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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
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
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
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
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
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
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]
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] |