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"""Schur decomposition functions."""
import numpy
from numpy import asarray_chkfinite, single
# Local imports.
import misc
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):
"""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.
a : array, shape (M, M)
Matrix to decompose
output : {'real', 'complex'}
Construct the real or complex Schur decomposition (for real matrices).
lwork : integer
Work array size. If None or -1, it is automatically computed.
overwrite_a : boolean
Whether to overwrite data in a (may improve performance)
sort : {None, callable, 'lhp', 'rhp', 'iuc', 'ouc'}
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).
T : array, shape (M, M)
Schur form of A. It is real-valued for the real Schur decomposition.
Z : array, shape (M, M)
An unitary Schur transformation matrix for A.
It is real-valued for the real Schur decomposition.
sdim : integer
If and only if sorting was requested, a third return value will
contain the number of eigenvalues satisfying the sort condition.
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
if not output in ['real','complex','r','c']:
raise ValueError("argument must be 'real', or 'complex'")
a1 = asarray_chkfinite(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'
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(
if sort is None:
sort_t = 0
sfunction = lambda x: None
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)
raise ValueError("sort parameter must be None, a callable, or " +
"one of ('lhp','rhp','iuc','ouc')")
result = gees(sfunction, a1, lwork=lwork, overwrite_a=overwrite_a,
info = result[-1]
if info < 0:
raise ValueError('illegal value in %d-th argument of internal gees'
% -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]
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(),)
cast_arrays = cast_arrays + (a.astype(type),)
if len(cast_arrays) == 1:
return cast_arrays[0]
return cast_arrays
def rsf2csf(T, Z):
"""Convert real Schur form to complex Schur form.
Convert a quasi-diagonal real-valued Schur form to the upper triangular
complex-valued Schur form.
T : array, shape (M, M)
Real Schur form of the original matrix
Z : array, shape (M, M)
Schur transformation matrix
T : array, shape (M, M)
Complex Schur form of the original matrix
Z : array, shape (M, M)
Schur transformation matrix corresponding to the complex form
See also
schur : Schur decompose a matrix
Z, T = map(asarray_chkfinite, (Z, T))
if len(Z.shape) != 2 or Z.shape[0] != Z.shape[1]:
raise ValueError("matrix must be square.")
if len(T.shape) != 2 or T.shape[0] != T.shape[1]:
raise ValueError("matrix must be square.")
if T.shape[0] != Z.shape[0]:
raise ValueError("matrices must be same dimension.")
N = T.shape[0]
arr = numpy.array
t = _commonType(Z, T, arr([3.0],'F'))
Z, T = _castCopy(t, Z, T)
conj = numpy.conj
dot =
r_ = numpy.r_
transp = numpy.transpose
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])):
k = slice(m-1, m+1)
mu = eigvals(T[k,k]) - T[m,m]
r = misc.norm([mu[0], T[m,m-1]])
c = mu[0] / r
s = T[m,m-1] / r
G = r_[arr([[conj(c), s]], dtype=t), arr([[-s, c]], dtype=t)]
Gc = conj(transp(G))
j = slice(m-1, N)
T[k,j] = dot(G, T[k,j])
i = slice(0, m+1)
T[i,k] = dot(T[i,k], Gc)
i = slice(0, N)
Z[i,k] = dot(Z[i,k], Gc)
T[m,m-1] = 0.0;
return T, Z
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