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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`.

  The documentation is written assuming array arguments are of specified “core” shapes. However, array argument(s) of this function may have additional “batch” dimensions prepended to the core shape. In this case, the array is treated as a batch of lower-dimensional slices; see Batched Linear Operations for details.

  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.

  Added 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