qr_insert(Q, R, u, k, which=u'row', rcond=None, overwrite_qru=False, check_finite=True)
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
.
This routine does not guarantee that the diagonal entries of R1
are positive.
Unitary/orthogonal matrix from the QR decomposition of A.
Upper triangular matrix from the QR decomposition of A.
Rows or columns to insert
Index before which u is to be inserted.
Determines if rows or columns will be inserted, defaults to 'row'
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.
If True, consume Q, R, and u, if possible, while performing the update, otherwise make copies as necessary. Defaults to False.
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.
If updating a (M,N) (N,N) factorization and the reciprocal condition number of Q augmented with u/
<SubstitutionRef: |value: '||u||' |>is smaller than rcond.
QR update on row or column insertions
>>> 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)) TrueSee :
The following pages refer to to this document either explicitly or contain code examples using this.
scipy.linalg._decomp_update.qr_insert
scipy.linalg._decomp_update.qr_delete
scipy.linalg._decomp_update.qr_update
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