scipy.sparse.linalg.norm

scipy.sparse.linalg.norm(x, ord=None, axis=None)

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:

xa 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:

nfloat 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/main/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	Spectral norm (the largest singular value)
-2	Not implemented
other	Not implemented

The Frobenius norm is given by [1]:

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

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

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

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