# Numerical Mathematics

## Operator Norm

$$
||F||=\sup\limits_{x\neq0}\frac{||F(x)||_w}{||x||_v}=\sup\limits_{||x||_v=1}||F(x)||_w
$$

## Condition numbers

$K_a := Absolute condition number$

$K_r := Relative conditioon number$

### Operator linear

$$
K_a=||F||_{v,w}\in[0, \infin[
K_r\leq\frac{||F||_{v,w}}{\inf\limits_{||x||_v=1}||F(x)||_w}\in[0,\infin[
$$

### F, A bijektive

$$
K_r\leq||F||_{v,w}||F^{-1}||_{v,w}
K_r(A)=||A||_\infin||A^{-1}||_\infin
$$

### Component based

$$
K_r^c=||||
$$