linear solve
       for `A`, which is easy to code since A is tridiagonal.

    References
    ----------
    .. [1] A. V. Knyazev (2001),
           Toward the Optimal Preconditioned Eigensolver: Locally Optimal
           Block Preconditioned Conjugate Gradient Method.
           SIAM Journal on Scientific Computing 23, no. 2,
           pp. 517-541. http://dx.doi.org/10.1137/S1064827500366124

    .. [2] A. V. Knyazev, I. Lashuk, M. E. Argentati, and E. Ovchinnikov
           (2007), Block Locally Optimal Preconditioned Eigenvalue Xolvers
           (BLOPEX) in hypre and PETSc. https://arxiv.org/abs/0705.2626

    .. [3] A. V. Knyazev's C and MATLAB implementations:
           https://bitbucket.org/joseroman/blopex

    Examples
    --------

    Solve ``A x = lambda x`` with constraints and preconditioning.

    >>> import numpy as np
    >>> from scipy.sparse import spdiags, issparse
    >>> from scipy.sparse.linalg import lobpcg, LinearOperator
    >>> n = 100
    >>> vals = np.arange(1, n + 1)
    >>> A = spdiags(vals, 0, n, n)
    >>> A.toarray()
    array([[  1.,   0.,   0., ...,   0.,   0.,   0.],
           [  0.,   2.,   0., ...,   0.,   0.,   0.],
           [  0.,   0.,   3., ...,   0.,   0.,   0.],
           ...,
           [  0.,   0.,   0., ...,  98.,   0.,   0.],
           [  0.,   0.,   0., ...,   0.,  99.,   0.],
           [  0.,   0.,   0., ...,   0.,   0., 100.]])

    Constraints:

    >>> Y = np.eye(n, 3)

    Initial guess for eigenvectors, should have linearly independent
    columns. Column dimension = number of requested eigenvalues.

    >>> X = np.random.rand(n, 3)

    Preconditioner in the inverse of A in this example:

    >>> invA = spdiags([1./vals], 0, n, n)

    The preconditiner must be defined by a function:

    >>> def precond( x ):
    ...     return invA @ x

    The argument x of the preconditioner function is a matrix inside `lobpcg`,
    thus the use of matrix-matrix product ``@``.

    The preconditioner function is passed to lobpcg as a `LinearOperator`:

    >>> M = LinearOperator(matvec=precond, matmat=precond,
    ...                    shape=(n, n), dtype=float)

    Let us now solve the eigenvalue problem for the matrix A:

    >>> eigenvalues, _ = lobpcg(A, X, Y=Y, M=M, largest=False)
    >>> eigenvalues
    array([4., 5., 6.])

    Note that the vectors passed in Y are the eigenvectors of the 3 smallest
    eigenvalues. The results returned are orthogonal to those.

    """
    blockVectorX = X
    blockVectorY = Y
    residualTolerance = tol
    maxIterations = maxiter

    if blockVectorY is not None:
        sizeY = blockVectorY.shape[1]
    else:
        sizeY = 0

    # Block size.
    if len(blockVectorX.shape) != 2:
        raise ValueError("expected rank-2 array for argument X")

    n, sizeX = blockVectorX.shape

    if verbosityLevel:
        aux = "Solving "
        if B is None:
            aux += "standard"
        else:
            aux += "generalized"
        aux += " eigenvalue problem with"
        if M is None:
            aux += "out"
        aux += " preconditioning\n\n"
        aux += "matrix size %d\n" % n
        aux += "block size %d\n\n" % sizeX
        if blockVectorY is None:
            aux += "No constraints\n\n"
        else:
            if sizeY > 1:
                aux += "%d constraints\n\n" % sizeY
            else:
                aux += "%d constraint\n\n" % sizeY
        print(aux)

    A = _makeOperator(A, (n, n))
    B = _makeOperator(B, (n, n))
    M = _makeOperator(M, (n, n))

    if (n - sizeY) < (5 * sizeX):
        # warn('The problem size is small compared to the block size.' \
        #        ' Using dense eigensolver instead of LOBPCG.')

        sizeX = min(sizeX, n)

        if blockVectorY is not None:
            raise NotImplementedError(
                "The dense eigensolver does not support constraints."
            )

        # Define the closed range of indices of eigenvalues to return.
        if largest:
            eigvals = (n - sizeX, n - 1)
        else:
            eigvals = (0, sizeX - 1)

        A_dense = A(np.eye(n, dtype=A.dtype))
        B_dense = None if B is None else B(np.eye(n, dtype=B.dtype))

        vals, vecs = eigh(A_dense, B_dense, eigvals=eigvals, check_finite=False)
        if largest:
            # Reverse order to be compatible with eigs() in 'LM' mode.
            vals = vals[::-1]
            vecs = vecs[:, ::-1]

        return vals, vecs

    if (residualTolerance is None) or (residualTolerance <= 0.0):
        residualTolerance = np.sqrt(1e-15) * n

    # Apply constraints to X.
    if blockVectorY is not None:
        if B is not None:
            blockVectorBY = B(blockVectorY)
        else:
            blockVectorBY = blockVectorY

        # gramYBY is a dense array.
        gramYBY = np.dot(blockVectorY.T.conj(), blockVectorBY)
        try:
            # gramYBY is a Cholesky factor from now on...
            gramYBY = cho_factor(gramYBY)
        except LinAlgError:
            raise ValueError("cannot handle linearly dependent constraints")

        _applyConstraints(blockVectorX, gramYBY, blockVectorBY, blockVectorY)

    ##
    # B-orthonormalize X.
    blockVectorX, blockVectorBX = _b_orthonormalize(B, blockVectorX)

    ##
    # Compute the initial Ritz vectors: solve the eigenproblem.
    blockVectorAX = A(blockVectorX)
    gramXAX = np.dot(blockVectorX.T.conj(), blockVectorAX)

    _lambda, eigBlockVector = eigh(gramXAX, check_finite=False)
    ii = _get_indx(_lambda, sizeX, largest)
    _lambda = _lambda[ii]

    eigBlockVector = np.asarray(eigBlockVector[:, ii])
    blockVectorX = np.dot(blockVectorX, eigBlockVector)
    blockVectorAX = np.dot(blockVectorAX, eigBlockVector)
    if B is not None:
        blockVectorBX = np.dot(blockVectorBX, eigBlockVector)

    ##
    # Active index set.
    activeMask = np.ones((sizeX,), dtype=bool)

    lambdaHistory = [_lambda]
    residualNormsHistory = []

    previousBlockSize = sizeX
    ident = np.eye(sizeX, dtype=A.dtype)
    ident0 = np.eye(sizeX, dtype=A.dtype)

    ##
    # Main iteration loop.

    blockVectorP = None  # set during iteration
    blockVectorAP = None
    blockVectorBP = None

    iterationNumber = -1
    restart = True
    while iterationNumber < maxIterations:
        iterationNumber += 1
        if verbosityLevel > 0:
            print("iteration %d" % iterationNumber)

        if B is not None:
            aux = blockVectorBX * _lambda[np.newaxis, :]
        else:
            aux = blockVectorX * _lambda[np.newaxis, :]

        blockVectorR = blockVectorAX - aux

        aux = np.sum(blockVectorR.conj() * blockVectorR, 0)
        residualNorms = np.sqrt(aux)

        residualNormsHistory.append(residualNorms)

        ii = np.where(residualNorms > residualTolerance, True, False)
        activeMask = activeMask & ii
        if verbosityLevel > 2:
            print(activeMask)

        currentBlockSize = activeMask.sum()
        if currentBlockSize != previousBlockSize:
            previousBlockSize = currentBlockSize
            ident = np.eye(currentBlockSize, dtype=A.dtype)

        if currentBlockSize == 0:
            break

        if verbosityLevel > 0:
            print("current block size:", currentBlockSize)
            print("eigenvalue:", _lambda)
            print("residual norms:", residualNorms)
        if verbosityLevel > 10:
            print(eigBlockVector)

        activeBlockVectorR = _as2d(blockVectorR[:, activeMask])

        if iterationNumber > 0:
            activeBlockVectorP = _as2d(blockVectorP[:, activeMask])
            activeBlockVectorAP = _as2d(blockVectorAP[:, activeMask])
            if B is not None:
                activeBlockVectorBP = _as2d(blockVectorBP[:, activeMask])

        if M is not None:
            # Apply preconditioner T to the active residuals.
            activeBlockVectorR = M(activeBlockVectorR)

        ##
        # Apply constraints to the preconditioned residuals.
        if blockVectorY is not None:
            _applyConstraints(activeBlockVectorR, gramYBY, blockVectorBY, blockVectorY)

        ##
        # B-orthogonalize the preconditioned residuals to X.
        if B is not None:
            activeBlockVectorR = activeBlockVectorR - np.matmul(
                blockVectorX, np.matmul(blockVectorBX.T.conj(), activeBlockVectorR)
            )
        else:
            activeBlockVectorR = activeBlockVectorR - np.matmul(
                blockVectorX, np.matmul(blockVectorX.T.conj(), activeBlockVectorR)
            )

        ##
        # B-orthonormalize the preconditioned residuals.
        aux = _b_orthonormalize(B, activeBlockVectorR)
        activeBlockVectorR, activeBlockVectorBR = aux

        activeBlockVectorAR = A(activeBlockVectorR)

        if iterationNumber > 0:
            if B is not None:
                aux = _b_orthonormalize(
                    B, activeBlockVectorP, activeBlockVectorBP, retInvR=True
                )
                activeBlockVectorP, activeBlockVectorBP, invR, normal = aux
            else:
                aux = _b_orthonormalize(B, activeBlockVectorP, retInvR=True)
                activeBlockVectorP, _, invR, normal = aux
            # Function _b_orthonormalize returns None if Cholesky fails
            if activeBlockVectorP is not None:
                activeBlockVectorAP = activeBlockVectorAP / normal
                activeBlockVectorAP = np.dot(activeBlockVectorAP, invR)
                restart = False
            else:
                restart = True

        ##
        # Perform the Rayleigh Ritz Procedure:
        # Compute symmetric Gram matrices:

        if activeBlockVectorAR.dtype == "float32":
            myeps = 1
        elif activeBlockVectorR.dtype == "float32":
            myeps = 1e-4
        else:
            myeps = 1e-8

        if residualNorms.max() > myeps:
            explicitGramFlag = False
        else:
            # suggested by Garrett Moran
            explicitGramFlag = True

        # Shared memory assingments to simplify the code
        if B is None:
            blockVectorBX = blockVectorX
            activeBlockVectorBR = activeBlockVectorR
            if not restart:
                activeBlockVectorBP = activeBlockVectorP

        # Common submatrices:
        gramXAR = np.dot(blockVectorX.T.conj(), activeBlockVectorAR)
        gramRAR = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorAR)

        if explicitGramFlag:
            gramXAX = np.dot(blockVectorX.T.conj(), blockVectorAX)
            gramXBX = np.dot(blockVectorX.T.conj(), blockVectorBX)
            gramRBR = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorBR)
            gramXBR = np.dot(blockVectorX.T.conj(), activeBlockVectorBR)
        else:
            gramXAX = np.diag(_lambda)
            gramXBX = ident0
            gramRBR = ident
            gramXBR = np.zeros((sizeX, currentBlockSize), dtype=A.dtype)

        if not restart:
            gramXAP = np.dot(blockVectorX.T.conj(), activeBlockVectorAP)
            gramRAP = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorAP)
            gramPAP = np.dot(activeBlockVectorP.T.conj(), activeBlockVectorAP)
            gramXBP = np.dot(blockVectorX.T.conj(), activeBlockVectorBP)
            gramRBP = np.dot(activeBlockVectorR.T.conj(), activeBlockVectorBP)
            if explicitGramFlag:
                gramPBP = np.dot(activeBlockVectorP.T.conj(), activeBlockVectorBP)
            else:
                gramPBP = ident

            gramA = bmat(
                [
                    [gramXAX, gramXAR, gramXAP],
                    [gramXAR.T.conj(), gramRAR, gramRAP],
                    [gramXAP.T.conj(), gramRAP.T.conj(), gramPAP],
                ]
            )
            gramB = bmat(
                [
                    [gramXBX, gramXBR, gramXBP],
                    [gramXBR.T.conj(), gramRBR, gramRBP],
                    [gramXBP.T.conj(), gramRBP.T.conj(), gramPBP],
                ]
            )
            if verbosityLevel > 0:
                _report_nonhermitian(gramA, "gramA")
                _report_nonhermitian(gramB, "gramB")
            if verbosityLevel > 10:
                _save(gramA, "gramA")
                _save(gramB, "gramB")

            try:
                _lambda, eigBlockVector = eigh(gramA, gramB, check_finite=False)
            except LinAlgError:
                # try again after dropping the direction vectors P from RR
                restart = True

        if restart:
            gramA = bmat([[gramXAX, gramXAR], [gramXAR.T.conj(), gramRAR]])
            gramB = bmat([[gramXBX, gramXBR], [gramXBR.T.conj(), gramRBR]])
            if verbosityLevel > 0:
                _report_nonhermitian(gramA, "gramA")
                _report_nonhermitian(gramB, "gramB")
            if verbosityLevel > 10:
                _save(gramA, "gramA")
                _save(gramB, "gramB")
            try:
                _lambda, eigBlockVector = eigh(gramA, gramB, check_finite=False)
            except LinAlgError:
                raise ValueError("eigh has failed in lobpcg iterations")

        ii = _get_indx(_lambda, sizeX, largest)
        if verbosityLevel > 10:
            print(ii)
            print(_lambda)

        _lambda = _lambda[ii]
        eigBlockVector = eigBlockVector[:, ii]

        lambdaHistory.append(_lambda)

        if verbosityLevel > 10:
            print("lambda:", _lambda)
        #         # Normalize eigenvectors!
        #         aux = np.sum( eigBlockVector.conj() * eigBlockVector, 0 )
        #         eigVecNorms = np.sqrt( aux )
        #         eigBlockVector = eigBlockVector / eigVecNorms[np.newaxis, :]
        #         eigBlockVector, aux = _b_orthonormalize( B, eigBlockVector )

        if verbosityLevel > 10:
            print(eigBlockVector)

        # Compute Ritz vectors.
        if B is not None:
            if not restart:
                eigBlockVectorX = eigBlockVector[:sizeX]
                eigBlockVectorR = eigBlockVector[sizeX : sizeX + currentBlockSize]
                eigBlockVectorP = eigBlockVector[sizeX + currentBlockSize :]

                pp = np.dot(activeBlockVectorR, eigBlockVectorR)
                pp += np.dot(activeBlockVectorP, eigBlockVectorP)

                app = np.dot(activeBlockVectorAR, eigBlockVectorR)
                app += np.dot(activeBlockVectorAP, eigBlockVectorP)

                bpp = np.dot(activeBlockVectorBR, eigBlockVectorR)
                bpp += np.dot(activeBlockVectorBP, eigBlockVectorP)
            else:
                eigBlockVectorX = eigBlockVector[:sizeX]
                eigBlockVectorR = eigBlockVector[sizeX:]

                pp = np.dot(activeBlockVectorR, eigBlockVectorR)
                app = np.dot(activeBlockVectorAR, eigBlockVectorR)
                bpp = np.dot(activeBlockVectorBR, eigBlockVectorR)

            if verbosityLevel > 10:
                print(pp)
                print(app)
                print(bpp)

            blockVectorX = np.dot(blockVectorX, eigBlockVectorX) + pp
            blockVectorAX = np.dot(blockVectorAX, eigBlockVectorX) + app
            blockVectorBX = np.dot(blockVectorBX, eigBlockVectorX) + bpp

            blockVectorP, blockVectorAP, blockVectorBP = pp, app, bpp

        else:
            if not restart:
                eigBlockVectorX = eigBlockVector[:sizeX]
                eigBlockVectorR = eigBlockVector[sizeX : sizeX + currentBlockSize]
                eigBlockVectorP = eigBlockVector[sizeX + currentBlockSize :]

                pp = np.dot(activeBlockVectorR, eigBlockVectorR)
                pp += np.dot(activeBlockVectorP, eigBlockVectorP)

                app = np.dot(activeBlockVectorAR, eigBlockVectorR)
                app += np.dot(activeBlockVectorAP, eigBlockVectorP)
            else:
                eigBlockVectorX = eigBlockVector[:sizeX]
                eigBlockVectorR = eigBlockVector[sizeX:]

                pp = np.dot(activeBlockVectorR, eigBlockVectorR)
                app = np.dot(activeBlockVectorAR, eigBlockVectorR)

            if verbosityLevel > 10:
                print(pp)
                print(app)

            blockVectorX = np.dot(blockVectorX, eigBlockVectorX) + pp
            blockVectorAX = np.dot(blockVectorAX, eigBlockVectorX) + app

            blockVectorP, blockVectorAP = pp, app

    if B is not None:
        aux = blockVectorBX * _lambda[np.newaxis, :]

    else:
        aux = blockVectorX * _lambda[np.newaxis, :]

    blockVectorR = blockVectorAX - aux

    aux = np.sum(blockVectorR.conj() * blockVectorR, 0)
    residualNorms = np.sqrt(aux)

    # Future work: Need to add Postprocessing here:
    # Making sure eigenvectors "exactly" satisfy the blockVectorY constrains?
    # Making sure eigenvecotrs are "exactly" othonormalized by final "exact" RR
    # Computing the actual true residuals

    if verbosityLevel > 0:
        print("final eigenvalue:", _lambda)
        print("final residual norms:", residualNorms)

    if retLambdaHistory:
        if retResidualNormsHistory:
            return _lambda, blockVectorX, lambdaHistory, residualNormsHistory
        else:
            return _lambda, blockVectorX, lambdaHistory
    else:
        if retResidualNormsHistory:
            return _lambda, blockVectorX, residualNormsHistory
        else:
            return _lambda, blockVectorX