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\begin{document}

\title{Numerical Methods I}
\author{Problem Set 9}
\date{due in class, November 26, 2003}
\maketitle


\begin{enumerate}


\item The Chebychev polynomial of degree $n$ is given by
\[
  T_n(x) = \cos(n \, \arccos x) \,.
\]

\begin{enumerate}
\item Show that the extrema of $T_n$ are located at
\[
  x_k=\cos \frac{k \pi}{n}
\]
for $k=0,\dots,n$, and that the zeros of $T_n$ are located at
\[
  x_k=\cos \frac{(1+2k) \pi}{2n}
\]
for $k=0,\dots,n-1$.
\item Show that $T_n$ and $T_m$ are orthogonal with respect to the
inner product
\[
  \langle f,g \rangle = \int_{-1}^1
  \frac{f(x) \, g(x)}{\sqrt{1-x^2}} \, \d x 
\]
if and only if $m \neq n$.

Hint: Use trigonometric substitution, then note that
\[
  2 \, \cos \alpha \, \cos \beta
  = \cos (\alpha+\beta) + \cos(\alpha - \beta) \,.
\]
\end{enumerate}

\item Solve the logistic differential equation
\begin{gather*}
  \dot y = y - y^2 \,,\\
  y(0) = y_0  \,.
\end{gather*}

\item Compute the matrix exponential $\exp (A)$ for
\begin{enumerate}
\item $\displaystyle A = 
\begin{pmatrix}
  1 & 2 \\ 0 & 3
\end{pmatrix} $,
\item $\displaystyle A = 
\begin{pmatrix}
  a & b \\ -b & a
\end{pmatrix} $.
\end{enumerate}

\item Give an example that $\exp(A+B) \neq \exp (A) \, \exp (B)$.

\item \textbf{Project:} Solve the logistic equation from Question~2 on
the interval $[0,1]$ by using the explicit Taylor methods of order $1$
and $2$.  Generate a doubly logarithmic error plot to demonstrate the
order of the methods.

\end{enumerate}

\end{document}
