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qr_update(Q, R, u, v, overwrite_qruv=False, check_finite=True)

If A = Q R is the QR factorization of A , return the QR factorization of A + u v**T for real A or A + u v**H for complex A .

Notes

This routine does not guarantee that the diagonal entries of :None:None:`R1` are real or positive.

versionadded

Parameters

Q : (M, M) or (M, N) array_like: Unitary/orthogonal matrix from the qr decomposition of A.
R : (M, N) or (N, N) array_like: Upper triangular matrix from the qr decomposition of A.
u : (M,) or (M, k) array_like: Left update vector
v : (N,) or (N, k) array_like: Right update vector
overwrite_qruv : bool, optional: If True, consume Q, R, u, and v, if possible, while performing the update, otherwise make copies as necessary. 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

Rank-k 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 q, r decomposition, perform a rank 1 update.

>>> u = np.array([7., -2., 4., 3., 5.])
... v = np.array([1., 3., -5.])
... q_up, r_up = linalg.qr_update(q, r, u, v, False)
... q_up
array([[ 0.54073807,  0.18645997,  0.81707661, -0.02136616,  0.06902409],  # may vary (signs)
       [ 0.21629523, -0.63257324,  0.06567893,  0.34125904, -0.65749222],
       [ 0.05407381,  0.64757787, -0.12781284, -0.20031219, -0.72198188],
       [ 0.48666426, -0.30466718, -0.27487277, -0.77079214,  0.0256951 ],
       [ 0.64888568,  0.23001   , -0.4859845 ,  0.49883891,  0.20253783]])

>>> r_up
array([[ 18.49324201,  24.11691794, -44.98940746],  # may vary (signs)
       [  0.        ,  31.95894662, -27.40998201],
       [  0.        ,   0.        ,  -9.25451794],
       [  0.        ,   0.        ,   0.        ],
       [  0.        ,   0.        ,   0.        ]])

The update is equivalent, but faster than the following.

>>> a_up = a + np.outer(u, v)
... q_direct, r_direct = linalg.qr(a_up)

Check that we have equivalent results:

>>> np.allclose(np.dot(q_up, r_up), a_up)
True

And the updated Q is still unitary:

>>> np.allclose(np.dot(q_up.T, q_up), np.eye(5))
True

Updating economic (reduced, thin) decompositions is also possible:

>>> qe, re = linalg.qr(a, mode='economic')
... qe_up, re_up = linalg.qr_update(qe, re, u, v, False)
... qe_up
array([[ 0.54073807,  0.18645997,  0.81707661],  # may vary (signs)
       [ 0.21629523, -0.63257324,  0.06567893],
       [ 0.05407381,  0.64757787, -0.12781284],
       [ 0.48666426, -0.30466718, -0.27487277],
       [ 0.64888568,  0.23001   , -0.4859845 ]])

>>> re_up
array([[ 18.49324201,  24.11691794, -44.98940746],  # may vary (signs)
       [  0.        ,  31.95894662, -27.40998201],
       [  0.        ,   0.        ,  -9.25451794]])

>>> np.allclose(np.dot(qe_up, re_up), a_up)
True

>>> np.allclose(np.dot(qe_up.T, qe_up), np.eye(3))
True

Similarly to the above, perform a rank 2 update.

>>> u2 = np.array([[ 7., -1,],
...                [-2.,  4.],
...                [ 4.,  2.],
...                [ 3., -6.],
...                [ 5.,  3.]])
... v2 = np.array([[ 1., 2.],
...                [ 3., 4.],
...                [-5., 2]])
... q_up2, r_up2 = linalg.qr_update(q, r, u2, v2, False)
... q_up2
array([[-0.33626508, -0.03477253,  0.61956287, -0.64352987, -0.29618884],  # may vary (signs)
       [-0.50439762,  0.58319694, -0.43010077, -0.33395279,  0.33008064],
       [-0.21016568, -0.63123106,  0.0582249 , -0.13675572,  0.73163206],
       [ 0.12609941,  0.49694436,  0.64590024,  0.31191919,  0.47187344],
       [-0.75659643, -0.11517748,  0.10284903,  0.5986227 , -0.21299983]])

>>> r_up2
array([[-23.79075451, -41.1084062 ,  24.71548348],  # may vary (signs)
       [  0.        , -33.83931057,  11.02226551],
       [  0.        ,   0.        ,  48.91476811],
       [  0.        ,   0.        ,   0.        ],
       [  0.        ,   0.        ,   0.        ]])

This update is also a valid qr decomposition of A + U V**T .

>>> a_up2 = a + np.dot(u2, v2.T)
... np.allclose(a_up2, np.dot(q_up2, r_up2))
True

>>> np.allclose(np.dot(q_up2.T, q_up2), np.eye(5))
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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