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Module « scipy.linalg »

Fonction qr - module scipy.linalg

Signature de la fonction qr

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

Description

help(scipy.linalg.qr)

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 if
    `overwrite_a` is set to True by reusing the existing input data
    structure rather than creating a new one.)
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. Equivalently, albeit less efficiently,
    an explicit P matrix may be formed explicitly by permuting the rows or columns
    (depending on the side of the equation on which it is to be used) of
    an identity matrix. See Examples.
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
    if ``mode='r'``. Replaced by tuple ``(Q, TAU)`` if ``mode='raw'``.
R : float or complex ndarray
    Of shape (M, N), or (K, N) for ``mode in ['economic', 'raw']``.
    ``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
--------
>>> import numpy as np
>>> from scipy import linalg
>>> rng = np.random.default_rng()
>>> a = rng.standard_normal((9, 6))

>>> q, r = linalg.qr(a)
>>> np.allclose(a, np.dot(q, r))
True
>>> q.shape, r.shape
((9, 9), (9, 6))

>>> r2 = linalg.qr(a, mode='r')
>>> np.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 = np.abs(np.diag(r4))
>>> np.all(d[1:] <= d[:-1])
True
>>> np.allclose(a[:, p4], np.dot(q4, r4))
True
>>> P = np.eye(p4.size)[p4]
>>> np.allclose(a, np.dot(q4, r4) @ P)
True
>>> np.allclose(a @ P.T, np.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,))
>>> P = np.eye(6)[:, p5]
>>> np.allclose(a @ P, np.dot(q5, r5))
True

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