the linear equation system $Ax=b$ has one approximation $\bar x$ and one exact $x^* \neq 0$ solutions. we also gives: $p>3, \|x^* - \bar x \| \leq 10^{-20} + \|A\| \|A^{-1}\| 10^{-p} \|x^*\| $ which $\|A\| \|A^{-1}\|=10^{4}$. The reliable digits of $\bar x$ for solutions of this system of equation is $0$.
this is the mathematica code for verification of the above example.
hm = HilbertMatrix[4];
sol = {1, 1, 1, 1};
mat = Round[N[hm], 1.*^-4];
bv = Round[N[hm.sol], 1.*^-4];
s = LinearSolve[mat, bv];
NumberForm[Norm[sol - s], 5]
The result is $0.6780$. I didn't understand what the meaning of this result. anyone could describe it the result of this code for me? (i.e what is the relation among this answer and the solved example ).
