Sensitivity analysis

It is defined as the forward error divided by the backward error:

$$\frac{\Vert f(x)-\tilde{f}(x)\Vert }{\Vert x-\tilde{x}\Vert }=\frac{\Vert f(x)-f(\tilde{x})\Vert }{\Vert x-\tilde{x}\Vert }=\mathrm{sensitivity}$$

$f$: exact map

$\tilde{f}$: numerical solution

$\tilde{x}$: defined as the input such that $\tilde{f}(x)=f(\tilde{x})$.

Conditioning = relative sensitivity.

This quantity is dimensionless.

$$\frac{\Vert f(x)-\tilde{f}(x)\Vert }{\Vert f(x)\Vert }\frac{\Vert x\Vert }{\Vert x-\tilde{x}\Vert }$$

The conditioning can be defined for any algorithm or map $f$.

Forward and backward error, Stability of the LU factorization