%Eigenvalues and eigenvectors %If you modify this file, please indicate here and in the footnote %Math410 \documentclass[12pt]{article} \usepackage{times} \pagestyle{empty} \addtolength{\textwidth}{1.2in} \addtolength{\textheight}{1.2in} \addtolength{\oddsidemargin}{-.58in} \addtolength{\evensidemargin}{-.58in} \renewcommand{\baselinestretch}{1.0} \parindent = 0cm \parskip = .1cm \begin{document} \begin{center} {\Large Eigenvalues by the QR Method\footnote{ Copyright \copyright 2002 Todd Young. All rights reserved. Please address comments to young@math.ohiou.edu.}} \end{center} \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. } \end{document}