108 lines
3.8 KiB
TeX
108 lines
3.8 KiB
TeX
% Created 2023-11-27 Mon 15:13
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% Intended LaTeX compiler: pdflatex
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\documentclass[11pt]{article}
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\usepackage[utf8]{inputenc}
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\usepackage[T1]{fontenc}
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\usepackage{graphicx}
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\usepackage{longtable}
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\usepackage{wrapfig}
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\usepackage{rotating}
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\usepackage[normalem]{ulem}
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage{capt-of}
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\usepackage{hyperref}
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\notindent \notag \usepackage{amsmath} \usepackage[a4paper,margin=1in,portrait]{geometry}
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\author{Elizabeth Hunt}
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\date{\today}
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\title{Homework 7}
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\hypersetup{
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pdfauthor={Elizabeth Hunt},
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pdftitle={Homework 7},
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pdfkeywords={},
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pdfsubject={},
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pdfcreator={Emacs 29.1 (Org mode 9.7-pre)},
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pdflang={English}}
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\begin{document}
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\maketitle
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\setlength\parindent{0pt}
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\section{Question One}
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\label{sec:org8ef0ee6}
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See \texttt{UTEST(eigen, dominant\_eigenvalue)} in \texttt{test/eigen.t.c} and the entry
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\texttt{Eigen-Adjacent -> dominant\_eigenvalue} in the LIZFCM API documentation.
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\section{Question Two}
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\label{sec:orgbdba5c1}
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See \texttt{UTEST(eigen, leslie\_matrix\_dominant\_eigenvalue)} in \texttt{test/eigen.t.c}
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and the entry \texttt{Eigen-Adjacent -> leslie\_matrix} in the LIZFCM API
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documentation.
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\section{Question Three}
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\label{sec:org19b04f4}
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See \texttt{UTEST(eigen, least\_dominant\_eigenvalue)} in \texttt{test/eigen.t.c} which
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finds the least dominant eigenvalue on the matrix:
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\begin{bmatrix}
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2 & 2 & 4 \\
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1 & 4 & 7 \\
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0 & 2 & 6
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\end{bmatrix}
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which has eigenvalues: \(5 + \sqrt{17}, 2, 5 - \sqrt{17}\) and should thus produce \(5 - \sqrt{17}\).
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See also the entry \texttt{Eigen-Adjacent -> least\_dominant\_eigenvalue} in the LIZFCM API
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documentation.
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\section{Question Four}
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\label{sec:orgc58d42d}
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See \texttt{UTEST(eigen, shifted\_eigenvalue)} in \texttt{test/eigen.t.c} which
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finds the least dominant eigenvalue on the matrix:
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\begin{bmatrix}
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2 & 2 & 4 \\
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1 & 4 & 7 \\
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0 & 2 & 6
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\end{bmatrix}
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which has eigenvalues: \(5 + \sqrt{17}, 2, 5 - \sqrt{17}\) and should thus produce \(2.0\).
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With the initial guess: \([0.5, 1.0, 0.75]\).
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See also the entry \texttt{Eigen-Adjacent -> shift\_inverse\_power\_eigenvalue} in the LIZFCM API
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documentation.
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\section{Question Five}
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\label{sec:orga369221}
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See \texttt{UTEST(eigen, partition\_find\_eigenvalues)} in \texttt{test/eigen.t.c} which
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finds the eigenvalues in a partition of 10 on the matrix:
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\begin{bmatrix}
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2 & 2 & 4 \\
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1 & 4 & 7 \\
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0 & 2 & 6
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\end{bmatrix}
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which has eigenvalues: \(5 + \sqrt{17}, 2, 5 - \sqrt{17}\), and should produce all three from
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the partitions when given the guesses \([0.5, 1.0, 0.75]\) from the questions above.
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See also the entry \texttt{Eigen-Adjacent -> partition\_find\_eigenvalues} in the LIZFCM API
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documentation.
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\section{Question Six}
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\label{sec:orgadc3078}
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Consider we have the results of two methods developed in this homework: \texttt{least\_dominant\_eigenvalue}, and \texttt{dominant\_eigenvalue}
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into \texttt{lambda\_0}, \texttt{lambda\_n}, respectively. Also assume that we have the method implemented as we've introduced,
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\texttt{shift\_inverse\_power\_eigenvalue}.
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Then, we begin at the midpoint of \texttt{lambda\_0} and \texttt{lambda\_n}, and compute the
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\texttt{new\_lambda = shift\_inverse\_power\_eigenvalue}
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with a shift at the midpoint, and some given initial guess.
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\begin{enumerate}
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\item If the result is equal (or within some tolerance) to \texttt{lambda\_n} then the closest eigenvalue to the midpoint
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is still the dominant eigenvalue, and thus the next most dominant will be on the left. Set \texttt{lambda\_n}
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to the midpoint and reiterate.
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\item If the result is greater or equal to \texttt{lambda\_0} we know an eigenvalue of greater or equal magnitude
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exists on the right. So, we set \texttt{lambda\_0} to this eigenvalue associated with the midpoint, and
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re-iterate.
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\item Continue re-iterating until we hit some given maximum number of iterations. Finally we will return
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\texttt{new\_lambda}.
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\end{enumerate}
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\end{document}
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