4.1.1.3.21. 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 a eigenvalue, returns a boolean denoting whether the eigenvalue should be sorted to the top-left (True). For real matrix pairs, the sort function takes three real arguments (alphar, alphai, beta). The eigenvalue x = (alphar + alphai*1j)/beta. For complex matrix pairs or output=’complex’, the sort function takes two complex arguments (alpha, beta). The eigenvalue x = (alpha/beta). 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) 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.