301 lines
10 KiB
Python
301 lines
10 KiB
Python
"""Schur decomposition functions."""
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import numpy
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from numpy import asarray_chkfinite, single, asarray, array
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from numpy.linalg import norm
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# Local imports.
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from ._misc import LinAlgError, _datacopied
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from .lapack import get_lapack_funcs
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from ._decomp import eigvals
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__all__ = ['schur', 'rsf2csf']
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_double_precision = ['i', 'l', 'd']
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def schur(a, output='real', lwork=None, overwrite_a=False, sort=None,
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check_finite=True):
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"""
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Compute Schur decomposition of a matrix.
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The Schur decomposition is::
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A = Z T Z^H
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where Z is unitary and T is either upper-triangular, or for real
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Schur decomposition (output='real'), quasi-upper triangular. In
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the quasi-triangular form, 2x2 blocks describing complex-valued
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eigenvalue pairs may extrude from the diagonal.
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Parameters
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----------
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a : (M, M) array_like
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Matrix to decompose
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output : {'real', 'complex'}, optional
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Construct the real or complex Schur decomposition (for real matrices).
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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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overwrite_a : bool, optional
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Whether to overwrite data in a (may improve performance).
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sort : {None, callable, 'lhp', 'rhp', 'iuc', 'ouc'}, optional
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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).
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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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Defaults to None (no sorting).
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check_finite : bool, optional
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Whether to check that the input matrix contains only finite numbers.
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Disabling may give a performance gain, but may result in problems
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(crashes, non-termination) if the inputs do contain infinities or NaNs.
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Returns
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-------
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T : (M, M) ndarray
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Schur form of A. It is real-valued for the real Schur decomposition.
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Z : (M, M) ndarray
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An unitary Schur transformation matrix for A.
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It is real-valued for the real Schur decomposition.
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sdim : int
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If and only if sorting was requested, a third return value will
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contain the number of eigenvalues satisfying the sort condition.
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Raises
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------
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LinAlgError
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Error raised under three conditions:
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1. The algorithm failed due to a failure of the QR algorithm to
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compute all eigenvalues.
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2. If eigenvalue sorting was requested, the eigenvalues could not be
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reordered due to a failure to separate eigenvalues, usually because
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of poor conditioning.
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3. If eigenvalue sorting was requested, roundoff errors caused the
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leading eigenvalues to no longer satisfy the sorting condition.
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See Also
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--------
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rsf2csf : Convert real Schur form to complex Schur form
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Examples
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--------
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>>> import numpy as np
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>>> from scipy.linalg import schur, eigvals
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>>> A = np.array([[0, 2, 2], [0, 1, 2], [1, 0, 1]])
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>>> T, Z = schur(A)
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>>> T
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array([[ 2.65896708, 1.42440458, -1.92933439],
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[ 0. , -0.32948354, -0.49063704],
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[ 0. , 1.31178921, -0.32948354]])
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>>> Z
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array([[0.72711591, -0.60156188, 0.33079564],
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[0.52839428, 0.79801892, 0.28976765],
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[0.43829436, 0.03590414, -0.89811411]])
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>>> T2, Z2 = schur(A, output='complex')
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>>> T2
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array([[ 2.65896708, -1.22839825+1.32378589j, 0.42590089+1.51937378j], # may vary
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[ 0. , -0.32948354+0.80225456j, -0.59877807+0.56192146j],
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[ 0. , 0. , -0.32948354-0.80225456j]])
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>>> eigvals(T2)
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array([2.65896708, -0.32948354+0.80225456j, -0.32948354-0.80225456j])
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An arbitrary custom eig-sorting condition, having positive imaginary part,
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which is satisfied by only one eigenvalue
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>>> T3, Z3, sdim = schur(A, output='complex', sort=lambda x: x.imag > 0)
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>>> sdim
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1
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"""
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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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else:
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a1 = asarray(a)
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if numpy.issubdtype(a1.dtype, numpy.integer):
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a1 = asarray(a, dtype=numpy.dtype("long"))
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if len(a1.shape) != 2 or (a1.shape[0] != a1.shape[1]):
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raise ValueError('expected square matrix')
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typ = a1.dtype.char
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if output in ['complex', 'c'] and typ not in ['F', 'D']:
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if typ in _double_precision:
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a1 = a1.astype('D')
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typ = 'D'
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else:
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a1 = a1.astype('F')
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typ = 'F'
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overwrite_a = overwrite_a or (_datacopied(a1, a))
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gees, = get_lapack_funcs(('gees',), (a1,))
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if lwork is None or lwork == -1:
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# get optimal work array
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result = gees(lambda x: None, a1, lwork=-1)
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lwork = result[-2][0].real.astype(numpy.int_)
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if sort is None:
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sort_t = 0
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def sfunction(x):
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return None
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else:
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sort_t = 1
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if callable(sort):
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sfunction = sort
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elif sort == 'lhp':
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def sfunction(x):
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return x.real < 0.0
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elif sort == 'rhp':
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def sfunction(x):
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return x.real >= 0.0
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elif sort == 'iuc':
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def sfunction(x):
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return abs(x) <= 1.0
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elif sort == 'ouc':
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def sfunction(x):
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return abs(x) > 1.0
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else:
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raise ValueError("'sort' parameter must either be 'None', or a "
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"callable, or one of ('lhp','rhp','iuc','ouc')")
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result = gees(sfunction, a1, lwork=lwork, overwrite_a=overwrite_a,
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sort_t=sort_t)
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info = result[-1]
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if info < 0:
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raise ValueError(f'illegal value in {-info}-th argument of internal gees')
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elif info == a1.shape[0] + 1:
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raise LinAlgError('Eigenvalues could not be separated for reordering.')
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elif info == a1.shape[0] + 2:
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raise LinAlgError('Leading eigenvalues do not satisfy sort condition.')
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elif info > 0:
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raise LinAlgError("Schur form not found. Possibly ill-conditioned.")
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if sort_t == 0:
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return result[0], result[-3]
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else:
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return result[0], result[-3], result[1]
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eps = numpy.finfo(float).eps
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feps = numpy.finfo(single).eps
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_array_kind = {'b': 0, 'h': 0, 'B': 0, 'i': 0, 'l': 0,
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'f': 0, 'd': 0, 'F': 1, 'D': 1}
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_array_precision = {'i': 1, 'l': 1, 'f': 0, 'd': 1, 'F': 0, 'D': 1}
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_array_type = [['f', 'd'], ['F', 'D']]
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def _commonType(*arrays):
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kind = 0
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precision = 0
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for a in arrays:
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t = a.dtype.char
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kind = max(kind, _array_kind[t])
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precision = max(precision, _array_precision[t])
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return _array_type[kind][precision]
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def _castCopy(type, *arrays):
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cast_arrays = ()
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for a in arrays:
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if a.dtype.char == type:
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cast_arrays = cast_arrays + (a.copy(),)
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else:
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cast_arrays = cast_arrays + (a.astype(type),)
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if len(cast_arrays) == 1:
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return cast_arrays[0]
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else:
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return cast_arrays
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def rsf2csf(T, Z, check_finite=True):
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"""
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Convert real Schur form to complex Schur form.
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Convert a quasi-diagonal real-valued Schur form to the upper-triangular
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complex-valued Schur form.
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Parameters
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----------
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T : (M, M) array_like
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Real Schur form of the original array
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Z : (M, M) array_like
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Schur transformation matrix
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check_finite : bool, optional
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Whether to check that the input arrays contain only finite numbers.
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Disabling may give a performance gain, but may result in problems
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(crashes, non-termination) if the inputs do contain infinities or NaNs.
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Returns
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-------
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T : (M, M) ndarray
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Complex Schur form of the original array
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Z : (M, M) ndarray
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Schur transformation matrix corresponding to the complex form
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See Also
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--------
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schur : Schur decomposition of an array
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Examples
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--------
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>>> import numpy as np
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>>> from scipy.linalg import schur, rsf2csf
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>>> A = np.array([[0, 2, 2], [0, 1, 2], [1, 0, 1]])
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>>> T, Z = schur(A)
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>>> T
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array([[ 2.65896708, 1.42440458, -1.92933439],
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[ 0. , -0.32948354, -0.49063704],
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[ 0. , 1.31178921, -0.32948354]])
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>>> Z
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array([[0.72711591, -0.60156188, 0.33079564],
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[0.52839428, 0.79801892, 0.28976765],
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[0.43829436, 0.03590414, -0.89811411]])
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>>> T2 , Z2 = rsf2csf(T, Z)
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>>> T2
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array([[2.65896708+0.j, -1.64592781+0.743164187j, -1.21516887+1.00660462j],
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[0.+0.j , -0.32948354+8.02254558e-01j, -0.82115218-2.77555756e-17j],
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[0.+0.j , 0.+0.j, -0.32948354-0.802254558j]])
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>>> Z2
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array([[0.72711591+0.j, 0.28220393-0.31385693j, 0.51319638-0.17258824j],
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[0.52839428+0.j, 0.24720268+0.41635578j, -0.68079517-0.15118243j],
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[0.43829436+0.j, -0.76618703+0.01873251j, -0.03063006+0.46857912j]])
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"""
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if check_finite:
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Z, T = map(asarray_chkfinite, (Z, T))
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else:
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Z, T = map(asarray, (Z, T))
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for ind, X in enumerate([Z, T]):
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if X.ndim != 2 or X.shape[0] != X.shape[1]:
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raise ValueError("Input '{}' must be square.".format('ZT'[ind]))
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if T.shape[0] != Z.shape[0]:
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message = f"Input array shapes must match: Z: {Z.shape} vs. T: {T.shape}"
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raise ValueError(message)
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N = T.shape[0]
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t = _commonType(Z, T, array([3.0], 'F'))
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Z, T = _castCopy(t, Z, T)
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for m in range(N-1, 0, -1):
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if abs(T[m, m-1]) > eps*(abs(T[m-1, m-1]) + abs(T[m, m])):
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mu = eigvals(T[m-1:m+1, m-1:m+1]) - T[m, m]
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r = norm([mu[0], T[m, m-1]])
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c = mu[0] / r
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s = T[m, m-1] / r
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G = array([[c.conj(), s], [-s, c]], dtype=t)
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T[m-1:m+1, m-1:] = G.dot(T[m-1:m+1, m-1:])
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T[:m+1, m-1:m+1] = T[:m+1, m-1:m+1].dot(G.conj().T)
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Z[:, m-1:m+1] = Z[:, m-1:m+1].dot(G.conj().T)
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T[m, m-1] = 0.0
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return T, Z
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