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Module « numpy.linalg »

Fonction inv - module numpy.linalg

Signature de la fonction inv

def inv(a) 

Description

help(numpy.linalg.inv)

Compute the inverse of a matrix.

Given a square matrix `a`, return the matrix `ainv` satisfying
``a @ ainv = ainv @ a = eye(a.shape[0])``.

Parameters
----------
a : (..., M, M) array_like
    Matrix to be inverted.

Returns
-------
ainv : (..., M, M) ndarray or matrix
    Inverse of the matrix `a`.

Raises
------
LinAlgError
    If `a` is not square or inversion fails.

See Also
--------
scipy.linalg.inv : Similar function in SciPy.
numpy.linalg.cond : Compute the condition number of a matrix.
numpy.linalg.svd : Compute the singular value decomposition of a matrix.

Notes
-----
Broadcasting rules apply, see the `numpy.linalg` documentation for
details.

If `a` is detected to be singular, a `LinAlgError` is raised. If `a` is
ill-conditioned, a `LinAlgError` may or may not be raised, and results may
be inaccurate due to floating-point errors.

References
----------
.. [1] Wikipedia, "Condition number",
       https://en.wikipedia.org/wiki/Condition_number

Examples
--------
>>> import numpy as np
>>> from numpy.linalg import inv
>>> a = np.array([[1., 2.], [3., 4.]])
>>> ainv = inv(a)
>>> np.allclose(a @ ainv, np.eye(2))
True
>>> np.allclose(ainv @ a, np.eye(2))
True

If a is a matrix object, then the return value is a matrix as well:

>>> ainv = inv(np.matrix(a))
>>> ainv
matrix([[-2. ,  1. ],
        [ 1.5, -0.5]])

Inverses of several matrices can be computed at once:

>>> a = np.array([[[1., 2.], [3., 4.]], [[1, 3], [3, 5]]])
>>> inv(a)
array([[[-2.  ,  1.  ],
        [ 1.5 , -0.5 ]],
       [[-1.25,  0.75],
        [ 0.75, -0.25]]])

If a matrix is close to singular, the computed inverse may not satisfy
``a @ ainv = ainv @ a = eye(a.shape[0])`` even if a `LinAlgError`
is not raised:

>>> a = np.array([[2,4,6],[2,0,2],[6,8,14]])
>>> inv(a)  # No errors raised
array([[-1.12589991e+15, -5.62949953e+14,  5.62949953e+14],
   [-1.12589991e+15, -5.62949953e+14,  5.62949953e+14],
   [ 1.12589991e+15,  5.62949953e+14, -5.62949953e+14]])
>>> a @ inv(a)
array([[ 0.   , -0.5  ,  0.   ],  # may vary
       [-0.5  ,  0.625,  0.25 ],
       [ 0.   ,  0.   ,  1.   ]])

To detect ill-conditioned matrices, you can use `numpy.linalg.cond` to
compute its *condition number* [1]_. The larger the condition number, the
more ill-conditioned the matrix is. As a rule of thumb, if the condition
number ``cond(a) = 10**k``, then you may lose up to ``k`` digits of
accuracy on top of what would be lost to the numerical method due to loss
of precision from arithmetic methods.

>>> from numpy.linalg import cond
>>> cond(a)
np.float64(8.659885634118668e+17)  # may vary

It is also possible to detect ill-conditioning by inspecting the matrix's
singular values directly. The ratio between the largest and the smallest
singular value is the condition number:

>>> from numpy.linalg import svd
>>> sigma = svd(a, compute_uv=False)  # Do not compute singular vectors
>>> sigma.max()/sigma.min()
8.659885634118668e+17  # may vary

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