Module « numpy.linalg »

Fonction norm - module numpy.linalg

Signature de la fonction norm

def norm(x, ord=None, axis=None, keepdims=False) 

Description

norm.__doc__

    Matrix or vector norm.

    This function is able to return one of eight different matrix norms,
    or one of an infinite number of vector norms (described below), depending
    on the value of the ``ord`` parameter.

    Parameters
    ----------
    x : array_like
        Input array.  If `axis` is None, `x` must be 1-D or 2-D, unless `ord`
        is None. If both `axis` and `ord` are None, the 2-norm of
        ``x.ravel`` will be returned.
    ord : {non-zero int, inf, -inf, 'fro', 'nuc'}, optional
        Order of the norm (see table under ``Notes``). inf means numpy's
        `inf` object. The default is None.
    axis : {None, int, 2-tuple of ints}, optional.
        If `axis` is an integer, it specifies the axis of `x` along which to
        compute the vector norms.  If `axis` is a 2-tuple, it specifies the
        axes that hold 2-D matrices, and the matrix norms of these matrices
        are computed.  If `axis` is None then either a vector norm (when `x`
        is 1-D) or a matrix norm (when `x` is 2-D) is returned. The default
        is None.

        .. versionadded:: 1.8.0

    keepdims : bool, optional
        If this is set to True, the axes which are normed over are left in the
        result as dimensions with size one.  With this option the result will
        broadcast correctly against the original `x`.

        .. versionadded:: 1.10.0

    Returns
    -------
    n : float or ndarray
        Norm of the matrix or vector(s).

    See Also
    --------
    scipy.linalg.norm : Similar function in SciPy.

    Notes
    -----
    For values of ``ord < 1``, the result is, strictly speaking, not a
    mathematical 'norm', but it may still be useful for various numerical
    purposes.

    The following norms can be calculated:

    =====  ============================  ==========================
    ord    norm for matrices             norm for vectors
    =====  ============================  ==========================
    None   Frobenius norm                2-norm
    'fro'  Frobenius norm                --
    'nuc'  nuclear norm                  --
    inf    max(sum(abs(x), axis=1))      max(abs(x))
    -inf   min(sum(abs(x), axis=1))      min(abs(x))
    0      --                            sum(x != 0)
    1      max(sum(abs(x), axis=0))      as below
    -1     min(sum(abs(x), axis=0))      as below
    2      2-norm (largest sing. value)  as below
    -2     smallest singular value       as below
    other  --                            sum(abs(x)**ord)**(1./ord)
    =====  ============================  ==========================

    The Frobenius norm is given by [1]_:

        :math:`||A||_F = [\sum_{i,j} abs(a_{i,j})^2]^{1/2}`

    The nuclear norm is the sum of the singular values.

    Both the Frobenius and nuclear norm orders are only defined for
    matrices and raise a ValueError when ``x.ndim != 2``.

    References
    ----------
    .. [1] G. H. Golub and C. F. Van Loan, *Matrix Computations*,
           Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15

    Examples
    --------
    >>> from numpy import linalg as LA
    >>> a = np.arange(9) - 4
    >>> a
    array([-4, -3, -2, ...,  2,  3,  4])
    >>> b = a.reshape((3, 3))
    >>> b
    array([[-4, -3, -2],
           [-1,  0,  1],
           [ 2,  3,  4]])

    >>> LA.norm(a)
    7.745966692414834
    >>> LA.norm(b)
    7.745966692414834
    >>> LA.norm(b, 'fro')
    7.745966692414834
    >>> LA.norm(a, np.inf)
    4.0
    >>> LA.norm(b, np.inf)
    9.0
    >>> LA.norm(a, -np.inf)
    0.0
    >>> LA.norm(b, -np.inf)
    2.0

    >>> LA.norm(a, 1)
    20.0
    >>> LA.norm(b, 1)
    7.0
    >>> LA.norm(a, -1)
    -4.6566128774142013e-010
    >>> LA.norm(b, -1)
    6.0
    >>> LA.norm(a, 2)
    7.745966692414834
    >>> LA.norm(b, 2)
    7.3484692283495345

    >>> LA.norm(a, -2)
    0.0
    >>> LA.norm(b, -2)
    1.8570331885190563e-016 # may vary
    >>> LA.norm(a, 3)
    5.8480354764257312 # may vary
    >>> LA.norm(a, -3)
    0.0

    Using the `axis` argument to compute vector norms:

    >>> c = np.array([[ 1, 2, 3],
    ...               [-1, 1, 4]])
    >>> LA.norm(c, axis=0)
    array([ 1.41421356,  2.23606798,  5.        ])
    >>> LA.norm(c, axis=1)
    array([ 3.74165739,  4.24264069])
    >>> LA.norm(c, ord=1, axis=1)
    array([ 6.,  6.])

    Using the `axis` argument to compute matrix norms:

    >>> m = np.arange(8).reshape(2,2,2)
    >>> LA.norm(m, axis=(1,2))
    array([  3.74165739,  11.22497216])
    >>> LA.norm(m[0, :, :]), LA.norm(m[1, :, :])
    (3.7416573867739413, 11.224972160321824)