Math 6610 • Analysis of Numerical Methods I

{Sensitivity Analysis}
For the linear system \(Ax=b\) where \(A\) is a square, real non-singular matrix, a
numerical solution \(\hat{x}\) actually solves a slightly different problem due to
roundoff errors:
\[
  (A+\delta A)\hat{x}=b+\delta b.
\]

\textbf{(a)} If we denote \(\delta x=\hat{x}-x\), derive an upper bound for the
relative error
\[
  E_1=\frac{\lVert\delta x\rVert}{\lVert\hat{x}\rVert}
\]
in terms of the condition number of \(A\), \(\kappa(A)\), and the norms of
\(A,\hat{x},\delta A,\delta b\), where \(\lVert\cdot\rVert\) is a suitable vector norm.

\textbf{(b)} We can define another relative error
\[
  E_2=\frac{\lVert\delta x\rVert}{\lVert x\rVert},
\]
based on the exact solution. Derive an upper bound for \(E_2\) in terms of
\(\kappa(A)\) and the norms of \(\delta A,A,\delta b\) and \(b\).

\textbf{(c)} Strictly speaking, the relative error is given by \(E_2\), even though in
many applications we simply use \(E_1\) since it is more readily available. Point out
cases where this approximation may not be appropriate. Explain the significance of
the condition number \(\kappa(A)\), and express the relative distance of \(A\) to its
nearest singular matrix in terms of \(\kappa(A)\).