The norm of a vector in two dimensions:

Norm of a three-dimensional vector:

Norm of a complex number:

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:

The norm is always non-negative:

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

For vectors, the default norm is the 2-norm:

This is also true for matrices:

is a decreasing function of :

The limiting value is the -norm, equal to Max[Abs[v]]:

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

This is also equal to the largest singular value of :

The Frobenius norm is the same as the norm of the vector made from the entries of the matrix:

For a matrix m, Norm[m,Infinity] can be computed as Max[Total/@Abs/@m]:

This is equivalent to computing the -norm of the rows and taking the maximum:

Norm[m,1] can be computed as Max[Total/@Abs/@Transpose[m]]:

This is equivalent to computing the -norm of the columns and taking the maximum:

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

If you need only an estimate, the -norm and -norm are very fast:

Norms of general vectors contain Abs:

Use Simplify and FullSimplify to get simpler answers assuming real parameters:

Unit balls for using 1, 2, 3, and norms:

Different norm functions: