Coverage for /pythoncovmergedfiles/medio/medio/usr/local/lib/python3.9/dist-packages/scipy/linalg/_decomp_qz.py: 12%
120 statements
« prev ^ index » next coverage.py v7.3.1, created at 2023-09-23 06:43 +0000
1import warnings
3import numpy as np
4from numpy import asarray_chkfinite
5from ._misc import LinAlgError, _datacopied, LinAlgWarning
6from .lapack import get_lapack_funcs
9__all__ = ['qz', 'ordqz']
11_double_precision = ['i', 'l', 'd']
14def _select_function(sort):
15 if callable(sort):
16 # assume the user knows what they're doing
17 sfunction = sort
18 elif sort == 'lhp':
19 sfunction = _lhp
20 elif sort == 'rhp':
21 sfunction = _rhp
22 elif sort == 'iuc':
23 sfunction = _iuc
24 elif sort == 'ouc':
25 sfunction = _ouc
26 else:
27 raise ValueError("sort parameter must be None, a callable, or "
28 "one of ('lhp','rhp','iuc','ouc')")
30 return sfunction
33def _lhp(x, y):
34 out = np.empty_like(x, dtype=bool)
35 nonzero = (y != 0)
36 # handles (x, y) = (0, 0) too
37 out[~nonzero] = False
38 out[nonzero] = (np.real(x[nonzero]/y[nonzero]) < 0.0)
39 return out
42def _rhp(x, y):
43 out = np.empty_like(x, dtype=bool)
44 nonzero = (y != 0)
45 # handles (x, y) = (0, 0) too
46 out[~nonzero] = False
47 out[nonzero] = (np.real(x[nonzero]/y[nonzero]) > 0.0)
48 return out
51def _iuc(x, y):
52 out = np.empty_like(x, dtype=bool)
53 nonzero = (y != 0)
54 # handles (x, y) = (0, 0) too
55 out[~nonzero] = False
56 out[nonzero] = (abs(x[nonzero]/y[nonzero]) < 1.0)
57 return out
60def _ouc(x, y):
61 out = np.empty_like(x, dtype=bool)
62 xzero = (x == 0)
63 yzero = (y == 0)
64 out[xzero & yzero] = False
65 out[~xzero & yzero] = True
66 out[~yzero] = (abs(x[~yzero]/y[~yzero]) > 1.0)
67 return out
70def _qz(A, B, output='real', lwork=None, sort=None, overwrite_a=False,
71 overwrite_b=False, check_finite=True):
72 if sort is not None:
73 # Disabled due to segfaults on win32, see ticket 1717.
74 raise ValueError("The 'sort' input of qz() has to be None and will be "
75 "removed in a future release. Use ordqz instead.")
77 if output not in ['real', 'complex', 'r', 'c']:
78 raise ValueError("argument must be 'real', or 'complex'")
80 if check_finite:
81 a1 = asarray_chkfinite(A)
82 b1 = asarray_chkfinite(B)
83 else:
84 a1 = np.asarray(A)
85 b1 = np.asarray(B)
87 a_m, a_n = a1.shape
88 b_m, b_n = b1.shape
89 if not (a_m == a_n == b_m == b_n):
90 raise ValueError("Array dimensions must be square and agree")
92 typa = a1.dtype.char
93 if output in ['complex', 'c'] and typa not in ['F', 'D']:
94 if typa in _double_precision:
95 a1 = a1.astype('D')
96 typa = 'D'
97 else:
98 a1 = a1.astype('F')
99 typa = 'F'
100 typb = b1.dtype.char
101 if output in ['complex', 'c'] and typb not in ['F', 'D']:
102 if typb in _double_precision:
103 b1 = b1.astype('D')
104 typb = 'D'
105 else:
106 b1 = b1.astype('F')
107 typb = 'F'
109 overwrite_a = overwrite_a or (_datacopied(a1, A))
110 overwrite_b = overwrite_b or (_datacopied(b1, B))
112 gges, = get_lapack_funcs(('gges',), (a1, b1))
114 if lwork is None or lwork == -1:
115 # get optimal work array size
116 result = gges(lambda x: None, a1, b1, lwork=-1)
117 lwork = result[-2][0].real.astype(np.int_)
119 def sfunction(x):
120 return 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(f"Illegal value in argument {-info} of gges")
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-by-n
151 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 See Also
221 --------
222 ordqz
224 Notes
225 -----
226 Q is transposed versus the equivalent function in Matlab.
228 .. versionadded:: 0.11.0
230 Examples
231 --------
232 >>> import numpy as np
233 >>> from scipy.linalg import qz
235 >>> A = np.array([[1, 2, -1], [5, 5, 5], [2, 4, -8]])
236 >>> B = np.array([[1, 1, -3], [3, 1, -1], [5, 6, -2]])
238 Compute the decomposition. The QZ decomposition is not unique, so
239 depending on the underlying library that is used, there may be
240 differences in the signs of coefficients in the following output.
242 >>> AA, BB, Q, Z = qz(A, B)
243 >>> AA
244 array([[-1.36949157, -4.05459025, 7.44389431],
245 [ 0. , 7.65653432, 5.13476017],
246 [ 0. , -0.65978437, 2.4186015 ]]) # may vary
247 >>> BB
248 array([[ 1.71890633, -1.64723705, -0.72696385],
249 [ 0. , 8.6965692 , -0. ],
250 [ 0. , 0. , 2.27446233]]) # may vary
251 >>> Q
252 array([[-0.37048362, 0.1903278 , 0.90912992],
253 [-0.90073232, 0.16534124, -0.40167593],
254 [ 0.22676676, 0.96769706, -0.11017818]]) # may vary
255 >>> Z
256 array([[-0.67660785, 0.63528924, -0.37230283],
257 [ 0.70243299, 0.70853819, -0.06753907],
258 [ 0.22088393, -0.30721526, -0.92565062]]) # may vary
260 Verify the QZ decomposition. With real output, we only need the
261 transpose of ``Z`` in the following expressions.
263 >>> Q @ AA @ Z.T # Should be A
264 array([[ 1., 2., -1.],
265 [ 5., 5., 5.],
266 [ 2., 4., -8.]])
267 >>> Q @ BB @ Z.T # Should be B
268 array([[ 1., 1., -3.],
269 [ 3., 1., -1.],
270 [ 5., 6., -2.]])
272 Repeat the decomposition, but with ``output='complex'``.
274 >>> AA, BB, Q, Z = qz(A, B, output='complex')
276 For conciseness in the output, we use ``np.set_printoptions()`` to set
277 the output precision of NumPy arrays to 3 and display tiny values as 0.
279 >>> np.set_printoptions(precision=3, suppress=True)
280 >>> AA
281 array([[-1.369+0.j , 2.248+4.237j, 4.861-5.022j],
282 [ 0. +0.j , 7.037+2.922j, 0.794+4.932j],
283 [ 0. +0.j , 0. +0.j , 2.655-1.103j]]) # may vary
284 >>> BB
285 array([[ 1.719+0.j , -1.115+1.j , -0.763-0.646j],
286 [ 0. +0.j , 7.24 +0.j , -3.144+3.322j],
287 [ 0. +0.j , 0. +0.j , 2.732+0.j ]]) # may vary
288 >>> Q
289 array([[ 0.326+0.175j, -0.273-0.029j, -0.886-0.052j],
290 [ 0.794+0.426j, -0.093+0.134j, 0.402-0.02j ],
291 [-0.2 -0.107j, -0.816+0.482j, 0.151-0.167j]]) # may vary
292 >>> Z
293 array([[ 0.596+0.32j , -0.31 +0.414j, 0.393-0.347j],
294 [-0.619-0.332j, -0.479+0.314j, 0.154-0.393j],
295 [-0.195-0.104j, 0.576+0.27j , 0.715+0.187j]]) # may vary
297 With complex arrays, we must use ``Z.conj().T`` in the following
298 expressions to verify the decomposition.
300 >>> Q @ AA @ Z.conj().T # Should be A
301 array([[ 1.-0.j, 2.-0.j, -1.-0.j],
302 [ 5.+0.j, 5.+0.j, 5.-0.j],
303 [ 2.+0.j, 4.+0.j, -8.+0.j]])
304 >>> Q @ BB @ Z.conj().T # Should be B
305 array([[ 1.+0.j, 1.+0.j, -3.+0.j],
306 [ 3.-0.j, 1.-0.j, -1.+0.j],
307 [ 5.+0.j, 6.+0.j, -2.+0.j]])
309 """
310 # output for real
311 # AA, BB, sdim, alphar, alphai, beta, vsl, vsr, work, info
312 # output for complex
313 # AA, BB, sdim, alpha, beta, vsl, vsr, work, info
314 result, _ = _qz(A, B, output=output, lwork=lwork, sort=sort,
315 overwrite_a=overwrite_a, overwrite_b=overwrite_b,
316 check_finite=check_finite)
317 return result[0], result[1], result[-4], result[-3]
320def ordqz(A, B, sort='lhp', output='real', overwrite_a=False,
321 overwrite_b=False, check_finite=True):
322 """QZ decomposition for a pair of matrices with reordering.
324 Parameters
325 ----------
326 A : (N, N) array_like
327 2-D array to decompose
328 B : (N, N) array_like
329 2-D array to decompose
330 sort : {callable, 'lhp', 'rhp', 'iuc', 'ouc'}, optional
331 Specifies whether the upper eigenvalues should be sorted. A
332 callable may be passed that, given an ordered pair ``(alpha,
333 beta)`` representing the eigenvalue ``x = (alpha/beta)``,
334 returns a boolean denoting whether the eigenvalue should be
335 sorted to the top-left (True). For the real matrix pairs
336 ``beta`` is real while ``alpha`` can be complex, and for
337 complex matrix pairs both ``alpha`` and ``beta`` can be
338 complex. The callable must be able to accept a NumPy
339 array. Alternatively, string parameters may be used:
341 - 'lhp' Left-hand plane (x.real < 0.0)
342 - 'rhp' Right-hand plane (x.real > 0.0)
343 - 'iuc' Inside the unit circle (x*x.conjugate() < 1.0)
344 - 'ouc' Outside the unit circle (x*x.conjugate() > 1.0)
346 With the predefined sorting functions, an infinite eigenvalue
347 (i.e., ``alpha != 0`` and ``beta = 0``) is considered to lie in
348 neither the left-hand nor the right-hand plane, but it is
349 considered to lie outside the unit circle. For the eigenvalue
350 ``(alpha, beta) = (0, 0)``, the predefined sorting functions
351 all return `False`.
352 output : str {'real','complex'}, optional
353 Construct the real or complex QZ decomposition for real matrices.
354 Default is 'real'.
355 overwrite_a : bool, optional
356 If True, the contents of A are overwritten.
357 overwrite_b : bool, optional
358 If True, the contents of B are overwritten.
359 check_finite : bool, optional
360 If true checks the elements of `A` and `B` are finite numbers. If
361 false does no checking and passes matrix through to
362 underlying algorithm.
364 Returns
365 -------
366 AA : (N, N) ndarray
367 Generalized Schur form of A.
368 BB : (N, N) ndarray
369 Generalized Schur form of B.
370 alpha : (N,) ndarray
371 alpha = alphar + alphai * 1j. See notes.
372 beta : (N,) ndarray
373 See notes.
374 Q : (N, N) ndarray
375 The left Schur vectors.
376 Z : (N, N) ndarray
377 The right Schur vectors.
379 See Also
380 --------
381 qz
383 Notes
384 -----
385 On exit, ``(ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N``, will be the
386 generalized eigenvalues. ``ALPHAR(j) + ALPHAI(j)*i`` and
387 ``BETA(j),j=1,...,N`` are the diagonals of the complex Schur form (S,T)
388 that would result if the 2-by-2 diagonal blocks of the real generalized
389 Schur form of (A,B) were further reduced to triangular form using complex
390 unitary transformations. If ALPHAI(j) is zero, then the jth eigenvalue is
391 real; if positive, then the ``j``\\ th and ``(j+1)``\\ st eigenvalues are a
392 complex conjugate pair, with ``ALPHAI(j+1)`` negative.
394 .. versionadded:: 0.17.0
396 Examples
397 --------
398 >>> import numpy as np
399 >>> from scipy.linalg import ordqz
400 >>> A = np.array([[2, 5, 8, 7], [5, 2, 2, 8], [7, 5, 6, 6], [5, 4, 4, 8]])
401 >>> B = np.array([[0, 6, 0, 0], [5, 0, 2, 1], [5, 2, 6, 6], [4, 7, 7, 7]])
402 >>> AA, BB, alpha, beta, Q, Z = ordqz(A, B, sort='lhp')
404 Since we have sorted for left half plane eigenvalues, negatives come first
406 >>> (alpha/beta).real < 0
407 array([ True, True, False, False], dtype=bool)
409 """
410 (AA, BB, _, *ab, Q, Z, _, _), typ = _qz(A, B, output=output, sort=None,
411 overwrite_a=overwrite_a,
412 overwrite_b=overwrite_b,
413 check_finite=check_finite)
415 if typ == 's':
416 alpha, beta = ab[0] + ab[1]*np.complex64(1j), ab[2]
417 elif typ == 'd':
418 alpha, beta = ab[0] + ab[1]*1.j, ab[2]
419 else:
420 alpha, beta = ab
422 sfunction = _select_function(sort)
423 select = sfunction(alpha, beta)
425 tgsen = get_lapack_funcs('tgsen', (AA, BB))
426 # the real case needs 4n + 16 lwork
427 lwork = 4*AA.shape[0] + 16 if typ in 'sd' else 1
428 AAA, BBB, *ab, QQ, ZZ, _, _, _, _, info = tgsen(select, AA, BB, Q, Z,
429 ijob=0,
430 lwork=lwork, liwork=1)
432 # Once more for tgsen output
433 if typ == 's':
434 alpha, beta = ab[0] + ab[1]*np.complex64(1j), ab[2]
435 elif typ == 'd':
436 alpha, beta = ab[0] + ab[1]*1.j, ab[2]
437 else:
438 alpha, beta = ab
440 if info < 0:
441 raise ValueError(f"Illegal value in argument {-info} of tgsen")
442 elif info == 1:
443 raise ValueError("Reordering of (A, B) failed because the transformed"
444 " matrix pair (A, B) would be too far from "
445 "generalized Schur form; the problem is very "
446 "ill-conditioned. (A, B) may have been partially "
447 "reordered.")
449 return AAA, BBB, alpha, beta, QQ, ZZ