def _eigsh(A, k=6, M=None, sigma=None, which='LM', v0=None,
           ncv=None, maxiter=None, tol=0, return_eigenvectors=True,
           Minv=None, OPinv=None, mode='normal'):
    """
    Find k eigenvalues and eigenvectors of the real symmetric square matrix
    or complex hermitian matrix A.

    Solves ``A * x[i] = w[i] * x[i]``, the standard eigenvalue problem for
    w[i] eigenvalues with corresponding eigenvectors x[i].

    If M is specified, solves ``A * x[i] = w[i] * M * x[i]``, the
    generalized eigenvalue problem for w[i] eigenvalues
    with corresponding eigenvectors x[i]

    Parameters
    ----------
    A : An N x N matrix, array, sparse matrix, or LinearOperator representing
        the operation A * x, where A is a real symmetric matrix
        For buckling mode (see below) A must additionally be positive-definite
    k : int, optional
        The number of eigenvalues and eigenvectors desired.
        `k` must be smaller than N. It is not possible to compute all
        eigenvectors of a matrix.

    Returns
    -------
    w : array
        Array of k eigenvalues
    v : array
        An array representing the `k` eigenvectors.  The column ``v[:, i]`` is
        the eigenvector corresponding to the eigenvalue ``w[i]``.

    Other Parameters
    ----------------
    M : An N x N matrix, array, sparse matrix, or linear operator representing
        the operation M * x for the generalized eigenvalue problem

            A * x = w * M * x.

        M must represent a real, symmetric matrix if A is real, and must
        represent a complex, hermitian matrix if A is complex. For best
        results, the data type of M should be the same as that of A.
        Additionally:

            If sigma is None, M is symmetric positive definite

            If sigma is specified, M is symmetric positive semi-definite

            In buckling mode, M is symmetric indefinite.

        If sigma is None, eigsh requires an operator to compute the solution
        of the linear equation ``M * x = b``. This is done internally via a
        (sparse) LU decomposition for an explicit matrix M, or via an
        iterative solver for a general linear operator.  Alternatively,
        the user can supply the matrix or operator Minv, which gives
        ``x = Minv * b = M^-1 * b``.
    sigma : real
        Find eigenvalues near sigma using shift-invert mode.  This requires
        an operator to compute the solution of the linear system
        `[A - sigma * M] x = b`, where M is the identity matrix if
        unspecified.  This is computed internally via a (sparse) LU
        decomposition for explicit matrices A & M, or via an iterative
        solver if either A or M is a general linear operator.
        Alternatively, the user can supply the matrix or operator OPinv,
        which gives ``x = OPinv * b = [A - sigma * M]^-1 * b``.
        Note that when sigma is specified, the keyword 'which' refers to
        the shifted eigenvalues ``w'[i]`` where:

            if mode == 'normal', ``w'[i] = 1 / (w[i] - sigma)``.

            if mode == 'cayley', ``w'[i] = (w[i] + sigma) / (w[i] - sigma)``.

            if mode == 'buckling', ``w'[i] = w[i] / (w[i] - sigma)``.

        (see further discussion in 'mode' below)
    v0 : ndarray, optional
        Starting vector for iteration.
        Default: random
    ncv : int, optional
        The number of Lanczos vectors generated ncv must be greater than k and
        smaller than n; it is recommended that ``ncv > 2*k``.
        Default: ``min(n, 2*k + 1)``
    which : str ['LM' | 'SM' | 'LA' | 'SA' | 'BE']
        If A is a complex hermitian matrix, 'BE' is invalid.
        Which `k` eigenvectors and eigenvalues to find:

            'LM' : Largest (in magnitude) eigenvalues

            'SM' : Smallest (in magnitude) eigenvalues

            'LA' : Largest (algebraic) eigenvalues

            'SA' : Smallest (algebraic) eigenvalues

            'BE' : Half (k/2) from each end of the spectrum

        When k is odd, return one more (k/2+1) from the high end.
        When sigma != None, 'which' refers to the shifted eigenvalues ``w'[i]``
        (see discussion in 'sigma', above).  ARPACK is generally better
        at finding large values than small values.  If small eigenvalues are
        desired, consider using shift-invert mode for better performance.
    maxiter : int, optional
        Maximum number of Arnoldi update iterations allowed
        Default: ``n*10``
    tol : float
        Relative accuracy for eigenvalues (stopping criterion).
        The default value of 0 implies machine precision.
    Minv : N x N matrix, array, sparse matrix, or LinearOperator
        See notes in M, above
    OPinv : N x N matrix, array, sparse matrix, or LinearOperator
        See notes in sigma, above.
    return_eigenvectors : bool
        Return eigenvectors (True) in addition to eigenvalues
    mode : string ['normal' | 'buckling' | 'cayley']
        Specify strategy to use for shift-invert mode.  This argument applies
        only for real-valued A and sigma != None.  For shift-invert mode,
        ARPACK internally solves the eigenvalue problem
        ``OP * x'[i] = w'[i] * B * x'[i]``
        and transforms the resulting Ritz vectors x'[i] and Ritz values w'[i]
        into the desired eigenvectors and eigenvalues of the problem
        ``A * x[i] = w[i] * M * x[i]``.
        The modes are as follows:

            'normal' :
                OP = [A - sigma * M]^-1 * M,
                B = M,
                w'[i] = 1 / (w[i] - sigma)

            'buckling' :
                OP = [A - sigma * M]^-1 * A,
                B = A,
                w'[i] = w[i] / (w[i] - sigma)

            'cayley' :
                OP = [A - sigma * M]^-1 * [A + sigma * M],
                B = M,
                w'[i] = (w[i] + sigma) / (w[i] - sigma)

        The choice of mode will affect which eigenvalues are selected by
        the keyword 'which', and can also impact the stability of
        convergence (see [2] for a discussion)

    Raises
    ------
    ArpackNoConvergence
        When the requested convergence is not obtained.

        The currently converged eigenvalues and eigenvectors can be found
        as ``eigenvalues`` and ``eigenvectors`` attributes of the exception
        object.

    See Also
    --------
    eigs : eigenvalues and eigenvectors for a general (nonsymmetric) matrix A
    svds : singular value decomposition for a matrix A

    Notes
    -----
    This function is a wrapper to the ARPACK [1]_ SSEUPD and DSEUPD
    functions which use the Implicitly Restarted Lanczos Method to
    find the eigenvalues and eigenvectors [2]_.

    References
    ----------
    .. [1] ARPACK Software, http://www.caam.rice.edu/software/ARPACK/
    .. [2] R. B. Lehoucq, D. C. Sorensen, and C. Yang,  ARPACK USERS GUIDE:
       Solution of Large Scale Eigenvalue Problems by Implicitly Restarted
       Arnoldi Methods. SIAM, Philadelphia, PA, 1998.

    Examples
    --------
    >>> import scipy.sparse as sparse
    >>> id = np.eye(13)
    >>> vals, vecs = sparse.linalg.eigsh(id, k=6)
    >>> vals
    array([ 1.,  1.,  1.,  1.,  1.,  1.])
    >>> vecs.shape
    (13, 6)

    """
    # complex hermitian matrices should be solved with eigs
    if np.issubdtype(A.dtype, np.complexfloating):
        if mode != 'normal':
            raise ValueError("mode=%s cannot be used with "
                             "complex matrix A" % mode)
        if which == 'BE':
            raise ValueError("which='BE' cannot be used with complex matrix A")
        elif which == 'LA':
            which = 'LR'
        elif which == 'SA':
            which = 'SR'
        ret = eigs(A, k, M=M, sigma=sigma, which=which, v0=v0,
                   ncv=ncv, maxiter=maxiter, tol=tol,
                   return_eigenvectors=return_eigenvectors, Minv=Minv,
                   OPinv=OPinv)

        if return_eigenvectors:
            return ret[0].real, ret[1]
        else:
            return ret.real

    if A.shape[0] != A.shape[1]:
        raise ValueError('expected square matrix (shape=%s)' % (A.shape,))
    if M is not None:
        if M.shape != A.shape:
            raise ValueError('wrong M dimensions %s, should be %s'
                             % (M.shape, A.shape))
        if np.dtype(M.dtype).char.lower() != np.dtype(A.dtype).char.lower():
            import warnings
            warnings.warn('M does not have the same type precision as A. '
                          'This may adversely affect ARPACK convergence')
    n = A.shape[0]

    if k <= 0 or k >= n:
        raise ValueError("k must be between 1 and the order of the "
                         "square input matrix.")

    if sigma is None:
        A = _aslinearoperator_with_dtype(A)
        matvec = A.matvec

        if OPinv is not None:
            raise ValueError("OPinv should not be specified "
                             "with sigma = None.")
        if M is None:
            # standard eigenvalue problem
            mode = 1
            M_matvec = None
            Minv_matvec = None
            if Minv is not None:
                raise ValueError("Minv should not be "
                                 "specified with M = None.")
        else:
            # general eigenvalue problem
            mode = 2
            if Minv is None:
                Minv_matvec = get_inv_matvec(M, symmetric=True, tol=tol)
            else:
                Minv = _aslinearoperator_with_dtype(Minv)
                Minv_matvec = Minv.matvec
            M_matvec = _aslinearoperator_with_dtype(M).matvec
    else:
        # sigma is not None: shift-invert mode
        if Minv is not None:
            raise ValueError("Minv should not be specified when sigma is")

        # normal mode
        if mode == 'normal':
            mode = 3
            matvec = None
            if OPinv is None:
                Minv_matvec = get_OPinv_matvec(A, M, sigma,
                                               symmetric=True, tol=tol)
            else:
                OPinv = _aslinearoperator_with_dtype(OPinv)
                Minv_matvec = OPinv.matvec
            if M is None:
                M_matvec = None
            else:
                M = _aslinearoperator_with_dtype(M)
                M_matvec = M.matvec

        # buckling mode
        elif mode == 'buckling':
            mode = 4
            if OPinv is None:
                Minv_matvec = get_OPinv_matvec(A, M, sigma,
                                               symmetric=True, tol=tol)
            else:
                Minv_matvec = _aslinearoperator_with_dtype(OPinv).matvec
            matvec = _aslinearoperator_with_dtype(A).matvec
            M_matvec = None

        # cayley-transform mode
        elif mode == 'cayley':
            mode = 5
            matvec = _aslinearoperator_with_dtype(A).matvec
            if OPinv is None:
                Minv_matvec = get_OPinv_matvec(A, M, sigma,
                                               symmetric=True, tol=tol)
            else:
                Minv_matvec = _aslinearoperator_with_dtype(OPinv).matvec
            if M is None:
                M_matvec = None
            else:
                M_matvec = _aslinearoperator_with_dtype(M).matvec

        # unrecognized mode
        else:
            raise ValueError("unrecognized mode '%s'" % mode)

    params = _SymmetricArpackParams(n, k, A.dtype.char, matvec, mode,
                                    M_matvec, Minv_matvec, sigma,
                                    ncv, v0, maxiter, which, tol)

    while not params.converged:
        params.iterate()

    return params.extract(return_eigenvectors)
dataset/ETHPy150Open scikit-learn/scikit-learn/sklearn/utils/arpack.py/_eigsh/original