qr_delete#
- scipy.linalg.qr_delete(Q, R, k, int p=1, which='row', overwrite_qr=False, check_finite=True)#
- QR downdate on row or column deletions - If - A = Q Ris the QR factorization of- A, return the QR factorization of- Awhere- prows or columns have been removed starting at row or column- k.- 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 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 appropriately 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 - 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., -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