%Eigenvalues and eigenvectors
%If you modify this file, please indicate here and in the footnote
%Math410

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{\Large
Eigenvalues by the QR Method\footnote{
Copyright \copyright 2002 Todd Young.
All rights reserved. Please address comments to young@math.ohiou.edu.}}
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\begin{enumerate}
 \item Enter the following sequence  of commands:\\
         \verb&   format long&\\
         \verb&   A = hilb(5)&\\
         \verb&   m = eig(A)&\\
         \verb&   m = flipud(m)&

\item Next enter the the following sequence:\\
        \verb&   [Q,R] = qr(A);&\\
        \verb&   A = R*Q;&\\
        \verb&   ma = diag(A);&\\
        \verb&   e = norm(m-ma)&

\item Record the value of $e$. 
	Repeat the steps in the above sequence until the value of
        \verb&e& stops changing. Assume that the errors satisfies 
	$e_{n+1} = Ke_n^r$
        and use the recorded data to solve for $r$ and $K$.

\item Repeat the above experiment for the Pascal matrix generated by: 
	\verb& A = pascal(5)&.

\item Repeat the experiment for a larger matrix.

\item How do the computed values of $r$ and $K$ vary in your experiments?

\item Using complete sentences and standard mathematical notation, write a brief report.
\end{enumerate}

\vfill

\noindent
\textsf{This demonstrates the simplest form of the QR method. Most modern software
including \textsc{Matlab}'s built-in function ``eig" use improved versions of this algorithm. }

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