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scipy.linalg.

# ordqz

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

  QZ decomposition for a pair of matrices with reordering.

  The documentation is written assuming array arguments are of specified “core” shapes. However, array argument(s) of this function may have additional “batch” dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see Batched Linear Operations for details.

  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.

  See also

  `qz`

  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.

  Added in version 0.17.0.

  Examples

      >>> import numpy as np
      >>> 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)