scipy.linalg.qr_multiply(a, c, mode='right', pivoting=False, conjugate=False, overwrite_a=False, overwrite_c=False)[source]#

Calculate the QR decomposition and multiply Q with a matrix.

Calculate the decomposition A = Q R where Q is unitary/orthogonal and R upper triangular. Multiply Q with a vector or a matrix c.

a(M, N), array_like

Input array


Input array to be multiplied by q.

mode{‘left’, ‘right’}, optional

Q @ c is returned if mode is ‘left’, c @ Q is returned if mode is ‘right’. The shape of c must be appropriate for the matrix multiplications, if mode is ‘left’, min(a.shape) == c.shape[0], if mode is ‘right’, a.shape[0] == c.shape[1].

pivotingbool, optional

Whether or not factorization should include pivoting for rank-revealing qr decomposition, see the documentation of qr.

conjugatebool, optional

Whether Q should be complex-conjugated. This might be faster than explicit conjugation.

overwrite_abool, optional

Whether data in a is overwritten (may improve performance)

overwrite_cbool, optional

Whether data in c is overwritten (may improve performance). If this is used, c must be big enough to keep the result, i.e. c.shape[0] = a.shape[0] if mode is ‘left’.


The product of Q and c.

R(K, N), ndarray

R array of the resulting QR factorization where K = min(M, N).

P(N,) ndarray

Integer pivot array. Only returned when pivoting=True.


Raised if QR decomposition fails.


This is an interface to the LAPACK routines ?GEQRF, ?ORMQR, ?UNMQR, and ?GEQP3.

Added in version 0.11.0.


>>> import numpy as np
>>> from scipy.linalg import qr_multiply, qr
>>> A = np.array([[1, 3, 3], [2, 3, 2], [2, 3, 3], [1, 3, 2]])
>>> qc, r1, piv1 = qr_multiply(A, 2*np.eye(4), pivoting=1)
>>> qc
array([[-1.,  1., -1.],
       [-1., -1.,  1.],
       [-1., -1., -1.],
       [-1.,  1.,  1.]])
>>> r1
array([[-6., -3., -5.            ],
       [ 0., -1., -1.11022302e-16],
       [ 0.,  0., -1.            ]])
>>> piv1
array([1, 0, 2], dtype=int32)
>>> q2, r2, piv2 = qr(A, mode='economic', pivoting=1)
>>> np.allclose(2*q2 - qc, np.zeros((4, 3)))