numpy.linalg.norm — NumPy v2.2 Manual (original) (raw)

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:

xarray_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 ofx.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’sinf 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.

keepdimsbool, 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:

nfloat 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:

The Frobenius norm is given by [1]:

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

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.

References

[1]

G. H. Golub and C. F. Van Loan, Matrix Computations, Baltimore, MD, Johns Hopkins University Press, 1985, pg. 15

Examples

import numpy as np from numpy import linalg as LA a = np.arange(9) - 4 a array([-4, -3, -2, ..., 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) 0.0 LA.norm(b, -2) 1.8570331885190563e-016 # may vary LA.norm(a, 3) 5.8480354764257312 # may vary LA.norm(a, -3) 0.0

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)