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\begin{document}

\title{Numerical Methods I -- Lab 4}
\author{Fall Semester 2005}
\date{October 4, 2005}
\maketitle

\noindent
Let
\[
  A = \begin{pmatrix}
        \varepsilon & 1 \\ 1 & 1
      \end{pmatrix} \,, \qquad
  \b = 
  \begin{pmatrix}
  1 + \varepsilon \\ 2
  \end{pmatrix} \,,
\]
so that the exact solution to the linear system $A \x = \b$ is
\[
  \x =
  \begin{pmatrix}
  1 \\ 1
  \end{pmatrix} \,.
\]

\begin{enumerate}

\item Use \textsf{Octave} to compute the condition number of $A$ with
respect to a matrix norm of your choice for $\varepsilon$ small.

Hint: \texttt{help norm}; \texttt{help inv}

\item Solve this equation using \textsf{Octave} via its $LU$
decomposition for $\varepsilon = 10^{-k}$ for $k=2,\ldots,18$.

Hint: \texttt{help logspace}

\item Divide the first equation by $\varepsilon$.  Does the accuracy
improve?  Does the condition number improve?  Will pivoting help?

\end{enumerate}
\end{document}
