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bundles / numpy 2.4.4 / numpy / linalg / norm

_ArrayFunctionDispatcher

numpy.linalg:norm

source: /numpy/linalg/_linalg.py :2598

Signature

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

Summary

Matrix or vector norm.

Extended Summary

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, unless ord is None. If both axis and ord are None, the 2-norm of x.ravel will be returned.

ord : {int, float, inf, -inf, 'fro', 'nuc'}, optional

Order of the norm (see table under Notes for what values are supported for matrices and vectors respectively). inf means numpy's inf object. The default is None.

axis : {None, int, 2-tuple of ints}, 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. The default is None.

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.

Returns

n : float or ndarray

Norm of the matrix or vector(s).

Notes

For values of ord < 1, 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 [1]:

The nuclear norm is the sum of the singular values.

Both the Frobenius and nuclear norm orders are only defined for matrices and raise a ValueError when x.ndim != 2.

Examples

import numpy as np
from numpy import linalg as LA
a = np.arange(9) - 4
a
b = a.reshape((3, 3))
b
✓
LA.norm(a)
LA.norm(b)
LA.norm(b, 'fro')
LA.norm(a, np.inf)
LA.norm(b, np.inf)
LA.norm(a, -np.inf)
LA.norm(b, -np.inf)
✗
LA.norm(a, 1)
LA.norm(b, 1)
LA.norm(a, -1)
LA.norm(b, -1)
LA.norm(a, 2)
LA.norm(b, 2)
✗
LA.norm(a, -2)
LA.norm(b, -2)
LA.norm(a, 3)
LA.norm(a, -3)
✗
Using the `axis` argument to compute vector norms: 
c = np.array([[ 1, 2, 3],
              [-1, 1, 4]])
✓
LA.norm(c, axis=0)
LA.norm(c, axis=1)
LA.norm(c, ord=1, axis=1)
✗
Using the `axis` argument to compute matrix norms: 
m = np.arange(8).reshape(2,2,2)
✓
LA.norm(m, axis=(1,2))
LA.norm(m[0, :, :]), LA.norm(m[1, :, :])
✗

See also

scipy.linalg.norm

Similar function in SciPy.

Aliases

  • numpy.linalg.norm

Referenced by

This package

Other packages