scipy.linalg.ordqz¶

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

QZ decomposition for a pair of matrices with reordering.

New in version 0.17.0.

Parameters: A : (N, N) array_like 2d array to decompose B : (N, N) array_like 2d 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. 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.

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 j-th eigenvalue is real; if positive, then the j-th and (j+1)-st eigenvalues are a complex conjugate pair, with ALPHAI(j+1) negative.

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)


scipy.linalg.qz

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scipy.linalg.schur