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

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