scipy.linalg.

qr_delete#

scipy.linalg.qr_delete(Q, R, k, int p=1, which=u'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.

kint

Index of the first row or column to delete.

pint, 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_qrbool, optional

If True, consume Q and R, overwriting their contents with their downdated versions, and returning appropriately sized views. Defaults to False.

check_finitebool, 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:
Q1ndarray

Updated unitary/orthogonal factor

R1ndarray

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
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