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def qz(A, B, output='real', lwork=None, sort=None, overwrite_a=False, overwrite_b=False, check_finite=True)
QZ decomposition for generalized eigenvalues of a pair of matrices.
The QZ, or generalized Schur, decomposition for a pair of n-by-n
matrices (A,B) is::
(A,B) = (Q @ AA @ Z*, Q @ BB @ Z*)
where AA, BB is in generalized Schur form if BB is upper-triangular
with non-negative diagonal and AA is upper-triangular, or for real QZ
decomposition (``output='real'``) block upper triangular with 1x1
and 2x2 blocks. In this case, the 1x1 blocks correspond to real
generalized eigenvalues and 2x2 blocks are 'standardized' by making
the corresponding elements of BB have the form::
[ a 0 ]
[ 0 b ]
and the pair of corresponding 2x2 blocks in AA and BB will have a complex
conjugate pair of generalized eigenvalues. If (``output='complex'``) or
A and B are complex matrices, Z' denotes the conjugate-transpose of Z.
Q and Z are unitary matrices.
Parameters
----------
A : (N, N) array_like
2-D array to decompose
B : (N, N) array_like
2-D array to decompose
output : {'real', 'complex'}, optional
Construct the real or complex QZ decomposition for real matrices.
Default is 'real'.
lwork : int, optional
Work array size. If None or -1, it is automatically computed.
sort : {None, callable, 'lhp', 'rhp', 'iuc', 'ouc'}, optional
NOTE: THIS INPUT IS DISABLED FOR NOW. Use ordqz instead.
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)
Defaults to None (no sorting).
overwrite_a : bool, optional
Whether to overwrite data in a (may improve performance)
overwrite_b : bool, optional
Whether to overwrite data in b (may improve performance)
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.
Q : (N, N) ndarray
The left Schur vectors.
Z : (N, N) ndarray
The right Schur vectors.
See Also
--------
ordqz
Notes
-----
Q is transposed versus the equivalent function in Matlab.
.. versionadded:: 0.11.0
Examples
--------
>>> import numpy as np
>>> from scipy.linalg import qz
>>> A = np.array([[1, 2, -1], [5, 5, 5], [2, 4, -8]])
>>> B = np.array([[1, 1, -3], [3, 1, -1], [5, 6, -2]])
Compute the decomposition. The QZ decomposition is not unique, so
depending on the underlying library that is used, there may be
differences in the signs of coefficients in the following output.
>>> AA, BB, Q, Z = qz(A, B)
>>> AA
array([[-1.36949157, -4.05459025, 7.44389431],
[ 0. , 7.65653432, 5.13476017],
[ 0. , -0.65978437, 2.4186015 ]]) # may vary
>>> BB
array([[ 1.71890633, -1.64723705, -0.72696385],
[ 0. , 8.6965692 , -0. ],
[ 0. , 0. , 2.27446233]]) # may vary
>>> Q
array([[-0.37048362, 0.1903278 , 0.90912992],
[-0.90073232, 0.16534124, -0.40167593],
[ 0.22676676, 0.96769706, -0.11017818]]) # may vary
>>> Z
array([[-0.67660785, 0.63528924, -0.37230283],
[ 0.70243299, 0.70853819, -0.06753907],
[ 0.22088393, -0.30721526, -0.92565062]]) # may vary
Verify the QZ decomposition. With real output, we only need the
transpose of ``Z`` in the following expressions.
>>> Q @ AA @ Z.T # Should be A
array([[ 1., 2., -1.],
[ 5., 5., 5.],
[ 2., 4., -8.]])
>>> Q @ BB @ Z.T # Should be B
array([[ 1., 1., -3.],
[ 3., 1., -1.],
[ 5., 6., -2.]])
Repeat the decomposition, but with ``output='complex'``.
>>> AA, BB, Q, Z = qz(A, B, output='complex')
For conciseness in the output, we use ``np.set_printoptions()`` to set
the output precision of NumPy arrays to 3 and display tiny values as 0.
>>> np.set_printoptions(precision=3, suppress=True)
>>> AA
array([[-1.369+0.j , 2.248+4.237j, 4.861-5.022j],
[ 0. +0.j , 7.037+2.922j, 0.794+4.932j],
[ 0. +0.j , 0. +0.j , 2.655-1.103j]]) # may vary
>>> BB
array([[ 1.719+0.j , -1.115+1.j , -0.763-0.646j],
[ 0. +0.j , 7.24 +0.j , -3.144+3.322j],
[ 0. +0.j , 0. +0.j , 2.732+0.j ]]) # may vary
>>> Q
array([[ 0.326+0.175j, -0.273-0.029j, -0.886-0.052j],
[ 0.794+0.426j, -0.093+0.134j, 0.402-0.02j ],
[-0.2 -0.107j, -0.816+0.482j, 0.151-0.167j]]) # may vary
>>> Z
array([[ 0.596+0.32j , -0.31 +0.414j, 0.393-0.347j],
[-0.619-0.332j, -0.479+0.314j, 0.154-0.393j],
[-0.195-0.104j, 0.576+0.27j , 0.715+0.187j]]) # may vary
With complex arrays, we must use ``Z.conj().T`` in the following
expressions to verify the decomposition.
>>> Q @ AA @ Z.conj().T # Should be A
array([[ 1.-0.j, 2.-0.j, -1.-0.j],
[ 5.+0.j, 5.+0.j, 5.-0.j],
[ 2.+0.j, 4.+0.j, -8.+0.j]])
>>> Q @ BB @ Z.conj().T # Should be B
array([[ 1.+0.j, 1.+0.j, -3.+0.j],
[ 3.-0.j, 1.-0.j, -1.+0.j],
[ 5.+0.j, 6.+0.j, -2.+0.j]])
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