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\begin{document}
\begin{center}
{\bf \hfill Seema Nanda\\}
{\bf Numerical Analysis \hfill Assignment 6 \hfill due: April 6, 2011
 in class}
\end{center}
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\begin{center}
Please follow the instructions for assignments and homework as given in the
course web page. You may discuss the problems and solutions with anyone but the work written up and submitted must be done on your own. Also programs must be written by you.
\end{center}
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% Homework starts here.
\begin{enumerate}
\item \textbf{Consistency:}
Consider the $m$ step multistep method: $y_0 = \alpha_0, \ldots, y_m=\alpha_m$\\
$y_{i+1} = a_{m}y_i + a_{m-1}y_{i-1}+\cdots + a_0y_{i-(m-1)}+ h \sum_{k=0}^m b_{m-k}f(t_{i-(k-1)},y_{i-(k-1)})$ 
where $a_0 ,\ldots a_m$ and $b_0 ,\ldots , b_m$ are constants. 

A multistep method  is consistent if $\lim_{h\downarrow 0}\alpha_i = y_i$ for $i = 1, . . . , m$ and if 
\begin{equation}
\tau_{i+1}(h) = \frac{1}{h} [y_{i+1} - \sum_{k=0}^m a_{m-k}y_{i-k} + h \sum_{k=0}^m b_{m-k}f(t_{i-(k-1)},y_{i-(k-1)})]
\end{equation}
goes to $0$ as $h\downarrow 0$ i.e.
\begin{equation}\label{truncErrorLimit}  
 \lim_{h\downarrow 0}max_{i>m}|\tau_i(h)| = 0 
\end{equation}
For multistep methods, there is an easy check for consistency: (\ref{truncErrorLimit})  holds if and only 
if $\sum_{k=0}^m a_k =1$ and $-\sum_{k=0}^m ka_k + \sum_{k=0}^m b_k = 1$.
Prove this.

 \item {\bf Analysis of multistep methods:}

Investigate the convergence of the following multistep methods by 
analysing the consistency and the stability. If convergent determine the order of convergence.

\begin{enumerate}

\item $y_{j+1}- y_{j-1} = 2hf(x_j , y_j)$ (explicit mid-point rule)
\item $y_{j+1}- y_{j-1} = hf(x_{j+1} , y_{j+1})$
\item $\frac{3}{2} y_{j+1}- 2y_j + \frac{1}{2} y_{j-1}= hf(x_{j+1} , y_{j+1})$
\end{enumerate}

\item {\bf Fibonacci series:}
The famous Fibonacci series is given by the recursion
$u_0 = 0, u_1 = 1, u_{n+1} = u_n + u_{n-1}$ for $n = 1, 2, 3,\ldots$
\begin{enumerate}
 \item  Determine a fundamental system of this linear difference equation. 
Use it to derive a formula for the elements in the Fibonacci series.
 \item Does this linear difference equation satisfy the root condition?
 \item Consider the perturbed initial values $\tilde u_0 = 1+\epsilon_1$ and 
$\tilde u_1 = 1 + \epsilon_2$.  
Prove via induction that the resulting series satisfies
$u_n = u_n (1 + \epsilon_2 ) + u_n\epsilon_1$ for $n = 2,3,4,\ldots$
What do you conclude?

\end{enumerate}

\item {\bf Region of stability}

Determine the region of stability for the methods in problem 2 above.
\end{enumerate}


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