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Module « numpy.linalg »

Fonction norm - module numpy.linalg

Signature de la fonction norm

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

Description

help(numpy.linalg.norm)

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 : {int, float, inf, -inf, 'fro', 'nuc'}, optional
    Order of the norm (see table under ``Notes`` for what values are
    supported for matrices and vectors respectively). 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.

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`.

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
--------

>>> import numpy as np
>>> 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)

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