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\begin{document}

\title{Numerical Methods I}
\author{Lab Session 1}
\date{September 11, 2003}
\maketitle

\noindent
Consider the sequences
\begin{gather}
  x_0=0, \qquad x_{k+1} = \cos(x_k) \\
  y_0=1, \qquad y_{k+1} = 1 - \cos(y_k) \\
  z_k = \sum_{j=1}^k \frac1{j^2}
\end{gather}

\begin{enumerate}

\item What are the limits $\xi$, $\eta$, and $\zeta$ for each of the
sequences?  Use \textsf{Octave} to compute the first few members of
each sequence.  Which ones converge fast, which ones converge slowly?

\item By plotting $k$ vs.\ $\log |\xi-x_k|$, etc., determine for each
of the sequences whether it converges linearly, sublinearly, or
superlinearly.

\item By performing a log-log plot of $|\eta-y_k|$ vs.\ 
$|\eta-y_{k+1}|$, demonstrate that the convergence of $y_n$ is in fact
quadratic.

\end{enumerate}
\end{document}

