Module « scipy.linalg »

Fonction ordqz - module scipy.linalg

Signature de la fonction ordqz

def ordqz(A, B, sort='lhp', output='real', overwrite_a=False, overwrite_b=False, check_finite=True) 

Description

ordqz.__doc__

QZ decomposition for a pair of matrices with reordering.

    Parameters
    ----------
    A : (N, N) array_like
        2-D array to decompose
    B : (N, N) array_like
        2-D array to decompose
    sort : {callable, 'lhp', 'rhp', 'iuc', 'ouc'}, optional
        Specifies whether the upper eigenvalues should be sorted. A
        callable may be passed that, given an ordered pair ``(alpha,
        beta)`` representing the eigenvalue ``x = (alpha/beta)``,
        returns a boolean denoting whether the eigenvalue should be
        sorted to the top-left (True). For the real matrix pairs
        ``beta`` is real while ``alpha`` can be complex, and for
        complex matrix pairs both ``alpha`` and ``beta`` can be
        complex. The callable must be able to accept a NumPy
        array. Alternatively, string parameters may be used:

            - 'lhp'   Left-hand plane (x.real < 0.0)
            - 'rhp'   Right-hand plane (x.real > 0.0)
            - 'iuc'   Inside the unit circle (x*x.conjugate() < 1.0)
            - 'ouc'   Outside the unit circle (x*x.conjugate() > 1.0)

        With the predefined sorting functions, an infinite eigenvalue
        (i.e., ``alpha != 0`` and ``beta = 0``) is considered to lie in
        neither the left-hand nor the right-hand plane, but it is
        considered to lie outside the unit circle. For the eigenvalue
        ``(alpha, beta) = (0, 0)``, the predefined sorting functions
        all return `False`.
    output : str {'real','complex'}, optional
        Construct the real or complex QZ decomposition for real matrices.
        Default is 'real'.
    overwrite_a : bool, optional
        If True, the contents of A are overwritten.
    overwrite_b : bool, optional
        If True, the contents of B are overwritten.
    check_finite : bool, optional
        If true checks the elements of `A` and `B` are finite numbers. If
        false does no checking and passes matrix through to
        underlying algorithm.

    Returns
    -------
    AA : (N, N) ndarray
        Generalized Schur form of A.
    BB : (N, N) ndarray
        Generalized Schur form of B.
    alpha : (N,) ndarray
        alpha = alphar + alphai * 1j. See notes.
    beta : (N,) ndarray
        See notes.
    Q : (N, N) ndarray
        The left Schur vectors.
    Z : (N, N) ndarray
        The right Schur vectors.

    Notes
    -----
    On exit, ``(ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N``, will be the
    generalized eigenvalues.  ``ALPHAR(j) + ALPHAI(j)*i`` and
    ``BETA(j),j=1,...,N`` are the diagonals of the complex Schur form (S,T)
    that would result if the 2-by-2 diagonal blocks of the real generalized
    Schur form of (A,B) were further reduced to triangular form using complex
    unitary transformations. If ALPHAI(j) is zero, then the jth eigenvalue is
    real; if positive, then the ``j``th and ``(j+1)``st eigenvalues are a
    complex conjugate pair, with ``ALPHAI(j+1)`` negative.

    .. versionadded:: 0.17.0

    See also
    --------
    qz

    Examples
    --------
    >>> from scipy.linalg import ordqz
    >>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]])
    >>> B = np.array([[0, 6, 0, 0], [5, 0, 2, 1], [5, 2, 6, 6], [4, 7, 7, 7]])
    >>> AA, BB, alpha, beta, Q, Z = ordqz(A, B, sort='lhp')

    Since we have sorted for left half plane eigenvalues, negatives come first

    >>> (alpha/beta).real < 0
    array([ True,  True, False, False], dtype=bool)