Math 6610 • Analysis of Numerical Methods I

{Sensitivity Analysis}

\textbf{(a)} Let \(X\in \mathbb{R}^{n\times n}\) such that \(\lVert X\rVert<1\) for some
induced norm. Show that \(I-X\) is nonsingular and
\[
  \lVert (I-X)^{-1}\rVert \le \frac{1}{1-\lVert X\rVert}.
\]

\textbf{(b)} Assume that \(A\in \mathbb{R}^{n\times n}\) is nonsingular, and the linear
system
\[
  Ax=b
\]
is perturbed to
\[
  (A+\delta A)(x+\delta x)=b+\delta b,
\]
and the perturbations are small enough so that
\[
  \lVert \delta A\rVert\le \epsilon\lVert A\rVert,\qquad
  \lVert \delta b\rVert\le \epsilon\lVert b\rVert,
\]
for some small \(\epsilon>0\), and \(\epsilon\kappa(A)=r<1\), where \(\kappa(A)\) is the
condition number of \(A\). Show that \(A+\delta A\) is nonsingular.

\textbf{(c)} Let \(\tilde{x}=x+\delta x\), show that
\[
  \lVert \tilde{x}\rVert \le \frac{1+r}{1-r}\lVert x\rVert;
\]
and
\[
  \frac{\lVert \delta x\rVert}{\lVert x\rVert}
  \le \frac{2\epsilon}{1-r}\kappa(A).
\]