Other Examples of Vector Norms

Using the standard Euclidean norm in the course of a vector-based numerical algorithm is not computationally wise. The square root function is the problem point. Suppose you wish to calculate the $sqrt{a}$. The following steps are involved.

1. The number $a$ is rescaled to a number $b$ such that $a = b 2^{f}$ with $1/4 \leq b \leq 1$ and $f$ an even integer. Many numerical methods use a scaling process to keep from working with numbers that are too large or too small.
2. Linear interpolation is used to produce an initial guess $x_{0}$.
3. The iteration process is based Newton's method and is easily seen to be
   $$x _{n+1} = 1/2(x_{n} + \frac{b}{x_{n}})$$

4. A certain number of iterations is used to produce an estimate of $\sqrt{b}$. Since $b$ was chosen to fit in a certain range, the number of iterations is know a priori.

5. $\sqrt{a} = \sqrt{b}2^{f/2}$.

These process can get expensive, particularly if it is enclosed in one or several loops in an iterative process.

So numerical modelers often appeal to vector norms other than the Euclidean norm. Once popular choice is the so-called maximum norm

$$\Vert \mathbf{x}\Vert _{\infty} = max_{1 \leq i \leq n} \vert x_{i} \vert$$

It measures the size of a vector in terms of its largest component.

Another common choice is the

$$\Vert \mathbf{x}\Vert _{1} = \sum_{i = 1}^{n} \vert x_{i} \vert$$

This avoids the root finding problem of the Euclidean norm.

Of course, these are not the only choice. There is the general family

$$\Vert \mathbf{x}\Vert _{p} = (\sum_{i = 1}^{n} \vert x_{i} \vert^{p})^{1/p}$$

where we have consider the cases where $p=1,2$. In general, these have the root finding problem attached to them except when $p=1$.