Commit 29a68c09 authored by Caoimhe Rooney's avatar Caoimhe Rooney
Browse files

Updated worksheet

parent 5ceaabba
...@@ -16,3 +16,6 @@ ...@@ -16,3 +16,6 @@
\newlabel{fig:ineff}{{3.1}{2}} \newlabel{fig:ineff}{{3.1}{2}}
\@writefile{toc}{\contentsline {section}{\numberline {4}Implementation}{2}} \@writefile{toc}{\contentsline {section}{\numberline {4}Implementation}{2}}
\@writefile{toc}{\contentsline {section}{\numberline {5}Results}{3}} \@writefile{toc}{\contentsline {section}{\numberline {5}Results}{3}}
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\@writefile{lof}{\contentsline {figure}{\numberline {5.2}{\ignorespaces Absolute error of the expected shortest path distance as the sample size increases.\relax }}{3}}
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{/usr/local/texlive/2016/texmf-dist/fonts/enc/dvips/base/8r.enc}</usr/local/tex {/usr/local/texlive/2016/texmf-dist/fonts/enc/dvips/base/8r.enc}</usr/local/tex
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...@@ -832,10 +898,10 @@ live/2016/texmf-dist/fonts/type1/adobe/utopia/putb8a.pfb></usr/local/texlive/20 ...@@ -832,10 +898,10 @@ live/2016/texmf-dist/fonts/type1/adobe/utopia/putb8a.pfb></usr/local/texlive/20
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Output written on worksheet_new.pdf (3 pages, 72189 bytes). Output written on worksheet_new.pdf (3 pages, 176894 bytes).
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...@@ -62,26 +62,39 @@ ...@@ -62,26 +62,39 @@
\newcommand{\horrule}[1]{\rule{\linewidth}{#1}} % Create horizontal rule command with 1 argument of height \newcommand{\horrule}[1]{\rule{\linewidth}{#1}} % Create horizontal rule command with 1 argument of height
\title{ %\title{
\normalfont \normalsize %\normalfont \normalsize
\textsc{Nick Trefethen's Problem Solving Squad} \\ [25pt] % Your university, school and/or department name(s) %\textsc{Nick Trefethen's Problem Solving Squad} \hfill \normalsize{Wednesday 10th May 2017} \\ [25pt] % Your university, school and/or department name(s)
\horrule{0.5pt} \\[0.4cm] % Thin top horizontal rule %\horrule{0.5pt} \\[0.2cm] % Thin top horizontal rule
\huge Problem 2 \\ % The assignment title %\large Problem 2 \hfill Thomas Roy and Caoimhe Rooney\\ % The assignment title
\horrule{2pt} \\[0.5cm] % Thick bottom horizontal rule %\horrule{2pt} \\[0.3cm] % Thick bottom horizontal rule
} %}
\author{Thomas and Caoimhe} % Your name
\date{\normalsize{Wednesday 10th May}} % Today's date or a custom date %\author{Thomas Roy and Caoimhe Rooney} % Your name
\begin{document} \begin{document}
{
\maketitle % Print the title \normalfont \normalsize
\textsc{Nick Trefethen's Problem Solving Squad} \hfill \normalsize{Wednesday 10th May 2017} \\ [10pt] % Your university, school and/or department name(s)
\horrule{0.5pt} \\[0.2cm] % Thin top horizontal rule
\large Problem 2 \hfill Thomas Roy and Caoimhe Rooney\\ % The assignment title
\horrule{2pt} \\[0.3cm] % Thick bottom horizontal rule
}
%---------------------------------------------------------------------------------------- %----------------------------------------------------------------------------------------
% PROBLEM 1 % PROBLEM 1
%---------------------------------------------------------------------------------------- %----------------------------------------------------------------------------------------
\begin{tabular}{cl}
\begin{tabular}{c}
\includegraphics[width=0.3\textwidth]{Unknown.jpg}
\end{tabular}
\begin{tabular}{l}
\parbox{0.5\linewidth}{\textit{"It does not matter how slowly you go as long as you do not stop."}\\
- Confucius}
\end{tabular} \\
\end{tabular}
\section{Problem Recap} \section{Problem Recap}
\textit{"n points are placed at random on the square $[0,1]^2$ (uniform measure). From each point after the first, the shortest possible line is joined to a previous point without crossing any of the lines already drawn. After $n=100$ points, what is the expected minimal length of a curve connecting $(0.0)$ to $(0.5,0.5)$ without crossing any lines?"} \textit{"n points are placed at random on the square $[0,1]^2$ (uniform measure). From each point after the first, the shortest possible line is joined to a previous point without crossing any of the lines already drawn. After $n=100$ points, what is the expected minimal length of a curve connecting $(0.0)$ to $(0.5,0.5)$ without crossing any lines?"}
...@@ -154,6 +167,26 @@ We wrote an algorithm in \texttt{MATLAB} which constructed a graph of $n=100$ no ...@@ -154,6 +167,26 @@ We wrote an algorithm in \texttt{MATLAB} which constructed a graph of $n=100$ no
\item Use Dijkstra's algorithm to find the shortest path. \item Use Dijkstra's algorithm to find the shortest path.
\end{enumerate} \end{enumerate}
\section{Results} \section{Results}
\begin{figure}[h]
\begin{subfigure}{0.5\textwidth}
\centering
\includegraphics[width=1\textwidth]{small_example.eps}
\caption{Example of shortest path for $n=20$.}
\end{subfigure}
\begin{subfigure}{0.5\textwidth}
\centering
\includegraphics[width=1\textwidth]{prob_dist.eps}
\caption{Probability distribution of the shortest path distance for $n=100$ from a sample size $N=5000$.}
\end{subfigure}
\caption{}
\label{fig:ineff}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=0.5\textwidth]{error.eps}
\caption{Absolute error of the expected shortest path distance as the sample size increases.}
\end{figure}
\end{document} \end{document}
\ No newline at end of file
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