\documentstyle[12pt]{article}
\begin{document}
\begin{center}
\begin{huge}
PROBLEM SET 5\\[.1in]
\end{huge}
Math 505-Fall 2002\\
Write a matlab script to plot the root locus for the stability 
polynomial
\[ P(z;\mu) := \rho(z) - \mu \sigma(z) \] for the following multistep methods,
with 
\[ \rho(z) = \sum_{l=0}^{k} \alpha_l z^l \ ,
\ \sigma(z) = \sum_{l=0}^{k} \beta_l z^l \ . \]
\end{center}
That is, plot $P(e^{i\theta};\mu)=0, 0\le \theta \le 2\pi$ 
in the complex $\mu$-plane.
Identify the region which corresponds to stable behavior.
\begin{enumerate}
\item Adams methods of order 1-4a (The $\alpha_l$ and $\beta_l$ are as given in
class).
\item BDF methods of order 1-6. These are defined by the formula
\[ \sum_{l=1}^k \frac{1}{l} \nabla^l y_{n+l} = h f_{n+k} \]
so that the characteristic polynomial becomes:
\[ \sum_{l=1}^k \frac{1}{l} (z-1)^l - \mu z^k = 0 \ . \]
Here $\nabla^n$ is the $n$-th difference operator:
\[ \nabla^n = (\nabla^{n-1})\nabla \ ,\  \nabla y_n = y_n - y_{n-1} \ . \]
\end{enumerate}
\end{document}
