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Module « numpy.linalg »

Fonction pinv - module numpy.linalg

Signature de la fonction pinv

def pinv(a, rcond=None, hermitian=False, *, rtol=<no value>) 

Description

help(numpy.linalg.pinv)

Compute the (Moore-Penrose) pseudo-inverse of a matrix.

Calculate the generalized inverse of a matrix using its
singular-value decomposition (SVD) and including all
*large* singular values.

Parameters
----------
a : (..., M, N) array_like
    Matrix or stack of matrices to be pseudo-inverted.
rcond : (...) array_like of float, optional
    Cutoff for small singular values.
    Singular values less than or equal to
    ``rcond * largest_singular_value`` are set to zero.
    Broadcasts against the stack of matrices. Default: ``1e-15``.
hermitian : bool, optional
    If True, `a` is assumed to be Hermitian (symmetric if real-valued),
    enabling a more efficient method for finding singular values.
    Defaults to False.
rtol : (...) array_like of float, optional
    Same as `rcond`, but it's an Array API compatible parameter name.
    Only `rcond` or `rtol` can be set at a time. If none of them are
    provided then NumPy's ``1e-15`` default is used. If ``rtol=None``
    is passed then the API standard default is used.

    .. versionadded:: 2.0.0

Returns
-------
B : (..., N, M) ndarray
    The pseudo-inverse of `a`. If `a` is a `matrix` instance, then so
    is `B`.

Raises
------
LinAlgError
    If the SVD computation does not converge.

See Also
--------
scipy.linalg.pinv : Similar function in SciPy.
scipy.linalg.pinvh : Compute the (Moore-Penrose) pseudo-inverse of a
                     Hermitian matrix.

Notes
-----
The pseudo-inverse of a matrix A, denoted :math:`A^+`, is
defined as: "the matrix that 'solves' [the least-squares problem]
:math:`Ax = b`," i.e., if :math:`\bar{x}` is said solution, then
:math:`A^+` is that matrix such that :math:`\bar{x} = A^+b`.

It can be shown that if :math:`Q_1 \Sigma Q_2^T = A` is the singular
value decomposition of A, then
:math:`A^+ = Q_2 \Sigma^+ Q_1^T`, where :math:`Q_{1,2}` are
orthogonal matrices, :math:`\Sigma` is a diagonal matrix consisting
of A's so-called singular values, (followed, typically, by
zeros), and then :math:`\Sigma^+` is simply the diagonal matrix
consisting of the reciprocals of A's singular values
(again, followed by zeros). [1]_

References
----------
.. [1] G. Strang, *Linear Algebra and Its Applications*, 2nd Ed., Orlando,
       FL, Academic Press, Inc., 1980, pp. 139-142.

Examples
--------
The following example checks that ``a * a+ * a == a`` and
``a+ * a * a+ == a+``:

>>> import numpy as np
>>> rng = np.random.default_rng()
>>> a = rng.normal(size=(9, 6))
>>> B = np.linalg.pinv(a)
>>> np.allclose(a, np.dot(a, np.dot(B, a)))
True
>>> np.allclose(B, np.dot(B, np.dot(a, B)))
True

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