Consider a system of equations $Ax=b$ and $A \tilde x = \tilde b$. Find upper and lower bounds on $\frac{\|x-\tilde x\|_{\infty}}{\|x\|_{\infty}}$

Question

Consider a system of equations $Ax=b$ and $A \tilde x = \tilde b$. Find upper and lower bounds on $\frac{\|x-\tilde x\|_{\infty}}{\|x\|_{\infty}}$ compared with $\frac{\|b-\tilde b\|_{\infty}}{\|b\|_{\infty}}$.

So far $\|Ax-A\tilde x\|_{\infty}\leq \|A\|\|x-\tilde x\|_{\infty}$ and $\|b\|_{\infty}\leq \|A\|\|x\|_{\infty}$ but failed to combine them. Any hints are appreciated.

Answer 1

Use also $x=A^{-1}b$. This should lead to a bound related to the condition number of the matrix $A$.