Module « scipy.linalg »

Fonction qr_insert - module scipy.linalg

Signature de la fonction qr_insert

Description

qr_insert.__doc__

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
    --------
    >>> 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