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scipy.linalg.

# qr_delete

scipy.linalg.qr_delete(Q, R, k, int p=1, which='row', overwrite_qr=False, check_finite=True)

  QR downdate on row or column deletions

  If `A = Q R` is the QR factorization of `A`, return the QR factorization of `A` where `p` rows or columns have been removed 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) or (M, N) array_like

      Unitary/orthogonal matrix from QR decomposition.

    R(M, N) or (N, N) array_like

      Upper triangular matrix from QR decomposition.

    k int

      Index of the first row or column to delete.

    p int, optional

      Number of rows or columns to delete, defaults to 1.

    which: {‘row’, ‘col’}, optional

      Determines if rows or columns will be deleted, defaults to ‘row’

    overwrite_qr bool, optional

      If True, consume Q and R, overwriting their contents with their downdated versions, and returning appropriately sized views. 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_insert`, `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.,  -9.,  -3.],
      ...               [ -3.,  10.,   1.],
      ...               [  6.,  -7.,   4.],
      ...               [  7.,   8.,  -6.]])
      >>> q, r = linalg.qr(a)

  Given this QR decomposition, update q and r when 2 rows are removed.

      >>> q1, r1 = linalg.qr_delete(q, r, 2, 2, 'row', False)
      >>> q1
      array([[ 0.30942637,  0.15347579,  0.93845645],  # may vary (signs)
             [ 0.61885275,  0.71680171, -0.32127338],
             [ 0.72199487, -0.68017681, -0.12681844]])
      >>> r1
      array([[  9.69535971,  -0.4125685 ,  -6.80738023],  # may vary (signs)
             [  0.        , -12.19958144,   1.62370412],
             [  0.        ,   0.        ,  -0.15218213]])

  The update is equivalent, but faster than the following.

      >>> a1 = np.delete(a, slice(2,4), 0)
      >>> a1
      array([[ 3., -2., -2.],
             [ 6., -9., -3.],
             [ 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., -9., -3.],
             [ 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(3))
      True