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Module « scipy.sparse.linalg »

Fonction norm - module scipy.sparse.linalg

Signature de la fonction norm

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

Description

help(scipy.sparse.linalg.norm)

Norm of a sparse matrix

This function is able to return one of seven different matrix norms,
depending on the value of the ``ord`` parameter.

Parameters
----------
x : a sparse array
    Input sparse array.
ord : {non-zero int, inf, -inf, 'fro'}, optional
    Order of the norm (see table under ``Notes``). inf means numpy's
    `inf` object.
axis : {int, 2-tuple of ints, None}, 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.

Returns
-------
n : float or ndarray

Notes
-----
Some of the ord are not implemented because some associated functions like,
_multi_svd_norm, are not yet available for sparse array.

This docstring is modified based on numpy.linalg.norm.
https://github.com/numpy/numpy/blob/main/numpy/linalg/linalg.py

The following norms can be calculated:

=====  ============================
ord    norm for sparse arrays
=====  ============================
None   Frobenius norm
'fro'  Frobenius norm
inf    max(sum(abs(x), axis=1))
-inf   min(sum(abs(x), axis=1))
0      abs(x).sum(axis=axis)
1      max(sum(abs(x), axis=0))
-1     min(sum(abs(x), axis=0))
2      Spectral norm (the largest singular value)
-2     Not implemented
other  Not implemented
=====  ============================

The Frobenius norm is given by [1]_:

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

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

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

>>> b = csr_array(b)
>>> norm(b)
7.745966692414834
>>> norm(b, 'fro')
7.745966692414834
>>> norm(b, np.inf)
9
>>> norm(b, -np.inf)
2
>>> norm(b, 1)
7
>>> norm(b, -1)
6

The matrix 2-norm or the spectral norm is the largest singular
value, computed approximately and with limitations.

>>> b = diags_array([-1, 1], [0, 1], shape=(9, 10))
>>> norm(b, 2)
1.9753...

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