scipy.linalg.qr

scipy.linalg.qr(a, overwrite_a=False, lwork=None, mode='full', pivoting=False, check_finite=True)

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 : boolean, 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 if mode='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 if pivoting=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), with K=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,))