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403 lines
14 KiB
Python
403 lines
14 KiB
Python
import warnings
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import numpy as np
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from numpy import asarray_chkfinite
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from .misc import LinAlgError, _datacopied, LinAlgWarning
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from .lapack import get_lapack_funcs
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__all__ = ['qz', 'ordqz']
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_double_precision = ['i', 'l', 'd']
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def _select_function(sort):
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if callable(sort):
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# assume the user knows what they're doing
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sfunction = sort
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elif sort == 'lhp':
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sfunction = _lhp
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elif sort == 'rhp':
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sfunction = _rhp
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elif sort == 'iuc':
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sfunction = _iuc
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elif sort == 'ouc':
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sfunction = _ouc
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else:
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raise ValueError("sort parameter must be None, a callable, or "
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"one of ('lhp','rhp','iuc','ouc')")
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return sfunction
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def _lhp(x, y):
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out = np.empty_like(x, dtype=bool)
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nonzero = (y != 0)
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# handles (x, y) = (0, 0) too
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out[~nonzero] = False
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out[nonzero] = (np.real(x[nonzero]/y[nonzero]) < 0.0)
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return out
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def _rhp(x, y):
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out = np.empty_like(x, dtype=bool)
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nonzero = (y != 0)
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# handles (x, y) = (0, 0) too
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out[~nonzero] = False
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out[nonzero] = (np.real(x[nonzero]/y[nonzero]) > 0.0)
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return out
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def _iuc(x, y):
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out = np.empty_like(x, dtype=bool)
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nonzero = (y != 0)
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# handles (x, y) = (0, 0) too
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out[~nonzero] = False
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out[nonzero] = (abs(x[nonzero]/y[nonzero]) < 1.0)
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return out
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def _ouc(x, y):
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out = np.empty_like(x, dtype=bool)
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xzero = (x == 0)
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yzero = (y == 0)
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out[xzero & yzero] = False
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out[~xzero & yzero] = True
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out[~yzero] = (abs(x[~yzero]/y[~yzero]) > 1.0)
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return out
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def _qz(A, B, output='real', lwork=None, sort=None, overwrite_a=False,
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overwrite_b=False, check_finite=True):
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if sort is not None:
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# Disabled due to segfaults on win32, see ticket 1717.
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raise ValueError("The 'sort' input of qz() has to be None and will be "
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"removed in a future release. Use ordqz instead.")
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if output not in ['real', 'complex', 'r', 'c']:
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raise ValueError("argument must be 'real', or 'complex'")
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if check_finite:
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a1 = asarray_chkfinite(A)
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b1 = asarray_chkfinite(B)
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else:
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a1 = np.asarray(A)
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b1 = np.asarray(B)
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a_m, a_n = a1.shape
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b_m, b_n = b1.shape
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if not (a_m == a_n == b_m == b_n):
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raise ValueError("Array dimensions must be square and agree")
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typa = a1.dtype.char
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if output in ['complex', 'c'] and typa not in ['F', 'D']:
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if typa in _double_precision:
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a1 = a1.astype('D')
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typa = 'D'
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else:
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a1 = a1.astype('F')
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typa = 'F'
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typb = b1.dtype.char
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if output in ['complex', 'c'] and typb not in ['F', 'D']:
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if typb in _double_precision:
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b1 = b1.astype('D')
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typb = 'D'
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else:
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b1 = b1.astype('F')
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typb = 'F'
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overwrite_a = overwrite_a or (_datacopied(a1, A))
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overwrite_b = overwrite_b or (_datacopied(b1, B))
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gges, = get_lapack_funcs(('gges',), (a1, b1))
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if lwork is None or lwork == -1:
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# get optimal work array size
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result = gges(lambda x: None, a1, b1, lwork=-1)
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lwork = result[-2][0].real.astype(np.int_)
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sfunction = lambda x: None
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result = gges(sfunction, a1, b1, lwork=lwork, overwrite_a=overwrite_a,
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overwrite_b=overwrite_b, sort_t=0)
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info = result[-1]
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if info < 0:
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raise ValueError("Illegal value in argument {} of gges".format(-info))
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elif info > 0 and info <= a_n:
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warnings.warn("The QZ iteration failed. (a,b) are not in Schur "
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"form, but ALPHAR(j), ALPHAI(j), and BETA(j) should be "
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"correct for J={},...,N".format(info-1), LinAlgWarning,
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stacklevel=3)
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elif info == a_n+1:
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raise LinAlgError("Something other than QZ iteration failed")
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elif info == a_n+2:
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raise LinAlgError("After reordering, roundoff changed values of some "
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"complex eigenvalues so that leading eigenvalues "
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"in the Generalized Schur form no longer satisfy "
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"sort=True. This could also be due to scaling.")
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elif info == a_n+3:
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raise LinAlgError("Reordering failed in <s,d,c,z>tgsen")
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return result, gges.typecode
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def qz(A, B, output='real', lwork=None, sort=None, overwrite_a=False,
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overwrite_b=False, check_finite=True):
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"""
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QZ decomposition for generalized eigenvalues of a pair of matrices.
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The QZ, or generalized Schur, decomposition for a pair of N x N
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nonsymmetric matrices (A,B) is::
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(A,B) = (Q*AA*Z', Q*BB*Z')
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where AA, BB is in generalized Schur form if BB is upper-triangular
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with non-negative diagonal and AA is upper-triangular, or for real QZ
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decomposition (``output='real'``) block upper triangular with 1x1
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and 2x2 blocks. In this case, the 1x1 blocks correspond to real
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generalized eigenvalues and 2x2 blocks are 'standardized' by making
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the corresponding elements of BB have the form::
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[ a 0 ]
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[ 0 b ]
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and the pair of corresponding 2x2 blocks in AA and BB will have a complex
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conjugate pair of generalized eigenvalues. If (``output='complex'``) or
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A and B are complex matrices, Z' denotes the conjugate-transpose of Z.
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Q and Z are unitary matrices.
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Parameters
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----------
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A : (N, N) array_like
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2-D array to decompose
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B : (N, N) array_like
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2-D array to decompose
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output : {'real', 'complex'}, optional
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Construct the real or complex QZ decomposition for real matrices.
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Default is 'real'.
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lwork : int, optional
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Work array size. If None or -1, it is automatically computed.
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sort : {None, callable, 'lhp', 'rhp', 'iuc', 'ouc'}, optional
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NOTE: THIS INPUT IS DISABLED FOR NOW. Use ordqz instead.
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Specifies whether the upper eigenvalues should be sorted. A callable
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may be passed that, given a eigenvalue, returns a boolean denoting
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whether the eigenvalue should be sorted to the top-left (True). For
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real matrix pairs, the sort function takes three real arguments
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(alphar, alphai, beta). The eigenvalue
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``x = (alphar + alphai*1j)/beta``. For complex matrix pairs or
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output='complex', the sort function takes two complex arguments
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(alpha, beta). The eigenvalue ``x = (alpha/beta)``. Alternatively,
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string parameters may be used:
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- 'lhp' Left-hand plane (x.real < 0.0)
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- 'rhp' Right-hand plane (x.real > 0.0)
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- 'iuc' Inside the unit circle (x*x.conjugate() < 1.0)
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- 'ouc' Outside the unit circle (x*x.conjugate() > 1.0)
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Defaults to None (no sorting).
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overwrite_a : bool, optional
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Whether to overwrite data in a (may improve performance)
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overwrite_b : bool, optional
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Whether to overwrite data in b (may improve performance)
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check_finite : bool, optional
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If true checks the elements of `A` and `B` are finite numbers. If
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false does no checking and passes matrix through to
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underlying algorithm.
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Returns
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-------
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AA : (N, N) ndarray
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Generalized Schur form of A.
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BB : (N, N) ndarray
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Generalized Schur form of B.
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Q : (N, N) ndarray
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The left Schur vectors.
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Z : (N, N) ndarray
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The right Schur vectors.
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Notes
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-----
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Q is transposed versus the equivalent function in Matlab.
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.. versionadded:: 0.11.0
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Examples
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--------
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>>> from scipy import linalg
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>>> np.random.seed(1234)
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>>> A = np.arange(9).reshape((3, 3))
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>>> B = np.random.randn(3, 3)
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>>> AA, BB, Q, Z = linalg.qz(A, B)
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>>> AA
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array([[-13.40928183, -4.62471562, 1.09215523],
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[ 0. , 0. , 1.22805978],
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[ 0. , 0. , 0.31973817]])
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>>> BB
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array([[ 0.33362547, -1.37393632, 0.02179805],
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[ 0. , 1.68144922, 0.74683866],
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[ 0. , 0. , 0.9258294 ]])
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>>> Q
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array([[ 0.14134727, -0.97562773, 0.16784365],
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[ 0.49835904, -0.07636948, -0.86360059],
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[ 0.85537081, 0.20571399, 0.47541828]])
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>>> Z
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array([[-0.24900855, -0.51772687, 0.81850696],
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[-0.79813178, 0.58842606, 0.12938478],
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[-0.54861681, -0.6210585 , -0.55973739]])
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See also
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--------
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ordqz
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"""
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# output for real
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# AA, BB, sdim, alphar, alphai, beta, vsl, vsr, work, info
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# output for complex
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# AA, BB, sdim, alpha, beta, vsl, vsr, work, info
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result, _ = _qz(A, B, output=output, lwork=lwork, sort=sort,
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overwrite_a=overwrite_a, overwrite_b=overwrite_b,
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check_finite=check_finite)
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return result[0], result[1], result[-4], result[-3]
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def ordqz(A, B, sort='lhp', output='real', overwrite_a=False,
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overwrite_b=False, check_finite=True):
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"""QZ decomposition for a pair of matrices with reordering.
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.. versionadded:: 0.17.0
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Parameters
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----------
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A : (N, N) array_like
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2-D array to decompose
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B : (N, N) array_like
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2-D array to decompose
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sort : {callable, 'lhp', 'rhp', 'iuc', 'ouc'}, optional
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Specifies whether the upper eigenvalues should be sorted. A
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callable may be passed that, given an ordered pair ``(alpha,
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beta)`` representing the eigenvalue ``x = (alpha/beta)``,
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returns a boolean denoting whether the eigenvalue should be
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sorted to the top-left (True). For the real matrix pairs
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``beta`` is real while ``alpha`` can be complex, and for
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complex matrix pairs both ``alpha`` and ``beta`` can be
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complex. The callable must be able to accept a NumPy
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array. Alternatively, string parameters may be used:
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- 'lhp' Left-hand plane (x.real < 0.0)
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- 'rhp' Right-hand plane (x.real > 0.0)
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- 'iuc' Inside the unit circle (x*x.conjugate() < 1.0)
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- 'ouc' Outside the unit circle (x*x.conjugate() > 1.0)
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With the predefined sorting functions, an infinite eigenvalue
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(i.e., ``alpha != 0`` and ``beta = 0``) is considered to lie in
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neither the left-hand nor the right-hand plane, but it is
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considered to lie outside the unit circle. For the eigenvalue
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``(alpha, beta) = (0, 0)``, the predefined sorting functions
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all return `False`.
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output : str {'real','complex'}, optional
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Construct the real or complex QZ decomposition for real matrices.
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Default is 'real'.
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overwrite_a : bool, optional
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If True, the contents of A are overwritten.
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overwrite_b : bool, optional
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If True, the contents of B are overwritten.
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check_finite : bool, optional
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If true checks the elements of `A` and `B` are finite numbers. If
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false does no checking and passes matrix through to
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underlying algorithm.
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Returns
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-------
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AA : (N, N) ndarray
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Generalized Schur form of A.
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BB : (N, N) ndarray
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Generalized Schur form of B.
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alpha : (N,) ndarray
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alpha = alphar + alphai * 1j. See notes.
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beta : (N,) ndarray
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See notes.
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Q : (N, N) ndarray
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The left Schur vectors.
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Z : (N, N) ndarray
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The right Schur vectors.
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Notes
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-----
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On exit, ``(ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N``, will be the
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generalized eigenvalues. ``ALPHAR(j) + ALPHAI(j)*i`` and
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``BETA(j),j=1,...,N`` are the diagonals of the complex Schur form (S,T)
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that would result if the 2-by-2 diagonal blocks of the real generalized
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Schur form of (A,B) were further reduced to triangular form using complex
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unitary transformations. If ALPHAI(j) is zero, then the jth eigenvalue is
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real; if positive, then the ``j``th and ``(j+1)``st eigenvalues are a
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complex conjugate pair, with ``ALPHAI(j+1)`` negative.
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See also
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--------
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qz
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Examples
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--------
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>>> from scipy.linalg import ordqz
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>>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]])
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>>> B = np.array([[0, 6, 0, 0], [5, 0, 2, 1], [5, 2, 6, 6], [4, 7, 7, 7]])
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>>> AA, BB, alpha, beta, Q, Z = ordqz(A, B, sort='lhp')
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Since we have sorted for left half plane eigenvalues, negatives come first
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>>> (alpha/beta).real < 0
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array([ True, True, False, False], dtype=bool)
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"""
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# NOTE: should users be able to set these?
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lwork = None
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result, typ = _qz(A, B, output=output, lwork=lwork, sort=None,
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overwrite_a=overwrite_a, overwrite_b=overwrite_b,
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check_finite=check_finite)
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AA, BB, Q, Z = result[0], result[1], result[-4], result[-3]
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if typ not in 'cz':
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alpha, beta = result[3] + result[4]*1.j, result[5]
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else:
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alpha, beta = result[3], result[4]
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sfunction = _select_function(sort)
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select = sfunction(alpha, beta)
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tgsen, = get_lapack_funcs(('tgsen',), (AA, BB))
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if lwork is None or lwork == -1:
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result = tgsen(select, AA, BB, Q, Z, lwork=-1)
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lwork = result[-3][0].real.astype(np.int_)
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# looks like wrong value passed to ZTGSYL if not
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lwork += 1
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liwork = None
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if liwork is None or liwork == -1:
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result = tgsen(select, AA, BB, Q, Z, liwork=-1)
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liwork = result[-2][0]
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result = tgsen(select, AA, BB, Q, Z, lwork=lwork, liwork=liwork)
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info = result[-1]
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if info < 0:
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raise ValueError("Illegal value in argument %d of tgsen" % -info)
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elif info == 1:
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raise ValueError("Reordering of (A, B) failed because the transformed"
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" matrix pair (A, B) would be too far from "
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"generalized Schur form; the problem is very "
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"ill-conditioned. (A, B) may have been partially "
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"reorded. If requested, 0 is returned in DIF(*), "
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"PL, and PR.")
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# for real results has a, b, alphar, alphai, beta, q, z, m, pl, pr, dif,
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# work, iwork, info
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if typ in ['f', 'd']:
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alpha = result[2] + result[3] * 1.j
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return (result[0], result[1], alpha, result[4], result[5], result[6])
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# for complex results has a, b, alpha, beta, q, z, m, pl, pr, dif, work,
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# iwork, info
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else:
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return result[0], result[1], result[2], result[3], result[4], result[5]
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