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\begin{document}

\title{Numerical Methods II}
\author{Lab Session 4}
\date{March 8, 2004}
\maketitle


\begin{enumerate}

\item ``Harden'' your favorite adaptive time-stepping scheme (e.g.\
  the embedded Runge--Kutta Prince--Dormand 4,5 pair from Lab~1) by
  making sure the code exits gracefully when the maximum number of
  steps is reached, or when the time step is getting too small.

\item Implement the Van Wijngaarden--Dekker--Brent root finding method
  from the \emph{Numerical Recipes},
  \begin{quote}
  \texttt{http://www.library.cornell.edu/nr/bookcpdf/c9-3.pdf}
  \end{quote}
  in \textsf{Octave}.

\item Use the ODE solver from Task~1 in combination with the Van
  Wijngaarden--Dekker--Brent method to solve the boundary value
  problem
\begin{gather*}
  y''(t) = 5 \, \sinh (5 \, y(t)) \,, \\
  y(0) = 0 \,, \\
  y(1) = 1 
\end{gather*}
by simple shooting.  If the ODE solver terminates prematurely, take
the last value computed as the value at the final time $t=1$.

\emph{Note:} You may start with using the bisection method for
verification of the shooting code before implementing the Van
Wijngaarden--Dekker--Brent root finding method.

\end{enumerate}

\end{document}
