Math 6610 • Analysis of Numerical Methods I

{Sensitivity Analysis}
In this problem, we consider a real, non-singular \(n\times n\) matrix \(A\) and vectors
\(f\), \(b\), \(x\in\mathbb{R}^n\).

\textbf{a)} (6\%) First we consider the conditioning of matrix-vector multiplication
\(f=Ax\), that is, find the smallest factor \(\kappa>0\) such that, when \(x\) is
perturbed by any small \(\delta x\), the perturbation for \(f\) satisfies
\[
  \frac{\lVert\delta f\rVert}{\lVert f\rVert}
  \le
  \kappa\frac{\lVert\delta x\rVert}{\lVert x\rVert}.
\]
Here we assume no perturbation in matrix \(A\).

\textbf{b)} (8\%) For the linear system \(Ax=b\), a numerical solution \(\hat{x}\)
actually solves a slightly different problem due to roundoff errors:
\[
  (A+\delta A)\hat{x}=b+\delta b.
\]
If we denote \(\delta x=\hat{x}-x\), derive an upper bound for the relative error
\[
  E=\frac{\lVert\delta x\rVert}{\lVert x\rVert}
\]
in terms of the condition number of \(A\), \(\kappa(A)\), and the norms of \(A\),
\(\delta A\), \(b\) and \(\delta b\), where \(\lVert\cdot\rVert\) is a suitable vector norm.

\textbf{c)} (6\%) In practice, \(x\) is not available so \(\hat{x}\) is often used in the
above bound to generate an approximate bound for \(\delta x\). Comment on this
practice as what the advantages and disadvantages are, point out cases where this
approximation may be inappropriate.