Chapter 7.4, Problem 35E

Given
$A=\left(\begin{array}{cc}3& 2\\ 1& 1\end{array}\right)$ and $b=\left(\begin{array}{c}5\\ 2\end{array}\right)$
If two-digit decimal floating-point arithmetic is used to solve the system $Ax=b,$ the computed solution will be $x={\left(1.1,0.88\right)}^T.$

1. Determine the residual vector r and the value of the relative residual ${\Vert r\Vert }_{\infty }/{\Vert b\Vert }_{\infty }.$
2. Find the value of ${\text{cond}}_{\infty }\left(A\right).$
3. Without computing the exact solution, use the results from parts (a) and (b) to obtain bounds for the relative error in the computed solution.
4. Compute the exact solution x and determine the actual relative error. Compare this to the bounds derived in part (c).