Module « scipy.linalg »

Fonction eigh_tridiagonal - module scipy.linalg

Signature de la fonction eigh_tridiagonal

def eigh_tridiagonal(d, e, eigvals_only=False, select='a', select_range=None, check_finite=True, tol=0.0, lapack_driver='auto') 

Description

eigh_tridiagonal.__doc__

    Solve eigenvalue problem for a real symmetric tridiagonal matrix.

    Find eigenvalues `w` and optionally right eigenvectors `v` of ``a``::

        a v[:,i] = w[i] v[:,i]
        v.H v    = identity

    For a real symmetric matrix ``a`` with diagonal elements `d` and
    off-diagonal elements `e`.

    Parameters
    ----------
    d : ndarray, shape (ndim,)
        The diagonal elements of the array.
    e : ndarray, shape (ndim-1,)
        The off-diagonal elements of the array.
    select : {'a', 'v', 'i'}, optional
        Which eigenvalues to calculate

        ======  ========================================
        select  calculated
        ======  ========================================
        'a'     All eigenvalues
        'v'     Eigenvalues in the interval (min, max]
        'i'     Eigenvalues with indices min <= i <= max
        ======  ========================================
    select_range : (min, max), optional
        Range of selected eigenvalues
    check_finite : bool, optional
        Whether to check that the input matrix contains only finite numbers.
        Disabling may give a performance gain, but may result in problems
        (crashes, non-termination) if the inputs do contain infinities or NaNs.
    tol : float
        The absolute tolerance to which each eigenvalue is required
        (only used when 'stebz' is the `lapack_driver`).
        An eigenvalue (or cluster) is considered to have converged if it
        lies in an interval of this width. If <= 0. (default),
        the value ``eps*|a|`` is used where eps is the machine precision,
        and ``|a|`` is the 1-norm of the matrix ``a``.
    lapack_driver : str
        LAPACK function to use, can be 'auto', 'stemr', 'stebz', 'sterf',
        or 'stev'. When 'auto' (default), it will use 'stemr' if ``select='a'``
        and 'stebz' otherwise. When 'stebz' is used to find the eigenvalues and
        ``eigvals_only=False``, then a second LAPACK call (to ``?STEIN``) is
        used to find the corresponding eigenvectors. 'sterf' can only be
        used when ``eigvals_only=True`` and ``select='a'``. 'stev' can only
        be used when ``select='a'``.

    Returns
    -------
    w : (M,) ndarray
        The eigenvalues, in ascending order, each repeated according to its
        multiplicity.
    v : (M, M) ndarray
        The normalized eigenvector corresponding to the eigenvalue ``w[i]`` is
        the column ``v[:,i]``.

    Raises
    ------
    LinAlgError
        If eigenvalue computation does not converge.

    See Also
    --------
    eigvalsh_tridiagonal : eigenvalues of symmetric/Hermitian tridiagonal
        matrices
    eig : eigenvalues and right eigenvectors for non-symmetric arrays
    eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays
    eig_banded : eigenvalues and right eigenvectors for symmetric/Hermitian
        band matrices

    Notes
    -----
    This function makes use of LAPACK ``S/DSTEMR`` routines.

    Examples
    --------
    >>> from scipy.linalg import eigh_tridiagonal
    >>> d = 3*np.ones(4)
    >>> e = -1*np.ones(3)
    >>> w, v = eigh_tridiagonal(d, e)
    >>> A = np.diag(d) + np.diag(e, k=1) + np.diag(e, k=-1)
    >>> np.allclose(A @ v - v @ np.diag(w), np.zeros((4, 4)))
    True