vecnorm

Vector-wise norm

Syntax

N = vecnorm(A)

N = vecnorm(A,p)

N = vecnorm(A,p,dim)

Description

N = vecnorm(A) returns the 2-norm or Euclidean norm of A:

• If A is a vector, then vecnorm returns the norm of the vector.

• If A is a matrix, then vecnorm returns the norm of each column.

• If A is a multidimensional array, then vecnorm returns the norm along the first array dimension whose size does not equal 1.

N = vecnorm(A,p) calculates the generalized vector p-norm.

N = vecnorm(A,p,dim) operates along dimension dim. The size of this dimension reduces to 1 while the sizes of all other dimensions remain the same.

Examples

1-Norm and 2-Norm of Vector

Calculate the 2-norm of a vector corresponding to the point (2,2,2) in 3-D space. The 2-norm is equal to the Euclidean length of the vector, $$\sqrt{12}$$.

x = [2 2 2];
n = vecnorm(x)

n = 3.4641

Calculate the 1-norm of the vector, which is the sum of the element magnitudes.

n = vecnorm(x,1)

n = 6

2-Norm of Matrix Columns

Calculate the 2-norm of the columns of a matrix.

A = [2 0 1;-1 1 0;-3 3 0]

A = 3×3

     2     0     1
    -1     1     0
    -3     3     0

n = vecnorm(A)

n = 1×3

    3.7417    3.1623    1.0000

As an alternative, you can use the norm function to calculate the 2-norm of the entire matrix.

Input Arguments

A — Input array

vector | matrix | multidimensional array

Input array, specified as a vector, matrix, or multidimensional array. By convention, vecnorm returns NaN values if the vector being operated on contains a NaN value.

Data Types: single | double
Complex Number Support: Yes

p — Norm type

2 (default) | positive scalar | Inf

Norm type, specified as 2 (default), a positive scalar, or Inf.

dim — Dimension to operate along

positive integer scalar

Dimension to operate along, specified as a positive integer scalar. If you do not specify a value, then the default is the first array dimension whose size does not equal 1.

Dimension dim indicates the dimension whose length reduces to 1. In other words, size(N,dim) is 1, while the sizes of all other dimensions remain the same.

Consider a two-dimensional input array, A:

• vecnorm(A,p,1) calculates the norm of each column.

  

• vecnorm(A,p,2) calculates the norm of each row.

  

  vecnorm returns abs(A) when dim is greater than ndims(A) or when size(A,dim) is 1.

Data Types: single | double | int8 | int16 | int32 | int64 | uint8 | uint16 | uint32 | uint64

More About

Euclidean Norm

The Euclidean norm (also called the vector magnitude, Euclidean length, or 2-norm) of a vector v with N elements is defined by

$$\Vert v\Vert =\sqrt{{\displaystyle \sum _{k=1}^N{\left|v_k\right|}^2}}.$$

General Vector Norm

The general definition for the p-norm of a vector v that has N elements is

$${\Vert v\Vert }_p={\left[{\displaystyle \sum _{k=1}^N{\left|v_k\right|}^p}\right]}^{1/p},$$

where p is any positive real value or Inf. Some interesting values of p are:

• If p = 1, then the resulting 1-norm is the sum of the absolute values of the vector elements.

• If p = 2, then the resulting 2-norm gives the vector magnitude or Euclidean length of the vector.

• If p = Inf, then $${\Vert v\Vert }_{\infty }={\max }_i\left(\left|v\left(i\right)\right|\right)$$.

Extended Capabilities

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

This function fully supports tall arrays. For more information, see Tall Arrays.

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• If you supply dim, then it must be constant.

• For limitations related to variable-size inputs, see Variable-Sizing Restrictions for Code Generation of Toolbox Functions (MATLAB Coder).

• Code generation does not support sparse matrix inputs for this function.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).