norm — SciPy v1.15.3 Manual (original) (raw)
scipy.sparse.linalg.
scipy.sparse.linalg.norm(x, ord=None, axis=None)[source]#
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 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:
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 array.
This docstring is modified based on numpy.linalg.norm.numpy/numpy
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]:
\(||A||_F = [\sum_{i,j} abs(a_{i,j})^2]^{1/2}\)
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...