Glossary

Vector and matrix norms

The size measures used to quantify errors and convergence in several dimensions.

Definition

The most used are $\|x\|_1=\sum|x_i|$ , $\|x\|_2=(\sum x_i^2)^{1/2}$ and $\|x\|_\infty=\max|x_i|$ . Each induces a matrix norm $\|A\|=\max_{x\neq0}\|Ax\|/\|x\|$ , which appears in error bounds and in the condition number $\kappa(A)$ .

How it is used

Fix the norm before comparing errors or tolerances: the same vector can pass a criterion in $\|\cdot\|_\infty$ and fail it in $\|\cdot\|_2$ .