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Module « scipy.linalg »

Fonction null_space - module scipy.linalg

Signature de la fonction null_space

def null_space(A, rcond=None, *, overwrite_a=False, check_finite=True, lapack_driver='gesdd') 

Description

help(scipy.linalg.null_space)

Construct an orthonormal basis for the null space of A using SVD

Parameters
----------
A : (M, N) array_like
    Input array
rcond : float, optional
    Relative condition number. Singular values ``s`` smaller than
    ``rcond * max(s)`` are considered zero.
    Default: floating point eps * max(M,N).
overwrite_a : bool, optional
    Whether to overwrite `a`; may improve performance.
    Default is False.
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.
lapack_driver : {'gesdd', 'gesvd'}, optional
    Whether to use the more efficient divide-and-conquer approach
    (``'gesdd'``) or general rectangular approach (``'gesvd'``)
    to compute the SVD. MATLAB and Octave use the ``'gesvd'`` approach.
    Default is ``'gesdd'``.

Returns
-------
Z : (N, K) ndarray
    Orthonormal basis for the null space of A.
    K = dimension of effective null space, as determined by rcond

See Also
--------
svd : Singular value decomposition of a matrix
orth : Matrix range

Examples
--------
1-D null space:

>>> import numpy as np
>>> from scipy.linalg import null_space
>>> A = np.array([[1, 1], [1, 1]])
>>> ns = null_space(A)
>>> ns * np.copysign(1, ns[0,0])  # Remove the sign ambiguity of the vector
array([[ 0.70710678],
       [-0.70710678]])

2-D null space:

>>> from numpy.random import default_rng
>>> rng = default_rng()
>>> B = rng.random((3, 5))
>>> Z = null_space(B)
>>> Z.shape
(5, 2)
>>> np.allclose(B.dot(Z), 0)
True

The basis vectors are orthonormal (up to rounding error):

>>> Z.T.dot(Z)
array([[  1.00000000e+00,   6.92087741e-17],
       [  6.92087741e-17,   1.00000000e+00]])

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