Built-in Mathematica Symbol

Norm

Norm[expr]
gives the norm of a number, vector or matrix.

Norm[expr, p]
gives the

-norm.

MORE INFORMATION

For complex numbers, Norm[z] is Abs[z].

For vectors, Norm[v] is Sqrt[v.Conjugate[v]]. »

For vectors, Norm[v, p] is Total[Abs[v]^p]^(1/p).

For vectors, Norm[v, Infinity] is the -norm given by Max[Abs[v]]. »

For matrices, Norm[m] gives the maximum singular value of m. »

Norm[m, "Frobenius"] gives the Frobenius norm of m. »

Norm can be used on SparseArray objects. »

EXAMPLES

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Basic Examples (2)

Norm of a vector:

Norm of a complex number:

Norm of a vector:

In[1]:=

Out[1]=

Norm of a complex number:

In[1]:=

Out[1]=

Scope (3)

v is a vector of integers:

Use exact arithmetic to compute the norm:

Use approximate machine-number arithmetic:

Use 35-digit precision arithmetic:

s is a SparseArray representation of v:

The norm is always real even when the input is complex:

TraditionalForm formatting:

Generalizations & Extensions (6)

The p-norm:

The

-norm:

Norm of a matrix, equal to the largest singular value:

The

-norm and

-norm, respectively, for matrices:

The Frobenius norm for matrices:

Symbolic matrix norms for a real parameter

:

Applications (3)

Estimate the mean distance from the origin to random points in the unit square:

Compare to the asymptotic result:

Solve an ill-conditioned linear system

with a known solution:

Get the norm of the residual:

Get the norm of the actual error:

Approximate the solution of

using n spatial points and k time steps:

Find two solutions with fixed n where the second has twice as many time steps:

Estimate the error by the norm of the difference:

Extrapolate to a better solution from the first-order convergence of the backward Euler method:

Compute a more accurate solution with NDSolve:

Compare the errors in the three solutions:

Properties & Relations (4)

The norm of v is equal to the square root of the Dot product

:

is a decreasing function of p:

The horizontal asymptote is the

-norm, equal to Max[Abs[v]]:

The matrix 2-norm is the maximum 2-norm of m.v for all unit vectors v:

This is also equal to the largest singular value of r:

The Frobenius norm is the same as the norm made up of the vector of the elements:

Possible Issues (1)

It is expensive to compute the 2-norm for large matrices:

If you need only an estimate, the 1- or

-norms are very fast:

Neat Examples (2)

Unit balls for using

,

and

norms:

Different norm functions: