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File Content: `_decomp_qz.py`

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', '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 = 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')")

    return sfunction


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 %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 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 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)

    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
    """
    #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]

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__pycache__	---	0755
tests	---	0755
__init__.py	6351 bytes	0644
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_cblas.cpython-35m-x86_64-linux-gnu.so	41224 bytes	0755
_clapack.cpython-35m-x86_64-linux-gnu.so	68984 bytes	0755
_cython_signature_generator.py	8369 bytes	0644
_cython_wrapper_generators.py	23463 bytes	0644
_decomp_polar.py	3623 bytes	0644
_decomp_qz.py	12986 bytes	0644
_decomp_update.cpython-35m-x86_64-linux-gnu.so	307856 bytes	0755
_expm_frechet.py	12182 bytes	0644
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_flapack.cpython-35m-x86_64-linux-gnu.so	934896 bytes	0755
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_interpolative_backend.py	44935 bytes	0644
_matfuncs_inv_ssq.py	28050 bytes	0644
_matfuncs_sqrtm.py	5867 bytes	0644
_procrustes.py	2375 bytes	0644
_solve_toeplitz.cpython-35m-x86_64-linux-gnu.so	183728 bytes	0755
_solvers.py	10185 bytes	0644
_testutils.py	1814 bytes	0644
basic.py	39332 bytes	0644
blas.py	6855 bytes	0644
calc_lwork.py	666 bytes	0644
cython_blas.cpython-35m-x86_64-linux-gnu.so	204424 bytes	0755
cython_blas.pxd	14416 bytes	0644
cython_lapack.cpython-35m-x86_64-linux-gnu.so	574600 bytes	0755
cython_lapack.pxd	169411 bytes	0644
decomp.py	31223 bytes	0644
decomp_cholesky.py	9601 bytes	0644
decomp_lu.py	5796 bytes	0644
decomp_qr.py	12675 bytes	0644
decomp_schur.py	8375 bytes	0644
decomp_svd.py	7226 bytes	0644
flinalg.py	1727 bytes	0644
interpolative.py	31146 bytes	0644
lapack.py	8103 bytes	0644
linalg_version.py	159 bytes	0644
matfuncs.py	19939 bytes	0644
misc.py	5881 bytes	0644
setup.py	5888 bytes	0644
special_matrices.py	29225 bytes	0644

File Content: _decomp_qz.py

File Content: `_decomp_qz.py`