Module « scipy.linalg »

Fonction eigvalsh - module scipy.linalg

Signature de la fonction eigvalsh

def eigvalsh(a, b=None, lower=True, overwrite_a=False, overwrite_b=False, turbo=True, eigvals=None, type=1, check_finite=True, subset_by_index=None, subset_by_value=None, driver=None) 

Description

eigvalsh.__doc__

    Solves a standard or generalized eigenvalue problem for a complex
    Hermitian or real symmetric matrix.

    Find eigenvalues array ``w`` of array ``a``, where ``b`` is positive
    definite such that for every eigenvalue λ (i-th entry of w) and its
    eigenvector vi (i-th column of v) satisfies::

                      a @ vi = λ * b @ vi
        vi.conj().T @ a @ vi = λ
        vi.conj().T @ b @ vi = 1

    In the standard problem, b is assumed to be the identity matrix.

    Parameters
    ----------
    a : (M, M) array_like
        A complex Hermitian or real symmetric matrix whose eigenvalues will
        be computed.
    b : (M, M) array_like, optional
        A complex Hermitian or real symmetric definite positive matrix in.
        If omitted, identity matrix is assumed.
    lower : bool, optional
        Whether the pertinent array data is taken from the lower or upper
        triangle of ``a`` and, if applicable, ``b``. (Default: lower)
    overwrite_a : bool, optional
        Whether to overwrite data in ``a`` (may improve performance). Default
        is False.
    overwrite_b : bool, optional
        Whether to overwrite data in ``b`` (may improve performance). Default
        is False.
    type : int, optional
        For the generalized problems, this keyword specifies the problem type
        to be solved for ``w`` and ``v`` (only takes 1, 2, 3 as possible
        inputs)::

            1 =>     a @ v = w @ b @ v
            2 => a @ b @ v = w @ v
            3 => b @ a @ v = w @ v

        This keyword is ignored for standard problems.
    check_finite : bool, optional
        Whether to check that the input matrices contain 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.
    subset_by_index : iterable, optional
        If provided, this two-element iterable defines the start and the end
        indices of the desired eigenvalues (ascending order and 0-indexed).
        To return only the second smallest to fifth smallest eigenvalues,
        ``[1, 4]`` is used. ``[n-3, n-1]`` returns the largest three. Only
        available with "evr", "evx", and "gvx" drivers. The entries are
        directly converted to integers via ``int()``.
    subset_by_value : iterable, optional
        If provided, this two-element iterable defines the half-open interval
        ``(a, b]`` that, if any, only the eigenvalues between these values
        are returned. Only available with "evr", "evx", and "gvx" drivers. Use
        ``np.inf`` for the unconstrained ends.
    driver: str, optional
        Defines which LAPACK driver should be used. Valid options are "ev",
        "evd", "evr", "evx" for standard problems and "gv", "gvd", "gvx" for
        generalized (where b is not None) problems. See the Notes section of
        `scipy.linalg.eigh`.
    turbo : bool, optional
        *Deprecated by ``driver=gvd`` option*. Has no significant effect for
        eigenvalue computations since no eigenvectors are requested.

        .. deprecated:: 1.5.0

    eigvals : tuple (lo, hi), optional
        *Deprecated by ``subset_by_index`` keyword*. Indexes of the smallest
        and largest (in ascending order) eigenvalues and corresponding
        eigenvectors to be returned: 0 <= lo <= hi <= M-1. If omitted, all
        eigenvalues and eigenvectors are returned.

        .. deprecated:: 1.5.0

    Returns
    -------
    w : (N,) ndarray
        The ``N`` (``1<=N<=M``) selected eigenvalues, in ascending order, each
        repeated according to its multiplicity.

    Raises
    ------
    LinAlgError
        If eigenvalue computation does not converge, an error occurred, or
        b matrix is not definite positive. Note that if input matrices are
        not symmetric or Hermitian, no error will be reported but results will
        be wrong.

    See Also
    --------
    eigh : eigenvalues and right eigenvectors for symmetric/Hermitian arrays
    eigvals : eigenvalues of general arrays
    eigvals_banded : eigenvalues for symmetric/Hermitian band matrices
    eigvalsh_tridiagonal : eigenvalues of symmetric/Hermitian tridiagonal
        matrices

    Notes
    -----
    This function does not check the input array for being Hermitian/symmetric
    in order to allow for representing arrays with only their upper/lower
    triangular parts.

    This function serves as a one-liner shorthand for `scipy.linalg.eigh` with
    the option ``eigvals_only=True`` to get the eigenvalues and not the
    eigenvectors. Here it is kept as a legacy convenience. It might be
    beneficial to use the main function to have full control and to be a bit
    more pythonic.

    Examples
    --------
    For more examples see `scipy.linalg.eigh`.

    >>> from scipy.linalg import eigvalsh
    >>> A = np.array([[6, 3, 1, 5], [3, 0, 5, 1], [1, 5, 6, 2], [5, 1, 2, 2]])
    >>> w = eigvalsh(A)
    >>> w
    array([-3.74637491, -0.76263923,  6.08502336, 12.42399079])