`scipy.sparse.linalg._norm:norm`

source: /scipy/sparse/linalg/_norm.py:20

Signature

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

Summary

Norm of a sparse matrix.

Extended Summary

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: The selected norm of x.

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]:

$∣∣ A ∣ ∣_{F} = [\sum_{i, j} ab s (a_{i, j})^{2}]^{1/2}$

Examples

from scipy.sparse import csr_array, diags_array
import numpy as np
from scipy.sparse.linalg import norm
a = np.arange(9) - 4

✓

✗

array([-4, -3, -2, -1,  0,  1,  2,  3,  4])

b = a.reshape((3, 3))

✓

✗

array([[-4, -3, -2],
       [-1,  0,  1],
       [ 2,  3,  4]])

b = csr_array(b)

✓

norm(b)
norm(b, 'fro')
norm(b, np.inf)
norm(b, -np.inf)
norm(b, 1)
norm(b, -1)

✗

np.float64(6.0)

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

b = diags_array([-1, 1], offsets=[0, 1], shape=(9, 10))

✓

norm(b, 2)

✗

np.float64(1.9753766811902758)

Aliases

scipy.sparse.linalg.norm