Math 6610 • Analysis of Numerical Methods I

Consider the (square) linear system \(Ax=b\). Suppose you have
obtained a numerical approximation \(\hat{x}\). Define the error and residual,
respectively, by
\[
  e=x-\hat{x}\qquad \text{and}\qquad r=b-A\hat{x}.
\]
Let \(\|\cdot\|\) denote a vector norm, and the induced matrix norm. Show that
\[
  \frac{1}{\|A\|\|A^{-1}\|}\frac{\|r\|}{\|b\|}
  \leq
  \frac{\|e\|}{\|x\|}
  \leq
  \|A\|\|A^{-1}\|\frac{\|r\|}{\|b\|}
\]
and discuss the significance of these inequalities.