scipy.linalg.qr_insert

scipy.linalg.qr_insert(Q, R, u, k, which='row', rcond=None, overwrite_qru=False, check_finite=True)

QR update on row or column insertions

If A = Q R is the QR factorization of A, return the QR factorization of A where rows or columns have been inserted starting at row or column k.

Parameters:

Q : (M, M) array_like

Unitary/orthogonal matrix from the QR decomposition of A.

R : (M, N) array_like

Upper triangular matrix from the QR decomposition of A.

u : (N,), (p, N), (M,), or (M, p) array_like

Rows or columns to insert

k : int

Index before which u is to be inserted.

which: {‘row’, ‘col’}, optional

Determines if rows or columns will be inserted, defaults to ‘row’

rcond : float

Lower bound on the reciprocal condition number of Q augmented with u/||u|| Only used when updating economic mode (thin, (M,N) (N,N)) decompositions. If None, machine precision is used. Defaults to None.

overwrite_qru : bool, optional

If True, consume Q, R, and u, if possible, while performing the update, otherwise make copies as necessary. Defaults to False.

check_finite : bool, optional

Whether to check that the input matrices contain 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

Raises:

LinAlgError :

If updating a (M,N) (N,N) factorization and the reciprocal condition number of Q augmented with u/||u|| is smaller than rcond.

See also

qr, qr_multiply, qr_delete, qr_update

Notes

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

New in version 0.16.0.

References

[R146]

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

[R147]

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).

[R148]

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.,  -7.,   4.],
...               [  7.,   8.,  -6.]])
>>> q, r = linalg.qr(a)

Given this QR decomposition, update q and r when 2 rows are inserted.

>>> u = np.array([[  6.,  -9.,  -3.],
...               [ -3.,  10.,   1.]])
>>> q1, r1 = linalg.qr_insert(q, r, u, 2, 'row')
>>> q1
array([[-0.25445668,  0.02246245,  0.18146236, -0.72798806,  0.60979671],  # may vary (signs)
       [-0.50891336,  0.23226178, -0.82836478, -0.02837033, -0.00828114],
       [-0.50891336,  0.35715302,  0.38937158,  0.58110733,  0.35235345],
       [ 0.25445668, -0.52202743, -0.32165498,  0.36263239,  0.65404509],
       [-0.59373225, -0.73856549,  0.16065817, -0.0063658 , -0.27595554]])
>>> r1
array([[-11.78982612,   6.44623587,   3.81685018],  # may vary (signs)
       [  0.        , -16.01393278,   3.72202865],
       [  0.        ,   0.        ,  -6.13010256],
       [  0.        ,   0.        ,   0.        ],
       [  0.        ,   0.        ,   0.        ]])

The update is equivalent, but faster than the following.

>>> a1 = np.insert(a, 2, u, 0)
>>> a1
array([[  3.,  -2.,  -2.],
       [  6.,  -7.,   4.],
       [  6.,  -9.,  -3.],
       [ -3.,  10.,   1.],
       [  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.,  -7.,   4.],
       [  6.,  -9.,  -3.],
       [ -3.,  10.,   1.],
       [  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(5))
True