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Server: Apache/2.4.46 (Win64) OpenSSL/1.1.1j PHP/8.4.25
System: Windows NT DESKTOP-4TAV2RJ 10.0 build 19045 (Windows 10) AMD64
User: fred (0)
PHP: 8.4.25
Disabled: NONE
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File: C:/Users/fred/anaconda3/Lib/site-packages/scipy/linalg/_decomp_update.cp312-win_amd64.pyd
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Routines for updating QR decompositions

.. versionadded:: 0.16.0

Expected (M,M) (M,N) or (M,N) (N,N) but found FImportErrorInput array too large for use with BLASInputs must be arraysLinAlgErrorMemoryErrorOne of the columns of u lies in the span of Q. Found reciprocal condition number of %s for Q augmented with u/||u||.Only arrays with dtypes float32, float64, complex64, and complex128 are supported.Q
    QR downdate on row or column deletions

    If ``A = Q R`` is the QR factorization of ``A``, return the QR
    factorization of ``A`` where ``p`` rows or columns have been removed
    starting at row or column ``k``.

    Parameters
    ----------
    Q : (M, M) or (M, N) array_like
        Unitary/orthogonal matrix from QR decomposition.
    R : (M, N) or (N, N) array_like
        Upper triangular matrix from QR decomposition.
    k : int
        Index of the first row or column to delete.
    p : int, optional
        Number of rows or columns to delete, defaults to 1.
    which: {'row', 'col'}, optional
        Determines if rows or columns will be deleted, defaults to 'row'
    overwrite_qr : bool, optional
        If True, consume Q and R, overwriting their contents with their
        downdated versions, and returning appropriately sized views.
        Defaults to False.
    check_finite : bool, optional
        Whether to check that the input matrix contains only finite numbers.
        Disabling may give a performance gain, but may result in problems
        (crashes, non-termination) if the inputs do contain infinities or NaNs.
        Default is True.

    Returns
    -------
    Q1 : ndarray
        Updated unitary/orthogonal factor
    R1 : ndarray
        Updated upper triangular factor

    See Also
    --------
    qr, qr_multiply, qr_insert, qr_update

    Notes
    -----
    This routine does not guarantee that the diagonal entries of ``R1`` are
    positive.

    .. versionadded:: 0.16.0

    References
    ----------
    .. [1] Golub, G. H. & Van Loan, C. F. Matrix Computations, 3rd Ed.
           (Johns Hopkins University Press, 1996).

    .. [2] Daniel, J. W., Gragg, W. B., Kaufman, L. & Stewart, G. W.
           Reorthogonalization and stable algorithms for updating the
           Gram-Schmidt QR factorization. Math. Comput. 30, 772-795 (1976).

    .. [3] Reichel, L. & Gragg, W. B. Algorithm 686: FORTRAN Subroutines for
           Updating the QR Decomposition. ACM Trans. Math. Softw. 16, 369-377
           (1990).

    Examples
    --------
    >>> import numpy as np
    >>> from scipy import linalg
    >>> a = np.array([[  3.,  -2.,  -2.],
    ...               [  6.,  -9.,  -3.],
    ...               [ -3.,  10.,   1.],
    ...               [  6.,  -7.,   4.],
    ...               [  7.,   8.,  -6.]])
    >>> q, r = linalg.qr(a)

    Given this QR decomposition, update q and r when 2 rows are removed.

    >>> q1, r1 = linalg.qr_delete(q, r, 2, 2, 'row', False)
    >>> q1
    array([[ 0.30942637,  0.15347579,  0.93845645],  # may vary (signs)
           [ 0.61885275,  0.71680171, -0.32127338],
           [ 0.72199487, -0.68017681, -0.12681844]])
    >>> r1
    array([[  9.69535971,  -0.4125685 ,  -6.80738023],  # may vary (signs)
           [  0.        , -12.19958144,   1.62370412],
           [  0.        ,   0.        ,  -0.15218213]])

    The update is equivalent, but faster than the following.

    >>> a1 = np.delete(a, slice(2,4), 0)
    >>> a1
    array([[ 3., -2., -2.],
           [ 6., -9., -3.],
           [ 7.,  8., -6.]])
    >>> q_direct, r_direct = linalg.qr(a1)

    Check that we have equivalent results:

    >>> np.dot(q1, r1)
    array([[ 3., -2., -2.],
           [ 6., -9., -3.],
           [ 7.,  8., -6.]])
    >>> np.allclose(np.dot(q1, r1), a1)
    True

    And the updated Q is still unitary:

    >>> np.allclose(np.dot(q1.T, q1), np.eye(3))
    True

    
    QR update on row or column insertions

    If ``A = Q R`` is the QR factorization of ``A``, return the QR
    factorization of ``A`` where rows or columns have been inserted starting
    at row or column ``k``.

    Parameters
    ----------
    Q : (M, M) array_like
        Unitary/orthogonal matrix from the QR decomposition of A.
    R : (M, N) array_like
        Upper triangular matrix from the QR decomposition of A.
    u : (N,), (p, N), (M,), or (M, p) array_like
        Rows or columns to insert
    k : int
        Index before which `u` is to be inserted.
    which: {'row', 'col'}, optional
        Determines if rows or columns will be inserted, defaults to 'row'
    rcond : float
        Lower bound on the reciprocal condition number of ``Q`` augmented with
        ``u/||u||`` Only used when updating economic mode (thin, (M,N) (N,N))
        decompositions.  If None, machine precision is used.  Defaults to
        None.
    overwrite_qru : bool, optional
        If True, consume Q, R, and u, if possible, while performing the update,
        otherwise make copies as necessary. Defaults to False.
    check_finite : bool, optional
        Whether to check that the input matrices contain only finite numbers.
        Disabling may give a performance gain, but may result in problems
        (crashes, non-termination) if the inputs do contain infinities or NaNs.
        Default is True.

    Returns
    -------
    Q1 : ndarray
        Updated unitary/orthogonal factor
    R1 : ndarray
        Updated upper triangular factor

    Raises
    ------
    LinAlgError :
        If updating a (M,N) (N,N) factorization and the reciprocal condition
        number of Q augmented with ``u/||u||`` is smaller than rcond.

    See Also
    --------
    qr, qr_multiply, qr_delete, qr_update

    Notes
    -----
    This routine does not guarantee that the diagonal entries of ``R1`` are
    positive.

    .. versionadded:: 0.16.0

    References
    ----------

    .. [1] Golub, G. H. & Van Loan, C. F. Matrix Computations, 3rd Ed.
           (Johns Hopkins University Press, 1996).

    .. [2] Daniel, J. W., Gragg, W. B., Kaufman, L. & Stewart, G. W.
           Reorthogonalization and stable algorithms for updating the
           Gram-Schmidt QR factorization. Math. Comput. 30, 772-795 (1976).

    .. [3] Reichel, L. & Gragg, W. B. Algorithm 686: FORTRAN Subroutines for
           Updating the QR Decomposition. ACM Trans. Math. Softw. 16, 369-377
           (1990).

    Examples
    --------
    >>> import numpy as np
    >>> from scipy import linalg
    >>> a = np.array([[  3.,  -2.,  -2.],
    ...               [  6.,  -7.,   4.],
    ...               [  7.,   8.,  -6.]])
    >>> q, r = linalg.qr(a)

    Given this QR decomposition, update q and r when 2 rows are inserted.

    >>> u = np.array([[  6.,  -9.,  -3.],
    ...               [ -3.,  10.,   1.]])
    >>> q1, r1 = linalg.qr_insert(q, r, u, 2, 'row')
    >>> q1
    array([[-0.25445668,  0.02246245,  0.18146236, -0.72798806,  0.60979671],  # may vary (signs)
           [-0.50891336,  0.23226178, -0.82836478, -0.02837033, -0.00828114],
           [-0.50891336,  0.35715302,  0.38937158,  0.58110733,  0.35235345],
           [ 0.25445668, -0.52202743, -0.32165498,  0.36263239,  0.65404509],
           [-0.59373225, -0.73856549,  0.16065817, -0.0063658 , -0.27595554]])
    >>> r1
    array([[-11.78982612,   6.44623587,   3.81685018],  # may vary (signs)
           [  0.        , -16.01393278,   3.72202865],
           [  0.        ,   0.        ,  -6.13010256],
           [  0.        ,   0.        ,   0.        ],
           [  0.        ,   0.        ,   0.        ]])

    The update is equivalent, but faster than the following.

    >>> a1 = np.insert(a, 2, u, 0)
    >>> a1
    array([[  3.,  -2.,  -2.],
           [  6.,  -7.,   4.],
           [  6.,  -9.,  -3.],
           [ -3.,  10.,   1.],
           [  7.,   8.,  -6.]])
    >>> q_direct, r_direct = linalg.qr(a1)

    Check that we have equivalent results:

    >>> np.dot(q1, r1)
    array([[  3.,  -2.,  -2.],
           [  6.,  -7.,   4.],
           [  6.,  -9.,  -3.],
           [ -3.,  10.,   1.],
           [  7.,   8.,  -6.]])

    >>> np.allclose(np.dot(q1, r1), a1)
    True

    And the updated Q is still unitary:

    >>> np.allclose(np.dot(q1.T, q1), np.eye(5))
    True

    Q and R do not have compatible shapes. Expected (M,M) (M,N) or (M,N) (N,N) but found Q and R must be 2-DQ and R must have the same dtypeR
    Rank-k QR update

    If ``A = Q R`` is the QR factorization of ``A``, return the QR
    factorization of ``A + u v**T`` for real ``A`` or ``A + u v**H``
    for complex ``A``.

    Parameters
    ----------
    Q : (M, M) or (M, N) array_like
        Unitary/orthogonal matrix from the qr decomposition of A.
    R : (M, N) or (N, N) array_like
        Upper triangular matrix from the qr decomposition of A.
    u : (M,) or (M, k) array_like
        Left update vector
    v : (N,) or (N, k) array_like
        Right update vector
    overwrite_qruv : bool, optional
        If True, consume Q, R, u, and v, if possible, while performing the
        update, otherwise make copies as necessary. Defaults to False.
    check_finite : bool, optional
        Whether to check that the input matrix contains only finite numbers.
        Disabling may give a performance gain, but may result in problems
        (crashes, non-termination) if the inputs do contain infinities or NaNs.
        Default is True.

    Returns
    -------
    Q1 : ndarray
        Updated unitary/orthogonal factor
    R1 : ndarray
        Updated upper triangular factor

    See Also
    --------
    qr, qr_multiply, qr_delete, qr_insert

    Notes
    -----
    This routine does not guarantee that the diagonal entries of `R1` are
    real or positive.

    .. versionadded:: 0.16.0

    References
    ----------
    .. [1] Golub, G. H. & Van Loan, C. F. Matrix Computations, 3rd Ed.
           (Johns Hopkins University Press, 1996).

    .. [2] Daniel, J. W., Gragg, W. B., Kaufman, L. & Stewart, G. W.
           Reorthogonalization and stable algorithms for updating the
           Gram-Schmidt QR factorization. Math. Comput. 30, 772-795 (1976).

    .. [3] Reichel, L. & Gragg, W. B. Algorithm 686: FORTRAN Subroutines for
           Updating the QR Decomposition. ACM Trans. Math. Softw. 16, 369-377
           (1990).

    Examples
    --------
    >>> import numpy as np
    >>> from scipy import linalg
    >>> a = np.array([[  3.,  -2.,  -2.],
    ...               [  6.,  -9.,  -3.],
    ...               [ -3.,  10.,   1.],
    ...               [  6.,  -7.,   4.],
    ...               [  7.,   8.,  -6.]])
    >>> q, r = linalg.qr(a)

    Given this q, r decomposition, perform a rank 1 update.

    >>> u = np.array([7., -2., 4., 3., 5.])
    >>> v = np.array([1., 3., -5.])
    >>> q_up, r_up = linalg.qr_update(q, r, u, v, False)
    >>> q_up
    array([[ 0.54073807,  0.18645997,  0.81707661, -0.02136616,  0.06902409],  # may vary (signs)
           [ 0.21629523, -0.63257324,  0.06567893,  0.34125904, -0.65749222],
           [ 0.05407381,  0.64757787, -0.12781284, -0.20031219, -0.72198188],
           [ 0.48666426, -0.30466718, -0.27487277, -0.77079214,  0.0256951 ],
           [ 0.64888568,  0.23001   , -0.4859845 ,  0.49883891,  0.20253783]])
    >>> r_up
    array([[ 18.49324201,  24.11691794, -44.98940746],  # may vary (signs)
           [  0.        ,  31.95894662, -27.40998201],
           [  0.        ,   0.        ,  -9.25451794],
           [  0.        ,   0.        ,   0.        ],
           [  0.        ,   0.        ,   0.        ]])

    The update is equivalent, but faster than the following.

    >>> a_up = a + np.outer(u, v)
    >>> q_direct, r_direct = linalg.qr(a_up)

    Check that we have equivalent results:

    >>> np.allclose(np.dot(q_up, r_up), a_up)
    True

    And the updated Q is still unitary:

    >>> np.allclose(np.dot(q_up.T, q_up), np.eye(5))
    True

    Updating economic (reduced, thin) decompositions is also possible:

    >>> qe, re = linalg.qr(a, mode='economic')
    >>> qe_up, re_up = linalg.qr_update(qe, re, u, v, False)
    >>> qe_up
    array([[ 0.54073807,  0.18645997,  0.81707661],  # may vary (signs)
           [ 0.21629523, -0.63257324,  0.06567893],
           [ 0.05407381,  0.64757787, -0.12781284],
           [ 0.48666426, -0.30466718, -0.27487277],
           [ 0.64888568,  0.23001   , -0.4859845 ]])
    >>> re_up
    array([[ 18.49324201,  24.11691794, -44.98940746],  # may vary (signs)
           [  0.        ,  31.95894662, -27.40998201],
           [  0.        ,   0.        ,  -9.25451794]])
    >>> np.allclose(np.dot(qe_up, re_up), a_up)
    True
    >>> np.allclose(np.dot(qe_up.T, qe_up), np.eye(3))
    True

    Similarly to the above, perform a rank 2 update.

    >>> u2 = np.array([[ 7., -1,],
    ...                [-2.,  4.],
    ...                [ 4.,  2.],
    ...                [ 3., -6.],
    ...                [ 5.,  3.]])
    >>> v2 = np.array([[ 1., 2.],
    ...                [ 3., 4.],
    ...                [-5., 2]])
    >>> q_up2, r_up2 = linalg.qr_update(q, r, u2, v2, False)
    >>> q_up2
    array([[-0.33626508, -0.03477253,  0.61956287, -0.64352987, -0.29618884],  # may vary (signs)
           [-0.50439762,  0.58319694, -0.43010077, -0.33395279,  0.33008064],
           [-0.21016568, -0.63123106,  0.0582249 , -0.13675572,  0.73163206],
           [ 0.12609941,  0.49694436,  0.64590024,  0.31191919,  0.47187344],
           [-0.75659643, -0.11517748,  0.10284903,  0.5986227 , -0.21299983]])
    >>> r_up2
    array([[-23.79075451, -41.1084062 ,  24.71548348],  # may vary (signs)
           [  0.        , -33.83931057,  11.02226551],
           [  0.        ,   0.        ,  48.91476811],
           [  0.        ,   0.        ,   0.        ],
           [  0.        ,   0.        ,   0.        ]])

    This update is also a valid qr decomposition of ``A + U V**T``.

    >>> a_up2 = a + np.dot(u2, v2.T)
    >>> np.allclose(a_up2, np.dot(q_up2, r_up2))
    True
    >>> np.allclose(np.dot(q_up2.T, q_up2), np.eye(5))
    True

    Reorthogonalization Failed, unable to perform row deletion.Second dimension of u and v must be the sameShape of u is incorrect, should be 1 <= u.ndim <= 2TThe {0}th argument to ?geqrf was invalidThe {0}th argument to ?ormqr/?unmqr was invalidUnable to allocate memory for array.Unable to allocate memory for arrayUpdate rank larger than np.dot(Q, R).ValueError .*?aall__all__array must not contain infs or NaNsasyncio.coroutinesbcheck_finitechkfinite__class_getitem__cline_in_tracebackcolconjconjugatecopydeletedoteconomic for Q and R respectively_form_qTuformatfull__import__info_initializinginsertintp_is_coroutineisfinitekk1'k' is out of boundslengthlinalgm__main__n__name__ndimnormnpnumpynumpy.core.multiarray failed to importnumpy.core.umath failed to importnumpy.linalgooverwriteoverwrite_qroverwrite_qruoverwrite_qruvpp1p_ecop_full'p' is out of rangeqq1qTuqTuptrqTusqTuvoidq and u must be a blas compatible type: f d F or Dq and u must have the same type.q must be either F or C contiguousqisFqnewqptrqr_deleteqr_delete (line 1444)qr_insertqr_insert (line 1689)qr_updateqr_update (line 2154)qsr1rangeravelrcond'rcond' is not used when updating full, (M,M) (M,N) decompositions and must be None.'rcond' is unused when inserting rows and must be Nonernewrowrptrrssscipy.linalg._decomp_updatescipy\linalg\_decomp_update.pyxshape__spec__sptrss__test__typecodeuu1u and v must have the same number of dimensionsu and v must have the same type as Q and R'u' must be either 1- or 2-Du must be either 1- or 2-D.'u' must have the same type as 'Q' and 'R''u' must have the same type as Q and Ru.shape[0] must equal Q.shape[0]'u' should be either (M,) or (M,p) when inserting columns. Found %s.'u' should be either (N,) or (p,N) when inserting rows. Found %s.uptrusuv_flagsvv1v must be either 1- or 2-D.v.shape[0] must equal R.shape[1]vptrvswhich'which' must be either 'row' or 'col' this function only exists to expose the cdef version below for testing
        purposes. Here we perform minimal input validation to ensure that the
        inputs meet the requirements below.
    qr_delete(Q, R, k, int p=1, which=u'row', overwrite_qr=False, check_finite=True)

    QR downdate on row or column deletions

    If ``A = Q R`` is the QR factorization of ``A``, return the QR
    factorization of ``A`` where ``p`` rows or columns have been removed
    starting at row or column ``k``.

    Parameters
    ----------
    Q : (M, M) or (M, N) array_like
        Unitary/orthogonal matrix from QR decomposition.
    R : (M, N) or (N, N) array_like
        Upper triangular matrix from QR decomposition.
    k : int
        Index of the first row or column to delete.
    p : int, optional
        Number of rows or columns to delete, defaults to 1.
    which: {'row', 'col'}, optional
        Determines if rows or columns will be deleted, defaults to 'row'
    overwrite_qr : bool, optional
        If True, consume Q and R, overwriting their contents with their
        downdated versions, and returning appropriately sized views.
        Defaults to False.
    check_finite : bool, optional
        Whether to check that the input matrix contains only finite numbers.
        Disabling may give a performance gain, but may result in problems
        (crashes, non-termination) if the inputs do contain infinities or NaNs.
        Default is True.

    Returns
    -------
    Q1 : ndarray
        Updated unitary/orthogonal factor
    R1 : ndarray
        Updated upper triangular factor

    See Also
    --------
    qr, qr_multiply, qr_insert, qr_update

    Notes
    -----
    This routine does not guarantee that the diagonal entries of ``R1`` are
    positive.

    .. versionadded:: 0.16.0

    References
    ----------
    .. [1] Golub, G. H. & Van Loan, C. F. Matrix Computations, 3rd Ed.
           (Johns Hopkins University Press, 1996).

    .. [2] Daniel, J. W., Gragg, W. B., Kaufman, L. & Stewart, G. W.
           Reorthogonalization and stable algorithms for updating the
           Gram-Schmidt QR factorization. Math. Comput. 30, 772-795 (1976).

    .. [3] Reichel, L. & Gragg, W. B. Algorithm 686: FORTRAN Subroutines for
           Updating the QR Decomposition. ACM Trans. Math. Softw. 16, 369-377
           (1990).

    Examples
    --------
    >>> import numpy as np
    >>> from scipy import linalg
    >>> a = np.array([[  3.,  -2.,  -2.],
    ...               [  6.,  -9.,  -3.],
    ...               [ -3.,  10.,   1.],
    ...               [  6.,  -7.,   4.],
    ...               [  7.,   8.,  -6.]])
    >>> q, r = linalg.qr(a)

    Given this QR decomposition, update q and r when 2 rows are removed.

    >>> q1, r1 = linalg.qr_delete(q, r, 2, 2, 'row', False)
    >>> q1
    array([[ 0.30942637,  0.15347579,  0.93845645],  # may vary (signs)
           [ 0.61885275,  0.71680171, -0.32127338],
           [ 0.72199487, -0.68017681, -0.12681844]])
    >>> r1
    array([[  9.69535971,  -0.4125685 ,  -6.80738023],  # may vary (signs)
           [  0.        , -12.19958144,   1.62370412],
           [  0.        ,   0.        ,  -0.15218213]])

    The update is equivalent, but faster than the following.

    >>> a1 = np.delete(a, slice(2,4), 0)
    >>> a1
    array([[ 3., -2., -2.],
           [ 6., -9., -3.],
           [ 7.,  8., -6.]])
    >>> q_direct, r_direct = linalg.qr(a1)

    Check that we have equivalent results:

    >>> np.dot(q1, r1)
    array([[ 3., -2., -2.],
           [ 6., -9., -3.],
           [ 7.,  8., -6.]])
    >>> np.allclose(np.dot(q1, r1), a1)
    True

    And the updated Q is still unitary:

    >>> np.allclose(np.dot(q1.T, q1), np.eye(3))
    True

    qr_insert(Q, R, u, k, which=u'row', rcond=None, overwrite_qru=False, check_finite=True)

    QR update on row or column insertions

    If ``A = Q R`` is the QR factorization of ``A``, return the QR
    factorization of ``A`` where rows or columns have been inserted starting
    at row or column ``k``.

    Parameters
    ----------
    Q : (M, M) array_like
        Unitary/orthogonal matrix from the QR decomposition of A.
    R : (M, N) array_like
        Upper triangular matrix from the QR decomposition of A.
    u : (N,), (p, N), (M,), or (M, p) array_like
        Rows or columns to insert
    k : int
        Index before which `u` is to be inserted.
    which: {'row', 'col'}, optional
        Determines if rows or columns will be inserted, defaults to 'row'
    rcond : float
        Lower bound on the reciprocal condition number of ``Q`` augmented with
        ``u/||u||`` Only used when updating economic mode (thin, (M,N) (N,N))
        decompositions.  If None, machine precision is used.  Defaults to
        None.
    overwrite_qru : bool, optional
        If True, consume Q, R, and u, if possible, while performing the update,
        otherwise make copies as necessary. Defaults to False.
    check_finite : bool, optional
        Whether to check that the input matrices contain only finite numbers.
        Disabling may give a performance gain, but may result in problems
        (crashes, non-termination) if the inputs do contain infinities or NaNs.
        Default is True.

    Returns
    -------
    Q1 : ndarray
        Updated unitary/orthogonal factor
    R1 : ndarray
        Updated upper triangular factor

    Raises
    ------
    LinAlgError :
        If updating a (M,N) (N,N) factorization and the reciprocal condition
        number of Q augmented with ``u/||u||`` is smaller than rcond.

    See Also
    --------
    qr, qr_multiply, qr_delete, qr_update

    Notes
    -----
    This routine does not guarantee that the diagonal entries of ``R1`` are
    positive.

    .. versionadded:: 0.16.0

    References
    ----------

    .. [1] Golub, G. H. & Van Loan, C. F. Matrix Computations, 3rd Ed.
           (Johns Hopkins University Press, 1996).

    .. [2] Daniel, J. W., Gragg, W. B., Kaufman, L. & Stewart, G. W.
           Reorthogonalization and stable algorithms for updating the
           Gram-Schmidt QR factorization. Math. Comput. 30, 772-795 (1976).

    .. [3] Reichel, L. & Gragg, W. B. Algorithm 686: FORTRAN Subroutines for
           Updating the QR Decomposition. ACM Trans. Math. Softw. 16, 369-377
           (1990).

    Examples
    --------
    >>> import numpy as np
    >>> from scipy import linalg
    >>> a = np.array([[  3.,  -2.,  -2.],
    ...               [  6.,  -7.,   4.],
    ...               [  7.,   8.,  -6.]])
    >>> q, r = linalg.qr(a)

    Given this QR decomposition, update q and r when 2 rows are inserted.

    >>> u = np.array([[  6.,  -9.,  -3.],
    ...               [ -3.,  10.,   1.]])
    >>> q1, r1 = linalg.qr_insert(q, r, u, 2, 'row')
    >>> q1
    array([[-0.25445668,  0.02246245,  0.18146236, -0.72798806,  0.60979671],  # may vary (signs)
           [-0.50891336,  0.23226178, -0.82836478, -0.02837033, -0.00828114],
           [-0.50891336,  0.35715302,  0.38937158,  0.58110733,  0.35235345],
           [ 0.25445668, -0.52202743, -0.32165498,  0.36263239,  0.65404509],
           [-0.59373225, -0.73856549,  0.16065817, -0.0063658 , -0.27595554]])
    >>> r1
    array([[-11.78982612,   6.44623587,   3.81685018],  # may vary (signs)
           [  0.        , -16.01393278,   3.72202865],
           [  0.        ,   0.        ,  -6.13010256],
           [  0.        ,   0.        ,   0.        ],
           [  0.        ,   0.        ,   0.        ]])

    The update is equivalent, but faster than the following.

    >>> a1 = np.insert(a, 2, u, 0)
    >>> a1
    array([[  3.,  -2.,  -2.],
           [  6.,  -7.,   4.],
           [  6.,  -9.,  -3.],
           [ -3.,  10.,   1.],
           [  7.,   8.,  -6.]])
    >>> q_direct, r_direct = linalg.qr(a1)

    Check that we have equivalent results:

    >>> np.dot(q1, r1)
    array([[  3.,  -2.,  -2.],
           [  6.,  -7.,   4.],
           [  6.,  -9.,  -3.],
           [ -3.,  10.,   1.],
           [  7.,   8.,  -6.]])

    >>> np.allclose(np.dot(q1, r1), a1)
    True

    And the updated Q is still unitary:

    >>> np.allclose(np.dot(q1.T, q1), np.eye(5))
    True

    qr_update(Q, R, u, v, overwrite_qruv=False, check_finite=True)

    Rank-k QR update

    If ``A = Q R`` is the QR factorization of ``A``, return the QR
    factorization of ``A + u v**T`` for real ``A`` or ``A + u v**H``
    for complex ``A``.

    Parameters
    ----------
    Q : (M, M) or (M, N) array_like
        Unitary/orthogonal matrix from the qr decomposition of A.
    R : (M, N) or (N, N) array_like
        Upper triangular matrix from the qr decomposition of A.
    u : (M,) or (M, k) array_like
        Left update vector
    v : (N,) or (N, k) array_like
        Right update vector
    overwrite_qruv : bool, optional
        If True, consume Q, R, u, and v, if possible, while performing the
        update, otherwise make copies as necessary. Defaults to False.
    check_finite : bool, optional
        Whether to check that the input matrix contains only finite numbers.
        Disabling may give a performance gain, but may result in problems
        (crashes, non-termination) if the inputs do contain infinities or NaNs.
        Default is True.

    Returns
    -------
    Q1 : ndarray
        Updated unitary/orthogonal factor
    R1 : ndarray
        Updated upper triangular factor

    See Also
    --------
    qr, qr_multiply, qr_delete, qr_insert

    Notes
    -----
    This routine does not guarantee that the diagonal entries of `R1` are
    real or positive.

    .. versionadded:: 0.16.0

    References
    ----------
    .. [1] Golub, G. H. & Van Loan, C. F. Matrix Computations, 3rd Ed.
           (Johns Hopkins University Press, 1996).

    .. [2] Daniel, J. W., Gragg, W. B., Kaufman, L. & Stewart, G. W.
           Reorthogonalization and stable algorithms for updating the
           Gram-Schmidt QR factorization. Math. Comput. 30, 772-795 (1976).

    .. [3] Reichel, L. & Gragg, W. B. Algorithm 686: FORTRAN Subroutines for
           Updating the QR Decomposition. ACM Trans. Math. Softw. 16, 369-377
           (1990).

    Examples
    --------
    >>> import numpy as np
    >>> from scipy import linalg
    >>> a = np.array([[  3.,  -2.,  -2.],
    ...               [  6.,  -9.,  -3.],
    ...               [ -3.,  10.,   1.],
    ...               [  6.,  -7.,   4.],
    ...               [  7.,   8.,  -6.]])
    >>> q, r = linalg.qr(a)

    Given this q, r decomposition, perform a rank 1 update.

    >>> u = np.array([7., -2., 4., 3., 5.])
    >>> v = np.array([1., 3., -5.])
    >>> q_up, r_up = linalg.qr_update(q, r, u, v, False)
    >>> q_up
    array([[ 0.54073807,  0.18645997,  0.81707661, -0.02136616,  0.06902409],  # may vary (signs)
           [ 0.21629523, -0.63257324,  0.06567893,  0.34125904, -0.65749222],
           [ 0.05407381,  0.64757787, -0.12781284, -0.20031219, -0.72198188],
           [ 0.48666426, -0.30466718, -0.27487277, -0.77079214,  0.0256951 ],
           [ 0.64888568,  0.23001   , -0.4859845 ,  0.49883891,  0.20253783]])
    >>> r_up
    array([[ 18.49324201,  24.11691794, -44.98940746],  # may vary (signs)
           [  0.        ,  31.95894662, -27.40998201],
           [  0.        ,   0.        ,  -9.25451794],
           [  0.        ,   0.        ,   0.        ],
           [  0.        ,   0.        ,   0.        ]])

    The update is equivalent, but faster than the following.

    >>> a_up = a + np.outer(u, v)
    >>> q_direct, r_direct = linalg.qr(a_up)

    Check that we have equivalent results:

    >>> np.allclose(np.dot(q_up, r_up), a_up)
    True

    And the updated Q is still unitary:

    >>> np.allclose(np.dot(q_up.T, q_up), np.eye(5))
    True

    Updating economic (reduced, thin) decompositions is also possible:

    >>> qe, re = linalg.qr(a, mode='economic')
    >>> qe_up, re_up = linalg.qr_update(qe, re, u, v, False)
    >>> qe_up
    array([[ 0.54073807,  0.18645997,  0.81707661],  # may vary (signs)
           [ 0.21629523, -0.63257324,  0.06567893],
           [ 0.05407381,  0.64757787, -0.12781284],
           [ 0.48666426, -0.30466718, -0.27487277],
           [ 0.64888568,  0.23001   , -0.4859845 ]])
    >>> re_up
    array([[ 18.49324201,  24.11691794, -44.98940746],  # may vary (signs)
           [  0.        ,  31.95894662, -27.40998201],
           [  0.        ,   0.        ,  -9.25451794]])
    >>> np.allclose(np.dot(qe_up, re_up), a_up)
    True
    >>> np.allclose(np.dot(qe_up.T, qe_up), np.eye(3))
    True

    Similarly to the above, perform a rank 2 update.

    >>> u2 = np.array([[ 7., -1,],
    ...                [-2.,  4.],
    ...                [ 4.,  2.],
    ...                [ 3., -6.],
    ...                [ 5.,  3.]])
    >>> v2 = np.array([[ 1., 2.],
    ...                [ 3., 4.],
    ...                [-5., 2]])
    >>> q_up2, r_up2 = linalg.qr_update(q, r, u2, v2, False)
    >>> q_up2
    array([[-0.33626508, -0.03477253,  0.61956287, -0.64352987, -0.29618884],  # may vary (signs)
           [-0.50439762,  0.58319694, -0.43010077, -0.33395279,  0.33008064],
           [-0.21016568, -0.63123106,  0.0582249 , -0.13675572,  0.73163206],
           [ 0.12609941,  0.49694436,  0.64590024,  0.31191919,  0.47187344],
           [-0.75659643, -0.11517748,  0.10284903,  0.5986227 , -0.21299983]])
    >>> r_up2
    array([[-23.79075451, -41.1084062 ,  24.71548348],  # may vary (signs)
           [  0.        , -33.83931057,  11.02226551],
           [  0.        ,   0.        ,  48.91476811],
           [  0.        ,   0.        ,   0.        ],
           [  0.        ,   0.        ,   0.        ]])

    This update is also a valid qr decomposition of ``A + U V**T``.

    >>> a_up2 = a + np.dot(u2, v2.T)
    >>> np.allclose(a_up2, np.dot(q_up2, r_up2))
    True
    >>> np.allclose(np.dot(q_up2.T, q_up2), np.eye(5))
    True

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The ability to return an instance of a strict subclass of int is deprecated, and may be removed in a future version of Python.Module '_decomp_update' has already been imported. Re-initialisation is not supported.module compiled against API version 0x%x but this version of numpy is 0x%x . 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