NORM Function

NORM (x, <, method> ) ;

The NORM function computes the vector or matrix norm of x. The norm depends on the metric specified by the method argument. The arguments are as follows:

x

specifies a numeric vector with $n$ elements or an $n\times p$ numeric matrix.

method

is an optional argument that specifies the method used to specify the norm. The method argument is either a numeric value, method$\geq 1$, or a case-insensitive character value. The valid options are given in the following sections.

Methods for Vector Norms

If x is a vector, then a vector norm is computed. The following are valid values of the method argument:

L1

specifies that the function compute the 1-norm: $\|  x \| _1 = \Sigma _ k |x_ k |$. An equivalent alias is CityBlock or Manhattan.

L2

specifies that the function compute the Euclidean 2-norm: $\|  x \| _2 = \sqrt (x^\prime x) = (\Sigma _ k |x_ k |^2 )^{1/2}$. This is the default value. An equivalent alias is Euclidean or Frobenius.

LInf

specifies that the function compute the $\infty $-norm: $\|  x \| _\infty = \max _ k |x_ k |$.

An equivalent alias is Chebyshev.

$p$

is a numeric value, $p \geq 1$, that specifies the $p$-norm: $\|  x \| _ p = (\Sigma _ k |x_ k |^ p )^{1/p}$, $p\geq 1$.

Methods for Matrix Norms

For an $n\times p$ matrix $A$ such that $n>1$ and $p>1$, the method argument has the following valid values:

Frobenius

specifies the Frobenius norm: $\|  A \| _ F = \left( \Sigma _{i=1}^ n \Sigma _{j=1}^ p |a_{ij} |^2 \right)^{1/2}$. This is the default value.

L1

specifies the matrix 1-norm: $\|  A \| _1 = \max _{1\leq j\leq p}\Sigma _{i=1}^ n |a_{ij} |$. This norm computes the maximum of the absolute column sums.

L2

specifies the matrix $2$-norm, which is equivalent to the square root of the largest eigenvalue of the $A^\prime A$ matrix. This quantity can be expensive to compute because the function internally computes eigenvalues.

LInf

specifies the $\infty $-norm: $\|  A \| _\infty = \max _{1\leq i\leq n}\Sigma _{j=1}^ p |a_{ij} |$. This norm computes the maximum of the absolute row sums.

The matrix $p$-norm is not available unless $p \in \{ 1,2, \infty \} $.

The following statements compute vector norms:

/* compute vector norms */
v = 1:5;
vn1 = norm(v, "L1");
vn2 = norm(v, "L2");
vnInf = norm(v, "LInf");
print vn1 vn2 vnInf;

Figure 24.241: Vector Norms

vn1 vn2 vnInf
15 7.4161985 5


You can also compute matrix norms, as follows:

x = {1 2, 3 4};
mn1 = norm(x, "L1");
mnF = norm(x, "Frobenius");
mnInf = norm(x, "LInf");
print mn1 mnF mnInf;

Figure 24.242: Matrix Norms

mn1 mnF mnInf
6 5.4772256 7


The NORM function returns a missing value if any element of the argument contains a missing value.