%Eigenvalue Power Method %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 Eigenvalue Power 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& v = ones(5,1)&\\ \verb& w = v./norm(v);& \item Next enter the following sequence:\\ \verb& v = A*w;&\\ \verb& w = v./norm(v);&\\ \verb& ma = w'*A*w& \item Repeat the steps in part 2 until the value of \verb&ma& stops changing. How many iteratations did it take? Is this number close to one of the eigenvalues? How close? \item Repeat the above experiment for the Pascal matrix generated by: \verb& A = pascal(5)&. \item Repeat the experiment for a larger matrix. \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}