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.

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 approriately 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.

New in version 0.16.0.

References

[R115]

Golub, G. H. & Van Loan, C. F. Matrix Computations, 3rd Ed. (Johns Hopkins University Press, 1996).

[R116]

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).

[R117]

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.,  -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