406 lines
14 KiB
Python
406 lines
14 KiB
Python
|
from __future__ import division, print_function, absolute_import
|
||
|
|
|
||
|
|
import warnings
|
||
|
|
|
||
|
|
import numpy as np
|
||
|
|
from numpy import asarray_chkfinite
|
||
|
|
|
||
|
|
from .misc import LinAlgError, _datacopied, LinAlgWarning
|
||
|
|
from .lapack import get_lapack_funcs
|
||
|
|
|
||
|
|
from scipy._lib.six import callable
|
||
|
|
|
||
|
|
__all__ = ['qz', 'ordqz']
|
||
|
|
|
||
|
|
_double_precision = ['i', 'l', 'd']
|
||
|
|
|
||
|
|
|
||
|
|
def _select_function(sort):
|
||
|
|
if callable(sort):
|
||
|
|
# assume the user knows what they're doing
|
||
|
|
sfunction = sort
|
||
|
|
elif sort == 'lhp':
|
||
|
|
sfunction = _lhp
|
||
|
|
elif sort == 'rhp':
|
||
|
|
sfunction = _rhp
|
||
|
|
elif sort == 'iuc':
|
||
|
|
sfunction = _iuc
|
||
|
|
elif sort == 'ouc':
|
||
|
|
sfunction = _ouc
|
||
|
|
else:
|
||
|
|
raise ValueError("sort parameter must be None, a callable, or "
|
||
|
|
"one of ('lhp','rhp','iuc','ouc')")
|
||
|
|
|
||
|
|
return sfunction
|
||
|
|
|
||
|
|
|
||
|
|
def _lhp(x, y):
|
||
|
|
out = np.empty_like(x, dtype=bool)
|
||
|
|
nonzero = (y != 0)
|
||
|
|
# handles (x, y) = (0, 0) too
|
||
|
|
out[~nonzero] = False
|
||
|
|
out[nonzero] = (np.real(x[nonzero]/y[nonzero]) < 0.0)
|
||
|
|
return out
|
||
|
|
|
||
|
|
|
||
|
|
def _rhp(x, y):
|
||
|
|
out = np.empty_like(x, dtype=bool)
|
||
|
|
nonzero = (y != 0)
|
||
|
|
# handles (x, y) = (0, 0) too
|
||
|
|
out[~nonzero] = False
|
||
|
|
out[nonzero] = (np.real(x[nonzero]/y[nonzero]) > 0.0)
|
||
|
|
return out
|
||
|
|
|
||
|
|
|
||
|
|
def _iuc(x, y):
|
||
|
|
out = np.empty_like(x, dtype=bool)
|
||
|
|
nonzero = (y != 0)
|
||
|
|
# handles (x, y) = (0, 0) too
|
||
|
|
out[~nonzero] = False
|
||
|
|
out[nonzero] = (abs(x[nonzero]/y[nonzero]) < 1.0)
|
||
|
|
return out
|
||
|
|
|
||
|
|
|
||
|
|
def _ouc(x, y):
|
||
|
|
out = np.empty_like(x, dtype=bool)
|
||
|
|
xzero = (x == 0)
|
||
|
|
yzero = (y == 0)
|
||
|
|
out[xzero & yzero] = False
|
||
|
|
out[~xzero & yzero] = True
|
||
|
|
out[~yzero] = (abs(x[~yzero]/y[~yzero]) > 1.0)
|
||
|
|
return out
|
||
|
|
|
||
|
|
|
||
|
|
def _qz(A, B, output='real', lwork=None, sort=None, overwrite_a=False,
|
||
|
|
overwrite_b=False, check_finite=True):
|
||
|
|
if sort is not None:
|
||
|
|
# Disabled due to segfaults on win32, see ticket 1717.
|
||
|
|
raise ValueError("The 'sort' input of qz() has to be None and will be "
|
||
|
|
"removed in a future release. Use ordqz instead.")
|
||
|
|
|
||
|
|
if output not in ['real', 'complex', 'r', 'c']:
|
||
|
|
raise ValueError("argument must be 'real', or 'complex'")
|
||
|
|
|
||
|
|
if check_finite:
|
||
|
|
a1 = asarray_chkfinite(A)
|
||
|
|
b1 = asarray_chkfinite(B)
|
||
|
|
else:
|
||
|
|
a1 = np.asarray(A)
|
||
|
|
b1 = np.asarray(B)
|
||
|
|
|
||
|
|
a_m, a_n = a1.shape
|
||
|
|
b_m, b_n = b1.shape
|
||
|
|
if not (a_m == a_n == b_m == b_n):
|
||
|
|
raise ValueError("Array dimensions must be square and agree")
|
||
|
|
|
||
|
|
typa = a1.dtype.char
|
||
|
|
if output in ['complex', 'c'] and typa not in ['F', 'D']:
|
||
|
|
if typa in _double_precision:
|
||
|
|
a1 = a1.astype('D')
|
||
|
|
typa = 'D'
|
||
|
|
else:
|
||
|
|
a1 = a1.astype('F')
|
||
|
|
typa = 'F'
|
||
|
|
typb = b1.dtype.char
|
||
|
|
if output in ['complex', 'c'] and typb not in ['F', 'D']:
|
||
|
|
if typb in _double_precision:
|
||
|
|
b1 = b1.astype('D')
|
||
|
|
typb = 'D'
|
||
|
|
else:
|
||
|
|
b1 = b1.astype('F')
|
||
|
|
typb = 'F'
|
||
|
|
|
||
|
|
overwrite_a = overwrite_a or (_datacopied(a1, A))
|
||
|
|
overwrite_b = overwrite_b or (_datacopied(b1, B))
|
||
|
|
|
||
|
|
gges, = get_lapack_funcs(('gges',), (a1, b1))
|
||
|
|
|
||
|
|
if lwork is None or lwork == -1:
|
||
|
|
# get optimal work array size
|
||
|
|
result = gges(lambda x: None, a1, b1, lwork=-1)
|
||
|
|
lwork = result[-2][0].real.astype(np.int)
|
||
|
|
|
||
|
|
sfunction = lambda x: None
|
||
|
|
result = gges(sfunction, a1, b1, lwork=lwork, overwrite_a=overwrite_a,
|
||
|
|
overwrite_b=overwrite_b, sort_t=0)
|
||
|
|
|
||
|
|
info = result[-1]
|
||
|
|
if info < 0:
|
||
|
|
raise ValueError("Illegal value in argument {} of gges".format(-info))
|
||
|
|
elif info > 0 and info <= a_n:
|
||
|
|
warnings.warn("The QZ iteration failed. (a,b) are not in Schur "
|
||
|
|
"form, but ALPHAR(j), ALPHAI(j), and BETA(j) should be "
|
||
|
|
"correct for J={},...,N".format(info-1), LinAlgWarning,
|
||
|
|
stacklevel=3)
|
||
|
|
elif info == a_n+1:
|
||
|
|
raise LinAlgError("Something other than QZ iteration failed")
|
||
|
|
elif info == a_n+2:
|
||
|
|
raise LinAlgError("After reordering, roundoff changed values of some "
|
||
|
|
"complex eigenvalues so that leading eigenvalues "
|
||
|
|
"in the Generalized Schur form no longer satisfy "
|
||
|
|
"sort=True. This could also be due to scaling.")
|
||
|
|
elif info == a_n+3:
|
||
|
|
raise LinAlgError("Reordering failed in <s,d,c,z>tgsen")
|
||
|
|
|
||
|
|
return result, gges.typecode
|
||
|
|
|
||
|
|
|
||
|
|
def qz(A, B, output='real', lwork=None, sort=None, overwrite_a=False,
|
||
|
|
overwrite_b=False, check_finite=True):
|
||
|
|
"""
|
||
|
|
QZ decomposition for generalized eigenvalues of a pair of matrices.
|
||
|
|
|
||
|
|
The QZ, or generalized Schur, decomposition for a pair of N x N
|
||
|
|
nonsymmetric matrices (A,B) is::
|
||
|
|
|
||
|
|
(A,B) = (Q*AA*Z', Q*BB*Z')
|
||
|
|
|
||
|
|
where AA, BB is in generalized Schur form if BB is upper-triangular
|
||
|
|
with non-negative diagonal and AA is upper-triangular, or for real QZ
|
||
|
|
decomposition (``output='real'``) block upper triangular with 1x1
|
||
|
|
and 2x2 blocks. In this case, the 1x1 blocks correspond to real
|
||
|
|
generalized eigenvalues and 2x2 blocks are 'standardized' by making
|
||
|
|
the corresponding elements of BB have the form::
|
||
|
|
|
||
|
|
[ a 0 ]
|
||
|
|
[ 0 b ]
|
||
|
|
|
||
|
|
and the pair of corresponding 2x2 blocks in AA and BB will have a complex
|
||
|
|
conjugate pair of generalized eigenvalues. If (``output='complex'``) or
|
||
|
|
A and B are complex matrices, Z' denotes the conjugate-transpose of Z.
|
||
|
|
Q and Z are unitary matrices.
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
A : (N, N) array_like
|
||
|
|
2d array to decompose
|
||
|
|
B : (N, N) array_like
|
||
|
|
2d array to decompose
|
||
|
|
output : {'real', 'complex'}, optional
|
||
|
|
Construct the real or complex QZ decomposition for real matrices.
|
||
|
|
Default is 'real'.
|
||
|
|
lwork : int, optional
|
||
|
|
Work array size. If None or -1, it is automatically computed.
|
||
|
|
sort : {None, callable, 'lhp', 'rhp', 'iuc', 'ouc'}, optional
|
||
|
|
NOTE: THIS INPUT IS DISABLED FOR NOW. Use ordqz instead.
|
||
|
|
|
||
|
|
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). For
|
||
|
|
real matrix pairs, the sort function takes three real arguments
|
||
|
|
(alphar, alphai, beta). The eigenvalue
|
||
|
|
``x = (alphar + alphai*1j)/beta``. For complex matrix pairs or
|
||
|
|
output='complex', the sort function takes two complex arguments
|
||
|
|
(alpha, beta). The eigenvalue ``x = (alpha/beta)``. 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).
|
||
|
|
overwrite_a : bool, optional
|
||
|
|
Whether to overwrite data in a (may improve performance)
|
||
|
|
overwrite_b : bool, optional
|
||
|
|
Whether to overwrite data in b (may improve performance)
|
||
|
|
check_finite : bool, optional
|
||
|
|
If true checks the elements of `A` and `B` are finite numbers. If
|
||
|
|
false does no checking and passes matrix through to
|
||
|
|
underlying algorithm.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
AA : (N, N) ndarray
|
||
|
|
Generalized Schur form of A.
|
||
|
|
BB : (N, N) ndarray
|
||
|
|
Generalized Schur form of B.
|
||
|
|
Q : (N, N) ndarray
|
||
|
|
The left Schur vectors.
|
||
|
|
Z : (N, N) ndarray
|
||
|
|
The right Schur vectors.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
Q is transposed versus the equivalent function in Matlab.
|
||
|
|
|
||
|
|
.. versionadded:: 0.11.0
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> from scipy import linalg
|
||
|
|
>>> np.random.seed(1234)
|
||
|
|
>>> A = np.arange(9).reshape((3, 3))
|
||
|
|
>>> B = np.random.randn(3, 3)
|
||
|
|
|
||
|
|
>>> AA, BB, Q, Z = linalg.qz(A, B)
|
||
|
|
>>> AA
|
||
|
|
array([[-13.40928183, -4.62471562, 1.09215523],
|
||
|
|
[ 0. , 0. , 1.22805978],
|
||
|
|
[ 0. , 0. , 0.31973817]])
|
||
|
|
>>> BB
|
||
|
|
array([[ 0.33362547, -1.37393632, 0.02179805],
|
||
|
|
[ 0. , 1.68144922, 0.74683866],
|
||
|
|
[ 0. , 0. , 0.9258294 ]])
|
||
|
|
>>> Q
|
||
|
|
array([[ 0.14134727, -0.97562773, 0.16784365],
|
||
|
|
[ 0.49835904, -0.07636948, -0.86360059],
|
||
|
|
[ 0.85537081, 0.20571399, 0.47541828]])
|
||
|
|
>>> Z
|
||
|
|
array([[-0.24900855, -0.51772687, 0.81850696],
|
||
|
|
[-0.79813178, 0.58842606, 0.12938478],
|
||
|
|
[-0.54861681, -0.6210585 , -0.55973739]])
|
||
|
|
|
||
|
|
See also
|
||
|
|
--------
|
||
|
|
ordqz
|
||
|
|
"""
|
||
|
|
# output for real
|
||
|
|
# AA, BB, sdim, alphar, alphai, beta, vsl, vsr, work, info
|
||
|
|
# output for complex
|
||
|
|
# AA, BB, sdim, alpha, beta, vsl, vsr, work, info
|
||
|
|
result, _ = _qz(A, B, output=output, lwork=lwork, sort=sort,
|
||
|
|
overwrite_a=overwrite_a, overwrite_b=overwrite_b,
|
||
|
|
check_finite=check_finite)
|
||
|
|
return result[0], result[1], result[-4], result[-3]
|
||
|
|
|
||
|
|
|
||
|
|
def ordqz(A, B, sort='lhp', output='real', overwrite_a=False,
|
||
|
|
overwrite_b=False, check_finite=True):
|
||
|
|
"""QZ decomposition for a pair of matrices with reordering.
|
||
|
|
|
||
|
|
.. versionadded:: 0.17.0
|
||
|
|
|
||
|
|
Parameters
|
||
|
|
----------
|
||
|
|
A : (N, N) array_like
|
||
|
|
2d array to decompose
|
||
|
|
B : (N, N) array_like
|
||
|
|
2d array to decompose
|
||
|
|
sort : {callable, 'lhp', 'rhp', 'iuc', 'ouc'}, optional
|
||
|
|
Specifies whether the upper eigenvalues should be sorted. A
|
||
|
|
callable may be passed that, given an ordered pair ``(alpha,
|
||
|
|
beta)`` representing the eigenvalue ``x = (alpha/beta)``,
|
||
|
|
returns a boolean denoting whether the eigenvalue should be
|
||
|
|
sorted to the top-left (True). For the real matrix pairs
|
||
|
|
``beta`` is real while ``alpha`` can be complex, and for
|
||
|
|
complex matrix pairs both ``alpha`` and ``beta`` can be
|
||
|
|
complex. The callable must be able to accept a numpy
|
||
|
|
array. 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)
|
||
|
|
|
||
|
|
With the predefined sorting functions, an infinite eigenvalue
|
||
|
|
(i.e. ``alpha != 0`` and ``beta = 0``) is considered to lie in
|
||
|
|
neither the left-hand nor the right-hand plane, but it is
|
||
|
|
considered to lie outside the unit circle. For the eigenvalue
|
||
|
|
``(alpha, beta) = (0, 0)`` the predefined sorting functions
|
||
|
|
all return `False`.
|
||
|
|
output : str {'real','complex'}, optional
|
||
|
|
Construct the real or complex QZ decomposition for real matrices.
|
||
|
|
Default is 'real'.
|
||
|
|
overwrite_a : bool, optional
|
||
|
|
If True, the contents of A are overwritten.
|
||
|
|
overwrite_b : bool, optional
|
||
|
|
If True, the contents of B are overwritten.
|
||
|
|
check_finite : bool, optional
|
||
|
|
If true checks the elements of `A` and `B` are finite numbers. If
|
||
|
|
false does no checking and passes matrix through to
|
||
|
|
underlying algorithm.
|
||
|
|
|
||
|
|
Returns
|
||
|
|
-------
|
||
|
|
AA : (N, N) ndarray
|
||
|
|
Generalized Schur form of A.
|
||
|
|
BB : (N, N) ndarray
|
||
|
|
Generalized Schur form of B.
|
||
|
|
alpha : (N,) ndarray
|
||
|
|
alpha = alphar + alphai * 1j. See notes.
|
||
|
|
beta : (N,) ndarray
|
||
|
|
See notes.
|
||
|
|
Q : (N, N) ndarray
|
||
|
|
The left Schur vectors.
|
||
|
|
Z : (N, N) ndarray
|
||
|
|
The right Schur vectors.
|
||
|
|
|
||
|
|
Notes
|
||
|
|
-----
|
||
|
|
On exit, ``(ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N``, will be the
|
||
|
|
generalized eigenvalues. ``ALPHAR(j) + ALPHAI(j)*i`` and
|
||
|
|
``BETA(j),j=1,...,N`` are the diagonals of the complex Schur form (S,T)
|
||
|
|
that would result if the 2-by-2 diagonal blocks of the real generalized
|
||
|
|
Schur form of (A,B) were further reduced to triangular form using complex
|
||
|
|
unitary transformations. If ALPHAI(j) is zero, then the j-th eigenvalue is
|
||
|
|
real; if positive, then the ``j``-th and ``(j+1)``-st eigenvalues are a
|
||
|
|
complex conjugate pair, with ``ALPHAI(j+1)`` negative.
|
||
|
|
|
||
|
|
See also
|
||
|
|
--------
|
||
|
|
qz
|
||
|
|
|
||
|
|
Examples
|
||
|
|
--------
|
||
|
|
>>> from scipy.linalg import ordqz
|
||
|
|
>>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]])
|
||
|
|
>>> B = np.array([[0, 6, 0, 0], [5, 0, 2, 1], [5, 2, 6, 6], [4, 7, 7, 7]])
|
||
|
|
>>> AA, BB, alpha, beta, Q, Z = ordqz(A, B, sort='lhp')
|
||
|
|
|
||
|
|
Since we have sorted for left half plane eigenvalues, negatives come first
|
||
|
|
|
||
|
|
>>> (alpha/beta).real < 0
|
||
|
|
array([ True, True, False, False], dtype=bool)
|
||
|
|
|
||
|
|
"""
|
||
|
|
# NOTE: should users be able to set these?
|
||
|
|
lwork = None
|
||
|
|
result, typ = _qz(A, B, output=output, lwork=lwork, sort=None,
|
||
|
|
overwrite_a=overwrite_a, overwrite_b=overwrite_b,
|
||
|
|
check_finite=check_finite)
|
||
|
|
AA, BB, Q, Z = result[0], result[1], result[-4], result[-3]
|
||
|
|
if typ not in 'cz':
|
||
|
|
alpha, beta = result[3] + result[4]*1.j, result[5]
|
||
|
|
else:
|
||
|
|
alpha, beta = result[3], result[4]
|
||
|
|
|
||
|
|
sfunction = _select_function(sort)
|
||
|
|
select = sfunction(alpha, beta)
|
||
|
|
|
||
|
|
tgsen, = get_lapack_funcs(('tgsen',), (AA, BB))
|
||
|
|
|
||
|
|
if lwork is None or lwork == -1:
|
||
|
|
result = tgsen(select, AA, BB, Q, Z, lwork=-1)
|
||
|
|
lwork = result[-3][0].real.astype(np.int)
|
||
|
|
# looks like wrong value passed to ZTGSYL if not
|
||
|
|
lwork += 1
|
||
|
|
|
||
|
|
liwork = None
|
||
|
|
if liwork is None or liwork == -1:
|
||
|
|
result = tgsen(select, AA, BB, Q, Z, liwork=-1)
|
||
|
|
liwork = result[-2][0]
|
||
|
|
|
||
|
|
result = tgsen(select, AA, BB, Q, Z, lwork=lwork, liwork=liwork)
|
||
|
|
|
||
|
|
info = result[-1]
|
||
|
|
if info < 0:
|
||
|
|
raise ValueError("Illegal value in argument %d of tgsen" % -info)
|
||
|
|
elif info == 1:
|
||
|
|
raise ValueError("Reordering of (A, B) failed because the transformed"
|
||
|
|
" matrix pair (A, B) would be too far from "
|
||
|
|
"generalized Schur form; the problem is very "
|
||
|
|
"ill-conditioned. (A, B) may have been partially "
|
||
|
|
"reorded. If requested, 0 is returned in DIF(*), "
|
||
|
|
"PL, and PR.")
|
||
|
|
|
||
|
|
# for real results has a, b, alphar, alphai, beta, q, z, m, pl, pr, dif,
|
||
|
|
# work, iwork, info
|
||
|
|
if typ in ['f', 'd']:
|
||
|
|
alpha = result[2] + result[3] * 1.j
|
||
|
|
return (result[0], result[1], alpha, result[4], result[5], result[6])
|
||
|
|
# for complex results has a, b, alpha, beta, q, z, m, pl, pr, dif, work,
|
||
|
|
# iwork, info
|
||
|
|
else:
|
||
|
|
return result[0], result[1], result[2], result[3], result[4], result[5]
|