Math 6610 • Analysis of Numerical Methods I

{Error Analysis} Consider the linear system
\[
  Ax=b \tag{1}
\]
where \(A\) is an invertible matrix. Let \(\hat{x}\) be an approximation of the
solution of (1) and let
\[
  e=x-\hat{x}\qquad \text{and}\qquad r=b-A\hat{x}.
\]
Let \(\|\cdot\|\) denote any vector norm or the corresponding induced matrix norm.
Show that
\[
  \frac{\|e\|}{\|x\|}
  \leq \|A\|\|A^{-1}\|\frac{\|r\|}{\|b\|}.
\]
Comment on the significance of the condition number \(\|A\|\|A^{-1}\|\) and give
a lower bound for it in terms of the eigenvalues of \(A\).