Module « scipy.sparse.linalg »

Fonction norm - module scipy.sparse.linalg

Signature de la fonction norm

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

Description

norm.__doc__

    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 matrix
        Input sparse matrix.
    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 matrix.

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

    The following norms can be calculated:

    =====  ============================
    ord    norm for sparse matrices
    =====  ============================
    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      Not implemented
    -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 *
    >>> 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_matrix(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