Math 6610 • Analysis of Numerical Methods I

{Sensitivity:}

\partitem{a} Suppose that \(A\) is an \(n\times n\) nonsingular real matrix. Analyze
the sensitivity of solutions of the system \(Ax=b\) to perturbations in \(b\). What
quantity related to \(A\) characterizes this sensitivity?

\partitem{b} Suppose \(\tilde{x}\) is an approximate solution to the linear system
\(Ax=b\), where \(A\) is an \(n\times n\) nonsingular real matrix. The residual is the
vector \(r=b-A\tilde{x}\). Derive inequalities relating the residual \(r\) to the
error \(e=x-\tilde{x}\).