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

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 .

Notes

This routine does not guarantee that the diagonal entries of R1 are positive.

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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

QR downdate on row or column deletions

See Also

qr
qr_insert
qr_multiply
qr_update

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
See :

Back References

The following pages refer to to this document either explicitly or contain code examples using this.

scipy.linalg._decomp_update.qr_insert scipy.linalg._decomp_update.qr_delete scipy.linalg._decomp_update.qr_update

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