# scipy.linalg.qr_insert

scipy.linalg.qr_insert(Q, R, u, k, which='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.

  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.,  -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