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 from __future__ import division, print_function, absolute_import import warnings import numpy as np from numpy import asarray_chkfinite from .misc import LinAlgError, _datacopied from .lapack import get_lapack_funcs from scipy.lib.six import callable __all__ = ['qz'] _double_precision = ['i','l','d'] def _select_function(sort, typ): if typ in ['F','D']: if callable(sort): # assume the user knows what they're doing sfunction = sort elif sort == 'lhp': sfunction = lambda x,y: (np.real(x/y) < 0.0) elif sort == 'rhp': sfunction = lambda x,y: (np.real(x/y) >= 0.0) elif sort == 'iuc': sfunction = lambda x,y: (abs(x/y) <= 1.0) elif sort == 'ouc': sfunction = lambda x,y: (abs(x/y) > 1.0) else: raise ValueError("sort parameter must be None, a callable, or " "one of ('lhp','rhp','iuc','ouc')") elif typ in ['f','d']: if callable(sort): # assume the user knows what they're doing sfunction = sort elif sort == 'lhp': sfunction = lambda x,y,z: (np.real((x+y*1j)/z) < 0.0) elif sort == 'rhp': sfunction = lambda x,y,z: (np.real((x+y*1j)/z) >= 0.0) elif sort == 'iuc': sfunction = lambda x,y,z: (abs((x+y*1j)/z) <= 1.0) elif sort == 'ouc': sfunction = lambda x,y,z: (abs((x+y*1j)/z) > 1.0) else: raise ValueError("sort parameter must be None, a callable, or " "one of ('lhp','rhp','iuc','ouc')") else: # to avoid an error later raise ValueError("dtype %s not understood" % typ) return sfunction def qz(A, B, output='real', lwork=None, sort=None, overwrite_a=False, overwrite_b=False, check_finite=True): """ QZ decompostion 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. .. versionadded:: 0.11.0 Parameters ---------- A : (N, N) array_like 2d array to decompose B : (N, N) array_like 2d array to decompose output : str {'real','complex'} 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, IT DOESN'T WORK WELL ON WINDOWS. 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). check_finite : boolean 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. sdim : int, optional If sorting was requested, a fifth return value will contain the number of eigenvalues for which the sort condition was True. Notes ----- Q is transposed versus the equivalent function in Matlab. 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]]) """ 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 (will " " change when this functionality is made more robust).") if not output 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 try: assert a_m == a_n == b_m == b_n except AssertionError: 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) if sort is None: sort_t = 0 sfunction = lambda x: None else: sort_t = 1 sfunction = _select_function(sort, typa) result = gges(sfunction, a1, b1, lwork=lwork, overwrite_a=overwrite_a, overwrite_b=overwrite_b, sort_t=sort_t) info = result[-1] if info < 0: raise ValueError("Illegal value in argument %d of gges" % -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=%d,...,N" % info-1, UserWarning) 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 caused due to scaling.") elif info == a_n+3: raise LinAlgError("Reordering failed in tgsen") # output for real # AA, BB, sdim, alphar, alphai, beta, vsl, vsr, work, info # output for complex # AA, BB, sdim, alphai, beta, vsl, vsr, work, info if sort_t == 0: return result[0], result[1], result[-4], result[-3] else: return result[0], result[1], result[-4], result[-3], result[2]
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