Skip to content


Subversion checkout URL

You can clone with
Download ZIP
Fetching contributors…

Cannot retrieve contributors at this time

168 lines (142 sloc) 5.239 kB
"""Schur decomposition functions."""
import numpy
from numpy import asarray_chkfinite, single
# Local imports.
import misc
from misc import LinAlgError, _datanotshared
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):
"""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)
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.
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 (_datanotshared(a1, a))
gees, = get_lapack_funcs(('gees',), (a1,))
if lwork is None or lwork == -1:
# get optimal work array
result = gees(lambda x: None, a, lwork=-1)
lwork = result[-2][0]
result = gees(lambda x: None, a, lwork=result[-2][0], overwrite_a=overwrite_a)
info = result[-1]
if info < 0:
raise ValueError('illegal value in %d-th argument of internal gees'
% -info)
elif info > 0:
raise LinAlgError("Schur form not found. Possibly ill-conditioned.")
return result[0], result[-3]
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
Jump to Line
Something went wrong with that request. Please try again.