Math 6610 • Analysis of Numerical Methods I

Sensitivity. For nonsingular \(A \in \mathbb{R}^{n \times n}\), and nonzero \(b \in \mathbb{R}^n\), consider the problems

\[
Ax = b,\qquad \hat{A}\hat{x} = b.
\]

\begin{enumerate}

\item
Show that
\[
\frac{\|x-\hat{x}\|}{\|x\|}
\le \|A\|\|A^{-1}\|\frac{\|A-\hat{A}\|}{\|A\|}.
\]



\item
Give a bound on \(\|A-\hat{A}\|_2\) which ensures that \(\hat{A}\) is nonsingular if \(A\) is nonsingular.

\end{enumerate}