scipy.linalg.qr¶
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scipy.linalg.
qr
(a, overwrite_a=False, lwork=None, mode='full', pivoting=False, check_finite=True)[source]¶ Compute QR decomposition of a matrix.
Calculate the decomposition
A = Q R
where Q is unitary/orthogonal and R upper triangular.Parameters: a : (M, N) array_like
Matrix to be decomposed
overwrite_a : bool, optional
Whether data in a is overwritten (may improve performance)
lwork : int, optional
Work array size, lwork >= a.shape[1]. If None or -1, an optimal size is computed.
mode : {‘full’, ‘r’, ‘economic’, ‘raw’}, optional
Determines what information is to be returned: either both Q and R (‘full’, default), only R (‘r’) or both Q and R but computed in economy-size (‘economic’, see Notes). The final option ‘raw’ (added in Scipy 0.11) makes the function return two matrices (Q, TAU) in the internal format used by LAPACK.
pivoting : bool, optional
Whether or not factorization should include pivoting for rank-revealing qr decomposition. If pivoting, compute the decomposition
A P = Q R
as above, but where P is chosen such that the diagonal of R is non-increasing.check_finite : bool, 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.
Returns: Q : float or complex ndarray
Of shape (M, M), or (M, K) for
mode='economic'
. Not returned ifmode='r'
.R : float or complex ndarray
Of shape (M, N), or (K, N) for
mode='economic'
.K = min(M, N)
.P : int ndarray
Of shape (N,) for
pivoting=True
. Not returned ifpivoting=False
.Raises: LinAlgError
Raised if decomposition fails
Notes
This is an interface to the LAPACK routines dgeqrf, zgeqrf, dorgqr, zungqr, dgeqp3, and zgeqp3.
If
mode=economic
, the shapes of Q and R are (M, K) and (K, N) instead of (M,M) and (M,N), withK=min(M,N)
.Examples
>>> from scipy import random, linalg, dot, diag, all, allclose >>> a = random.randn(9, 6)
>>> q, r = linalg.qr(a) >>> allclose(a, np.dot(q, r)) True >>> q.shape, r.shape ((9, 9), (9, 6))
>>> r2 = linalg.qr(a, mode='r') >>> allclose(r, r2) True
>>> q3, r3 = linalg.qr(a, mode='economic') >>> q3.shape, r3.shape ((9, 6), (6, 6))
>>> q4, r4, p4 = linalg.qr(a, pivoting=True) >>> d = abs(diag(r4)) >>> all(d[1:] <= d[:-1]) True >>> allclose(a[:, p4], dot(q4, r4)) True >>> q4.shape, r4.shape, p4.shape ((9, 9), (9, 6), (6,))
>>> q5, r5, p5 = linalg.qr(a, mode='economic', pivoting=True) >>> q5.shape, r5.shape, p5.shape ((9, 6), (6, 6), (6,))