scipy.sparse.linalg.norm¶
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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
ordparameter.- 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/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]:
\(||A||_F = [\sum_{i,j} abs(a_{i,j})^2]^{1/2}\)
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