##### Item1.3.5.6.b

Some types of linear systems are particularly sensitive to errors resulting from computers' approximate arithmetic. For instance, suppose we are interested in the linear system

\begin{equation*} \begin{alignedat}{3} x \amp {}+{} \amp y \amp {}={} \amp 2 \\ x \amp {}+{} 1.001\amp y \amp {}={} \amp 2 \\ \end{alignedat} \end{equation*}

Find the solution to this linear system.

Suppose now that the computer has accumulated some error in one of the entries of this system so that it incorrectly stores the system as

\begin{equation*} \begin{alignedat}{3} x \amp {}+{} \amp y \amp {}={} \amp 2 \\ x \amp {}+{} 1.001\amp y \amp {}={} \amp 2.001 \\ \end{alignedat} \end{equation*}

Find the solution to this linear system.

Notice how a small error in one of the entries in the linear system leads to a solution that has a dramatically large error. Fortunately, this is an issue that has been well studied, and there are techniques that mitigate this type of behavior.

in-context