qr_multiply#

rlinalg.qr_multiply(a: numpy.typing.ArrayLike, c: numpy.typing.ArrayLike, mode: Literal['left', 'right'] = 'right', tol=1e-07, check_finite=True, overwrite_a=False, overwrite_c=False) → QRMultiplyResult#

Calculate the QR decomposition and multiply Q with a matrix.

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

Parameters:

  • a ((M, N), array_like) – Input array

  • c (array_like) – 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].

  • overwrite_a (bool, optional) – Whether data in a is overwritten (may improve performance)

  • 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.

  • tol (float) – The absolute tolerance to which each column norm is required. An column is considered negligible when its norm falls under this value.

Returns:

  • CQ (ndarray) – 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.

  • rank (int) – The rank of the matrix.

Raises:

ValueError – When parameters are not compatible.

Notes

This is an interface to the R-modified LINPACK routine dqrdc2 and dqrsl.

See also

None

The SciPy implementation based on LAPACK.

Examples

>>> import numpy as np
>>> from rlinalg import qr, qr_multiply
>>> A = np.array([[1, 3, 3], [2, 3, 2], [2, 3, 3], [1, 3, 2]])
>>> qc, r1, piv1, rank = qr_multiply(A, 2*np.eye(4))
>>> qc
array([[-0.63245553,  1.26491106, -1.        ],
       [-1.26491106, -0.63245553,  1.        ],
       [-1.26491106, -0.63245553, -1.        ],
       [-0.63245553,  1.26491106,  1.        ]])
>>> r1
array([[-3.16227766, -5.69209979, -4.74341649],
       [ 0.        ,  1.8973666 ,  1.58113883],
       [ 0.        ,  0.        , -1.        ]])
>>> piv1
array([0, 1, 2], dtype=int32)
>>> q2, r2, piv2, rank = qr(A, mode='economic')
>>> np.allclose(2*q2 - qc, np.zeros((4, 3)))
True
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