SciPy

scipy.linalg.norm

scipy.linalg.norm(a, ord=None, axis=None, keepdims=False)[source]

Matrix or vector norm.

This function is able to return one of seven different matrix norms, or one of an infinite number of vector norms (described below), depending on the value of the ord parameter.

Parameters:

a : (M,) or (M, N) array_like

Input array. If axis is None, a must be 1-D or 2-D.

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 a 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 a is 1-D) or a matrix norm (when a is 2-D) is returned.

keepdims : bool, optional

If this is set to True, the axes which are normed over are left in the result as dimensions with size one. With this option the result will broadcast correctly against the original a.

Returns:

n : float or ndarray

Norm of the matrix or vector(s).

Notes

For values of ord <= 0, the result is, strictly speaking, not a mathematical ‘norm’, but it may still be useful for various numerical purposes.

The following norms can be calculated:

ord norm for matrices norm for vectors
None Frobenius norm 2-norm
‘fro’ Frobenius norm
inf max(sum(abs(x), axis=1)) max(abs(x))
-inf min(sum(abs(x), axis=1)) min(abs(x))
0 sum(x != 0)
1 max(sum(abs(x), axis=0)) as below
-1 min(sum(abs(x), axis=0)) as below
2 2-norm (largest sing. value) as below
-2 smallest singular value as below
other sum(abs(x)**ord)**(1./ord)

The Frobenius norm is given by [R110]:

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

The axis and keepdims arguments are passed directly to numpy.linalg.norm and are only usable if they are supported by the version of numpy in use.

References

[R110](1, 2) G. H. Golub and C. F. Van Loan, Matrix Computations, Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15

Examples

>>> from scipy.linalg import norm
>>> a = np.arange(9) - 4.0
>>> 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.]])
>>> norm(a)
7.745966692414834
>>> norm(b)
7.745966692414834
>>> norm(b, 'fro')
7.745966692414834
>>> norm(a, np.inf)
4
>>> norm(b, np.inf)
9
>>> norm(a, -np.inf)
0
>>> norm(b, -np.inf)
2
>>> norm(a, 1)
20
>>> norm(b, 1)
7
>>> norm(a, -1)
-4.6566128774142013e-010
>>> norm(b, -1)
6
>>> norm(a, 2)
7.745966692414834
>>> norm(b, 2)
7.3484692283495345
>>> norm(a, -2)
0
>>> norm(b, -2)
1.8570331885190563e-016
>>> norm(a, 3)
5.8480354764257312
>>> norm(a, -3)
0

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