scipy.linalg.

qr_update#

scipy.linalg.qr_update(Q, R, u, v, overwrite_qruv=False, check_finite=True)#

Rank-k QR update

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.

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 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_qruvbool, optional: If True, consume Q, R, u, and v, if possible, while performing the update, otherwise make copies as necessary. 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_delete, qr_insert

Notes

This routine does not guarantee that the diagonal entries of R1 are real or 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 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