%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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