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# scipy.linalg.qr_update

scipy.linalg.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.

  New in version 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