296 lines
10 KiB
Python
296 lines
10 KiB
Python
|
"""Schur decomposition functions."""
|
||
|
|
from __future__ import division, print_function, absolute_import
|
||
|
|
|
||
|
|
import numpy
|
||
|
|
from numpy import asarray_chkfinite, single, asarray, array
|
||
|
|
from numpy.linalg import norm
|
||
|
|
|
||
|
|
from scipy._lib.six import callable
|
||
|
|
|
||
|
|
# 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']
|
||
|
|
|
||
|
|
|
||
|
|
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
|