Module « scipy.linalg »

Fonction qr_update - module scipy.linalg

Signature de la fonction qr_update

Description

qr_update.__doc__

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
    --------
    >>> 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