let $Ax = b$ be a linear n by n system
and $x^*$ is a numerical solution of this system
I noticed that the relative error related to the solution of the system is necessarily small if $||b-Ax^*||_2$ is small is that true in general ?
I don't know how I can try to prove this any help or hints would be appreciated.
I'll assume there is a unique true solution $x = A^{-1} b$ (in particular, $A$ is invertible), and $b \ne 0$ (the relative error with respect to a true solution of $0$ would be meaningless). The relative error is $$RE = \frac{\|x^* - x\|}{\|x\|}$$
Now $$\frac{\|A x^* - b\|}{\|x\|} = \frac{\|A(x^* - x)\|}{\|x\|} \le \frac{\|A\| \|x^* - x\|}{\|x\|} = \|A\| RE $$ while in the other direct
Suppose $x^*$ is the approximate solution and $x$ the actual solution. ion $$ RE = \frac{\| A^{-1} (A x^* - b) \|}{\|x\|} \le \|A^{-1}\| \frac{\|A x^* - b \|}{\|x\|} $$
Thus if neither $\|A\|$ nor $\|A^{-1}\|$ is too big or too small, smallness of the relative error is associated with smallness of $\|A x^*-b\|/\|x\|$.