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.

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.

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 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)