numpy.linalg.norm — NumPy v1.11 Manual (original) (raw)

numpy.linalg.norm(x, ord=None, axis=None, keepdims=False)[source]¶

Matrix or vector norm.

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

Parameters:	x : array_like Input array. If axis is None, x must be 1-D or 2-D. ord : {non-zero int, inf, -inf, ‘fro’, ‘nuc’}, 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. 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 x. New in version 1.10.0.
Returns:	n : float or ndarray Norm of the matrix or vector(s).

Parameters:

x : array_like Input array. If axis is None, x must be 1-D or 2-D. ord : {non-zero int, inf, -inf, ‘fro’, ‘nuc’}, 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. 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 x. New in version 1.10.0.

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	–
‘nuc’	nuclear 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 [R41]:

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

The nuclear norm is the sum of the singular values.

References

[R41]	(1, 2) G. H. Golub and C. F. Van Loan, Matrix Computations, Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15

Examples

from numpy import linalg as LA 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]])

LA.norm(a) 7.745966692414834 LA.norm(b) 7.745966692414834 LA.norm(b, 'fro') 7.745966692414834 LA.norm(a, np.inf) 4.0 LA.norm(b, np.inf) 9.0 LA.norm(a, -np.inf) 0.0 LA.norm(b, -np.inf) 2.0

LA.norm(a, 1) 20.0 LA.norm(b, 1) 7.0 LA.norm(a, -1) -4.6566128774142013e-010 LA.norm(b, -1) 6.0 LA.norm(a, 2) 7.745966692414834 LA.norm(b, 2) 7.3484692283495345

LA.norm(a, -2) nan LA.norm(b, -2) 1.8570331885190563e-016 LA.norm(a, 3) 5.8480354764257312 LA.norm(a, -3) nan

Using the axis argument to compute vector norms:

c = np.array([[ 1, 2, 3], ... [-1, 1, 4]]) LA.norm(c, axis=0) array([ 1.41421356, 2.23606798, 5. ]) LA.norm(c, axis=1) array([ 3.74165739, 4.24264069]) LA.norm(c, ord=1, axis=1) array([ 6., 6.])

Using the axis argument to compute matrix norms:

m = np.arange(8).reshape(2,2,2) LA.norm(m, axis=(1,2)) array([ 3.74165739, 11.22497216]) LA.norm(m[0, :, :]), LA.norm(m[1, :, :]) (3.7416573867739413, 11.224972160321824)