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 ofA
, return the QR factorization ofA
wherep
rows or columns have been removed starting at row or columnk
.- 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 approriately 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
Notes
This routine does not guarantee that the diagonal entries of
R1
are positive.New 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
>>> 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