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 Ris the QR factorization of- A, return the QR factorization of- Awhere 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 
- kint
- Index before which u is to be inserted. 
- which: {‘row’, ‘col’}, optional
- Determines if rows or columns will be inserted, defaults to ‘row’ 
- rcondfloat
- Lower bound on the reciprocal condition number of - Qaugmented 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_qrubool, optional
- If True, consume Q, R, and u, if possible, while performing the update, otherwise make copies as necessary. Defaults to False. 
- check_finitebool, 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:
- Q1ndarray
- Updated unitary/orthogonal factor 
- R1ndarray
- 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 - Notes - This routine does not guarantee that the diagonal entries of - R1are 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., -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