Module « numpy.linalg »

Fonction pinv - module numpy.linalg

Signature de la fonction pinv

def pinv(a, rcond=1e-15, hermitian=False) 

Description

pinv.__doc__

    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.

    .. versionchanged:: 1.14
       Can now operate on stacks of matrices

    Parameters
    ----------
    a : (..., M, N) array_like
        Matrix or stack of matrices to be pseudo-inverted.
    rcond : (...) array_like of float
        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.
    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.

        .. versionadded:: 1.17.0

    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.pinv2 : Similar function in SciPy (SVD-based).
    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+``:

    >>> a = np.random.randn(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