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