so I am currently working on a homework problem. The goal is to find with a given matrix (where $\delta=2^{-17}$)$$ {\bf A}=\begin{pmatrix} 1 & 0 & 0 & 0 \\ 1 & \delta & 0 & 0 \\ 1 & 0 & \delta^2 & 0 \\ 1 & 0 & 0 & \delta^3 \end{pmatrix}$$ and the linear system $$ {\bf Ax}={\bf b} $$ , I want to find the exact error in input $\bf b$ such that there the solution $\bf x$ has the least amount of significant figures when calculated in double precision. From class, I know that the condition number a this linear system is $$ cond=\|{\bf A}\|\cdot \|{\bf A}^{-1}\|$$ which is equivalent to the definition $$cond= \sup_{{\bf b}\neq{\bf 0},\delta{\bf b}\neq {\bf 0}}\frac{\|\delta {\bf x}\|/\|{\bf x}\|}{\|\delta {\bf b}\|/\|{\bf b}\|}$$, where where $\delta{\bf b}$ is an error in the input ${\bf b}$ and $\delta{\bf x}$ is the error in the output ${\bf x}$. And then I calculated $A^{-1}$ as $$ {\bf A}^{-1}=\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1/\delta & 0 & 0 \\ 0 & 0 & 1/\delta^2 & 0 \\ 0 & 0 & 0 & 1/\delta^3 \end{pmatrix}. $$, and by taking the ${\bf A}$ with the $\ell_\infty$ vector norm $\|\cdot\|_\infty.$, I get the condition number to be $(1+\delta)*1/(\delta^3)$, so with the condition number I lost 10 sig figs of the solution $x$. But now, is there a way for me to know what is the exact error in input $\bf b$ such that I lost 10 sig figs? My professor told us that we should refer to the definition of matrix norm induced by the $\ell_\infty$ vector norm $\|\cdot\|_\infty.$, but I really don't know how to proceed.
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