BUILT-IN MATHEMATICA SYMBOL

Norm

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

Norm

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 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 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)

is a vector of integers:

Use exact arithmetic to compute the norm:

Use approximate machine-number arithmetic:

Use 35-digit precision arithmetic:

is a SparseArray representation of

:

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

TraditionalForm formatting:

Generalizations & Extensions (6)

The

-norm:

The

-norm:

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

The 1-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

spatial points and

time steps:

Find two solutions with fixed

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

is equal to the square root of the Dot product

:

is a decreasing function of

:

The horizontal asymptote is the

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

The matrix 2-norm is the maximum 2-norm of

for all unit vectors

:

This is also equal to the largest singular value of

:

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-norm or

-norm are very fast:

Neat Examples (2)

Unit balls for using 1, 2, 3, and

norms:

Different norm functions: