scipy.linalg.qr

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

Compute QR decomposition of a matrix.

Calculate the decomposition :lm:`A = Q R` where Q is unitary/orthogonal and R upper triangular.

Parameters :

a : array, shape (M, N)

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’}

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).

pivoting : bool, optional

Whether or not factorization should include pivoting for rank-revealing qr decomposition. If pivoting, compute the decomposition :lm:`A P = Q R` as above, but where P is chosen such that the diagonal of R is non-increasing.

Returns :

Q : double or complex ndarray

Of shape (M, M), or (M, K) for mode='economic'. Not returned if mode='r'.

R : double or complex ndarray

Of shape (M, N), or (K, N) for mode='economic'. K = min(M, N).

P : integer 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, 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,))