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1import warnings
3import numpy as np
4from numpy import asarray_chkfinite
6from .misc import LinAlgError, _datacopied, LinAlgWarning
7from .lapack import get_lapack_funcs
10__all__ = ['qz', 'ordqz']
12_double_precision = ['i', 'l', 'd']
15def _select_function(sort):
16 if callable(sort):
17 # assume the user knows what they're doing
18 sfunction = sort
19 elif sort == 'lhp':
20 sfunction = _lhp
21 elif sort == 'rhp':
22 sfunction = _rhp
23 elif sort == 'iuc':
24 sfunction = _iuc
25 elif sort == 'ouc':
26 sfunction = _ouc
27 else:
28 raise ValueError("sort parameter must be None, a callable, or "
29 "one of ('lhp','rhp','iuc','ouc')")
31 return sfunction
34def _lhp(x, y):
35 out = np.empty_like(x, dtype=bool)
36 nonzero = (y != 0)
37 # handles (x, y) = (0, 0) too
38 out[~nonzero] = False
39 out[nonzero] = (np.real(x[nonzero]/y[nonzero]) < 0.0)
40 return out
43def _rhp(x, y):
44 out = np.empty_like(x, dtype=bool)
45 nonzero = (y != 0)
46 # handles (x, y) = (0, 0) too
47 out[~nonzero] = False
48 out[nonzero] = (np.real(x[nonzero]/y[nonzero]) > 0.0)
49 return out
52def _iuc(x, y):
53 out = np.empty_like(x, dtype=bool)
54 nonzero = (y != 0)
55 # handles (x, y) = (0, 0) too
56 out[~nonzero] = False
57 out[nonzero] = (abs(x[nonzero]/y[nonzero]) < 1.0)
58 return out
61def _ouc(x, y):
62 out = np.empty_like(x, dtype=bool)
63 xzero = (x == 0)
64 yzero = (y == 0)
65 out[xzero & yzero] = False
66 out[~xzero & yzero] = True
67 out[~yzero] = (abs(x[~yzero]/y[~yzero]) > 1.0)
68 return out
71def _qz(A, B, output='real', lwork=None, sort=None, overwrite_a=False,
72 overwrite_b=False, check_finite=True):
73 if sort is not None:
74 # Disabled due to segfaults on win32, see ticket 1717.
75 raise ValueError("The 'sort' input of qz() has to be None and will be "
76 "removed in a future release. Use ordqz instead.")
78 if output not in ['real', 'complex', 'r', 'c']:
79 raise ValueError("argument must be 'real', or 'complex'")
81 if check_finite:
82 a1 = asarray_chkfinite(A)
83 b1 = asarray_chkfinite(B)
84 else:
85 a1 = np.asarray(A)
86 b1 = np.asarray(B)
88 a_m, a_n = a1.shape
89 b_m, b_n = b1.shape
90 if not (a_m == a_n == b_m == b_n):
91 raise ValueError("Array dimensions must be square and agree")
93 typa = a1.dtype.char
94 if output in ['complex', 'c'] and typa not in ['F', 'D']:
95 if typa in _double_precision:
96 a1 = a1.astype('D')
97 typa = 'D'
98 else:
99 a1 = a1.astype('F')
100 typa = 'F'
101 typb = b1.dtype.char
102 if output in ['complex', 'c'] and typb not in ['F', 'D']:
103 if typb in _double_precision:
104 b1 = b1.astype('D')
105 typb = 'D'
106 else:
107 b1 = b1.astype('F')
108 typb = 'F'
110 overwrite_a = overwrite_a or (_datacopied(a1, A))
111 overwrite_b = overwrite_b or (_datacopied(b1, B))
113 gges, = get_lapack_funcs(('gges',), (a1, b1))
115 if lwork is None or lwork == -1:
116 # get optimal work array size
117 result = gges(lambda x: None, a1, b1, lwork=-1)
118 lwork = result[-2][0].real.astype(np.int)
120 sfunction = lambda x: None
121 result = gges(sfunction, a1, b1, lwork=lwork, overwrite_a=overwrite_a,
122 overwrite_b=overwrite_b, sort_t=0)
124 info = result[-1]
125 if info < 0:
126 raise ValueError("Illegal value in argument {} of gges".format(-info))
127 elif info > 0 and info <= a_n:
128 warnings.warn("The QZ iteration failed. (a,b) are not in Schur "
129 "form, but ALPHAR(j), ALPHAI(j), and BETA(j) should be "
130 "correct for J={},...,N".format(info-1), LinAlgWarning,
131 stacklevel=3)
132 elif info == a_n+1:
133 raise LinAlgError("Something other than QZ iteration failed")
134 elif info == a_n+2:
135 raise LinAlgError("After reordering, roundoff changed values of some "
136 "complex eigenvalues so that leading eigenvalues "
137 "in the Generalized Schur form no longer satisfy "
138 "sort=True. This could also be due to scaling.")
139 elif info == a_n+3:
140 raise LinAlgError("Reordering failed in <s,d,c,z>tgsen")
142 return result, gges.typecode
145def qz(A, B, output='real', lwork=None, sort=None, overwrite_a=False,
146 overwrite_b=False, check_finite=True):
147 """
148 QZ decomposition for generalized eigenvalues of a pair of matrices.
150 The QZ, or generalized Schur, decomposition for a pair of N x N
151 nonsymmetric matrices (A,B) is::
153 (A,B) = (Q*AA*Z', Q*BB*Z')
155 where AA, BB is in generalized Schur form if BB is upper-triangular
156 with non-negative diagonal and AA is upper-triangular, or for real QZ
157 decomposition (``output='real'``) block upper triangular with 1x1
158 and 2x2 blocks. In this case, the 1x1 blocks correspond to real
159 generalized eigenvalues and 2x2 blocks are 'standardized' by making
160 the corresponding elements of BB have the form::
162 [ a 0 ]
163 [ 0 b ]
165 and the pair of corresponding 2x2 blocks in AA and BB will have a complex
166 conjugate pair of generalized eigenvalues. If (``output='complex'``) or
167 A and B are complex matrices, Z' denotes the conjugate-transpose of Z.
168 Q and Z are unitary matrices.
170 Parameters
171 ----------
172 A : (N, N) array_like
173 2-D array to decompose
174 B : (N, N) array_like
175 2-D array to decompose
176 output : {'real', 'complex'}, optional
177 Construct the real or complex QZ decomposition for real matrices.
178 Default is 'real'.
179 lwork : int, optional
180 Work array size. If None or -1, it is automatically computed.
181 sort : {None, callable, 'lhp', 'rhp', 'iuc', 'ouc'}, optional
182 NOTE: THIS INPUT IS DISABLED FOR NOW. Use ordqz instead.
184 Specifies whether the upper eigenvalues should be sorted. A callable
185 may be passed that, given a eigenvalue, returns a boolean denoting
186 whether the eigenvalue should be sorted to the top-left (True). For
187 real matrix pairs, the sort function takes three real arguments
188 (alphar, alphai, beta). The eigenvalue
189 ``x = (alphar + alphai*1j)/beta``. For complex matrix pairs or
190 output='complex', the sort function takes two complex arguments
191 (alpha, beta). The eigenvalue ``x = (alpha/beta)``. Alternatively,
192 string parameters may be used:
194 - 'lhp' Left-hand plane (x.real < 0.0)
195 - 'rhp' Right-hand plane (x.real > 0.0)
196 - 'iuc' Inside the unit circle (x*x.conjugate() < 1.0)
197 - 'ouc' Outside the unit circle (x*x.conjugate() > 1.0)
199 Defaults to None (no sorting).
200 overwrite_a : bool, optional
201 Whether to overwrite data in a (may improve performance)
202 overwrite_b : bool, optional
203 Whether to overwrite data in b (may improve performance)
204 check_finite : bool, optional
205 If true checks the elements of `A` and `B` are finite numbers. If
206 false does no checking and passes matrix through to
207 underlying algorithm.
209 Returns
210 -------
211 AA : (N, N) ndarray
212 Generalized Schur form of A.
213 BB : (N, N) ndarray
214 Generalized Schur form of B.
215 Q : (N, N) ndarray
216 The left Schur vectors.
217 Z : (N, N) ndarray
218 The right Schur vectors.
220 Notes
221 -----
222 Q is transposed versus the equivalent function in Matlab.
224 .. versionadded:: 0.11.0
226 Examples
227 --------
228 >>> from scipy import linalg
229 >>> np.random.seed(1234)
230 >>> A = np.arange(9).reshape((3, 3))
231 >>> B = np.random.randn(3, 3)
233 >>> AA, BB, Q, Z = linalg.qz(A, B)
234 >>> AA
235 array([[-13.40928183, -4.62471562, 1.09215523],
236 [ 0. , 0. , 1.22805978],
237 [ 0. , 0. , 0.31973817]])
238 >>> BB
239 array([[ 0.33362547, -1.37393632, 0.02179805],
240 [ 0. , 1.68144922, 0.74683866],
241 [ 0. , 0. , 0.9258294 ]])
242 >>> Q
243 array([[ 0.14134727, -0.97562773, 0.16784365],
244 [ 0.49835904, -0.07636948, -0.86360059],
245 [ 0.85537081, 0.20571399, 0.47541828]])
246 >>> Z
247 array([[-0.24900855, -0.51772687, 0.81850696],
248 [-0.79813178, 0.58842606, 0.12938478],
249 [-0.54861681, -0.6210585 , -0.55973739]])
251 See also
252 --------
253 ordqz
254 """
255 # output for real
256 # AA, BB, sdim, alphar, alphai, beta, vsl, vsr, work, info
257 # output for complex
258 # AA, BB, sdim, alpha, beta, vsl, vsr, work, info
259 result, _ = _qz(A, B, output=output, lwork=lwork, sort=sort,
260 overwrite_a=overwrite_a, overwrite_b=overwrite_b,
261 check_finite=check_finite)
262 return result[0], result[1], result[-4], result[-3]
265def ordqz(A, B, sort='lhp', output='real', overwrite_a=False,
266 overwrite_b=False, check_finite=True):
267 """QZ decomposition for a pair of matrices with reordering.
269 .. versionadded:: 0.17.0
271 Parameters
272 ----------
273 A : (N, N) array_like
274 2-D array to decompose
275 B : (N, N) array_like
276 2-D array to decompose
277 sort : {callable, 'lhp', 'rhp', 'iuc', 'ouc'}, optional
278 Specifies whether the upper eigenvalues should be sorted. A
279 callable may be passed that, given an ordered pair ``(alpha,
280 beta)`` representing the eigenvalue ``x = (alpha/beta)``,
281 returns a boolean denoting whether the eigenvalue should be
282 sorted to the top-left (True). For the real matrix pairs
283 ``beta`` is real while ``alpha`` can be complex, and for
284 complex matrix pairs both ``alpha`` and ``beta`` can be
285 complex. The callable must be able to accept a NumPy
286 array. Alternatively, string parameters may be used:
288 - 'lhp' Left-hand plane (x.real < 0.0)
289 - 'rhp' Right-hand plane (x.real > 0.0)
290 - 'iuc' Inside the unit circle (x*x.conjugate() < 1.0)
291 - 'ouc' Outside the unit circle (x*x.conjugate() > 1.0)
293 With the predefined sorting functions, an infinite eigenvalue
294 (i.e., ``alpha != 0`` and ``beta = 0``) is considered to lie in
295 neither the left-hand nor the right-hand plane, but it is
296 considered to lie outside the unit circle. For the eigenvalue
297 ``(alpha, beta) = (0, 0)``, the predefined sorting functions
298 all return `False`.
299 output : str {'real','complex'}, optional
300 Construct the real or complex QZ decomposition for real matrices.
301 Default is 'real'.
302 overwrite_a : bool, optional
303 If True, the contents of A are overwritten.
304 overwrite_b : bool, optional
305 If True, the contents of B are overwritten.
306 check_finite : bool, optional
307 If true checks the elements of `A` and `B` are finite numbers. If
308 false does no checking and passes matrix through to
309 underlying algorithm.
311 Returns
312 -------
313 AA : (N, N) ndarray
314 Generalized Schur form of A.
315 BB : (N, N) ndarray
316 Generalized Schur form of B.
317 alpha : (N,) ndarray
318 alpha = alphar + alphai * 1j. See notes.
319 beta : (N,) ndarray
320 See notes.
321 Q : (N, N) ndarray
322 The left Schur vectors.
323 Z : (N, N) ndarray
324 The right Schur vectors.
326 Notes
327 -----
328 On exit, ``(ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N``, will be the
329 generalized eigenvalues. ``ALPHAR(j) + ALPHAI(j)*i`` and
330 ``BETA(j),j=1,...,N`` are the diagonals of the complex Schur form (S,T)
331 that would result if the 2-by-2 diagonal blocks of the real generalized
332 Schur form of (A,B) were further reduced to triangular form using complex
333 unitary transformations. If ALPHAI(j) is zero, then the jth eigenvalue is
334 real; if positive, then the ``j``th and ``(j+1)``st eigenvalues are a
335 complex conjugate pair, with ``ALPHAI(j+1)`` negative.
337 See also
338 --------
339 qz
341 Examples
342 --------
343 >>> from scipy.linalg import ordqz
344 >>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]])
345 >>> B = np.array([[0, 6, 0, 0], [5, 0, 2, 1], [5, 2, 6, 6], [4, 7, 7, 7]])
346 >>> AA, BB, alpha, beta, Q, Z = ordqz(A, B, sort='lhp')
348 Since we have sorted for left half plane eigenvalues, negatives come first
350 >>> (alpha/beta).real < 0
351 array([ True, True, False, False], dtype=bool)
353 """
354 # NOTE: should users be able to set these?
355 lwork = None
356 result, typ = _qz(A, B, output=output, lwork=lwork, sort=None,
357 overwrite_a=overwrite_a, overwrite_b=overwrite_b,
358 check_finite=check_finite)
359 AA, BB, Q, Z = result[0], result[1], result[-4], result[-3]
360 if typ not in 'cz':
361 alpha, beta = result[3] + result[4]*1.j, result[5]
362 else:
363 alpha, beta = result[3], result[4]
365 sfunction = _select_function(sort)
366 select = sfunction(alpha, beta)
368 tgsen, = get_lapack_funcs(('tgsen',), (AA, BB))
370 if lwork is None or lwork == -1:
371 result = tgsen(select, AA, BB, Q, Z, lwork=-1)
372 lwork = result[-3][0].real.astype(np.int)
373 # looks like wrong value passed to ZTGSYL if not
374 lwork += 1
376 liwork = None
377 if liwork is None or liwork == -1:
378 result = tgsen(select, AA, BB, Q, Z, liwork=-1)
379 liwork = result[-2][0]
381 result = tgsen(select, AA, BB, Q, Z, lwork=lwork, liwork=liwork)
383 info = result[-1]
384 if info < 0:
385 raise ValueError("Illegal value in argument %d of tgsen" % -info)
386 elif info == 1:
387 raise ValueError("Reordering of (A, B) failed because the transformed"
388 " matrix pair (A, B) would be too far from "
389 "generalized Schur form; the problem is very "
390 "ill-conditioned. (A, B) may have been partially "
391 "reorded. If requested, 0 is returned in DIF(*), "
392 "PL, and PR.")
394 # for real results has a, b, alphar, alphai, beta, q, z, m, pl, pr, dif,
395 # work, iwork, info
396 if typ in ['f', 'd']:
397 alpha = result[2] + result[3] * 1.j
398 return (result[0], result[1], alpha, result[4], result[5], result[6])
399 # for complex results has a, b, alpha, beta, q, z, m, pl, pr, dif, work,
400 # iwork, info
401 else:
402 return result[0], result[1], result[2], result[3], result[4], result[5]