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Fixed images and results

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PolishPigeon 2021-12-03 05:56:09 +01:00
parent 774a999868
commit 9355dc9c70
71 changed files with 12021 additions and 7020 deletions
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9
ENUME/projectB/Code/task1Newton.m
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@ -1,12 +1,17 @@
interval = [-5, 10];
rootBrackets = rootBracketing(@taskFunction, interval(1), interval(2));
printGraph(@taskFunction, 'Newton', @newtonMethod, interval, rootBrackets, 'Approximate zeros of function for method of ');
% printGraph(@taskFunction, 'Newton', @newtonMethod, interval, rootBrackets, 'Approximate zeros of function for method of ');
printGraph(@polynomial, 'Newton', @newtonMethod, interval, rootBrackets, 'Approximate zeros of function for method of ');
function y = taskFunction(x)
y = -2.1 + 0.3*x - x*exp(1)^(-x);
end
function y = polynomial(x)
y = -2 * x^4 + 12 * x^3 + 4* x^2 + 1 * x + 3;
end
function [zero, iterations] = newtonMethod(taskFunction, a, b, tolerance)
[iterations, iteration, zero] = initialize(a, b);
[zero, iterations] = newtonLoop(iterations, iteration, zero, a, b, tolerance, taskFunction);

2
ENUME/projectB/Code/task2MM1.m
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@ -3,7 +3,7 @@ rootBrackets = rootBracketing(@polynomial, interval(1), interval(2));
printGraph(@polynomial, 'MM1', @mm1, interval, rootBrackets, 'Approximate zeros of function for method of ');
printComplexGraph(@polynomial, 'MM1', @mm1, [-1 + i, 0], 'Aproximate complex roots of polynomial');
printComplexGraph(@polynomial, 'MM1', @mm1, [-1 - i, 1 + i], 'Aproximate complex roots of polynomial');
function y = polynomial(x)
y = -2 * x^4 + 12 * x^3 + 4* x^2 + 1 * x + 3;

2
ENUME/projectB/Code/task2MM2.m
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@ -3,7 +3,7 @@ rootBrackets = rootBracketing(@polynomial, interval(1), interval(2));
printGraph(@polynomial, 'MM2', @mm2, interval, rootBrackets, 'Approximate zeros of function for method of ');
printComplexGraph(@polynomial, 'MM2', @mm2, [-1 + i, 0], 'Aproximate complex roots of polynomial');
printComplexGraph(@polynomial, 'MM2', @mm2, [-1 - i, 1 + i], 'Aproximate complex roots of polynomial');
% the polynomial function for task 2
function y = polynomial(x)

2
ENUME/projectB/Code/task3.m
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@ -3,7 +3,7 @@ rootBrackets = rootBracketing(@polynomial, interval(1), interval(2));
printGraph(@polynomial, 'Laguerre', @laguerre, interval, rootBrackets, 'Approximate zeros of function for method of ');
printComplexGraph(@polynomial, 'Laguerre', @laguerre, [-1 + i, 0], 'Aproximate complex roots of polynomial');
printComplexGraph(@polynomial, 'Laguerre', @laguerre, [-1 - i, 1 + i], 'Aproximate complex roots of polynomial');
function y = polynomial(x)
y = -2 * x^4 + 12 * x^3 + 4* x^2 + 1 * x + 3;

179
ENUME/projectB/Report/report.aux
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Chapter 2.
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Chapter 3.
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Chapter 4.
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@ -7,31 +7,32 @@
\BOOKMARK [2][-]{subsection.1.2.1}{Problem}{section.1.2}% 7
\BOOKMARK [2][-]{subsection.1.2.2}{Theoretical Introduction}{section.1.2}% 8
\BOOKMARK [2][-]{subsection.1.2.3}{Results}{section.1.2}% 9
\BOOKMARK [0][-]{chapter.2}{Find real and complex roots of the polynomial}{}% 10
\BOOKMARK [1][-]{section.2.1}{Problem}{chapter.2}% 11
\BOOKMARK [1][-]{section.2.2}{Theoretical Introduction}{chapter.2}% 12
\BOOKMARK [2][-]{subsection.2.2.1}{MM1}{section.2.2}% 13
\BOOKMARK [2][-]{subsection.2.2.2}{MM2}{section.2.2}% 14
\BOOKMARK [1][-]{section.2.3}{Results}{chapter.2}% 15
\BOOKMARK [2][-]{subsection.2.3.1}{Graphs for MM1}{section.2.3}% 16
\BOOKMARK [2][-]{subsection.2.3.2}{Graphs for MM2}{section.2.3}% 17
\BOOKMARK [2][-]{subsection.2.3.3}{Tables for MM1}{section.2.3}% 18
\BOOKMARK [2][-]{subsection.2.3.4}{Tables for MM2}{section.2.3}% 19
\BOOKMARK [2][-]{subsection.2.3.5}{Comparison of results between MM1 and MM2}{section.2.3}% 20
\BOOKMARK [2][-]{subsection.2.3.6}{Comparison of results between Newton's method and MM2}{section.2.3}% 21
\BOOKMARK [0][-]{chapter.3}{Find real and complex roots of the polynomial using Laguerre's method}{}% 22
\BOOKMARK [1][-]{section.3.1}{Problem}{chapter.3}% 23
\BOOKMARK [1][-]{section.3.2}{Theoretical Introduction}{chapter.3}% 24
\BOOKMARK [1][-]{section.3.3}{Results}{chapter.3}% 25
\BOOKMARK [2][-]{subsection.3.3.1}{Comparison of results between MM1 and MM2}{section.3.3}% 26
\BOOKMARK [0][-]{chapter.4}{Code appendix}{}% 27
\BOOKMARK [1][-]{section.4.1}{Task 1}{chapter.4}% 28
\BOOKMARK [2][-]{subsection.4.1.1}{task1Bisection.m}{section.4.1}% 29
\BOOKMARK [2][-]{subsection.4.1.2}{task1Newton.m}{section.4.1}% 30
\BOOKMARK [1][-]{section.4.2}{Task 2}{chapter.4}% 31
\BOOKMARK [2][-]{subsection.4.2.1}{task2MM1.m}{section.4.2}% 32
\BOOKMARK [2][-]{subsection.4.2.2}{task2MM2.m}{section.4.2}% 33
\BOOKMARK [1][-]{section.4.3}{Task 3}{chapter.4}% 34
\BOOKMARK [2][-]{subsection.4.3.1}{rootBrackering.m}{section.4.3}% 35
\BOOKMARK [2][-]{subsection.4.3.2}{printGraph.m}{section.4.3}% 36
\BOOKMARK [2][-]{subsection.4.3.3}{printComplexGraph.m}{section.4.3}% 37
\BOOKMARK [2][-]{subsection.1.2.4}{Discussion of results}{section.1.2}% 10
\BOOKMARK [0][-]{chapter.2}{Find real and complex roots of the polynomial}{}% 11
\BOOKMARK [1][-]{section.2.1}{Problem}{chapter.2}% 12
\BOOKMARK [1][-]{section.2.2}{Theoretical Introduction}{chapter.2}% 13
\BOOKMARK [2][-]{subsection.2.2.1}{MM1}{section.2.2}% 14
\BOOKMARK [2][-]{subsection.2.2.2}{MM2}{section.2.2}% 15
\BOOKMARK [1][-]{section.2.3}{Results}{chapter.2}% 16
\BOOKMARK [2][-]{subsection.2.3.1}{Graphs for MM1}{section.2.3}% 17
\BOOKMARK [2][-]{subsection.2.3.2}{Graphs for MM2}{section.2.3}% 18
\BOOKMARK [2][-]{subsection.2.3.3}{Tables for MM1}{section.2.3}% 19
\BOOKMARK [2][-]{subsection.2.3.4}{Tables for MM2}{section.2.3}% 20
\BOOKMARK [2][-]{subsection.2.3.5}{Comparison of results between MM1 and MM2}{section.2.3}% 21
\BOOKMARK [2][-]{subsection.2.3.6}{Comparison of results between Newton's method and MM2}{section.2.3}% 22
\BOOKMARK [0][-]{chapter.3}{Find real and complex roots of the polynomial using Laguerre's method}{}% 23
\BOOKMARK [1][-]{section.3.1}{Problem}{chapter.3}% 24
\BOOKMARK [1][-]{section.3.2}{Theoretical Introduction}{chapter.3}% 25
\BOOKMARK [1][-]{section.3.3}{Results}{chapter.3}% 26
\BOOKMARK [2][-]{subsection.3.3.1}{Comparison of results between MM1 and MM2}{section.3.3}% 27
\BOOKMARK [0][-]{chapter.4}{Code appendix}{}% 28
\BOOKMARK [1][-]{section.4.1}{Task 1}{chapter.4}% 29
\BOOKMARK [2][-]{subsection.4.1.1}{task1Bisection.m}{section.4.1}% 30
\BOOKMARK [2][-]{subsection.4.1.2}{task1Newton.m}{section.4.1}% 31
\BOOKMARK [1][-]{section.4.2}{Task 2}{chapter.4}% 32
\BOOKMARK [2][-]{subsection.4.2.1}{task2MM1.m}{section.4.2}% 33
\BOOKMARK [2][-]{subsection.4.2.2}{task2MM2.m}{section.4.2}% 34
\BOOKMARK [1][-]{section.4.3}{Task 3}{chapter.4}% 35
\BOOKMARK [2][-]{subsection.4.3.1}{rootBrackering.m}{section.4.3}% 36
\BOOKMARK [2][-]{subsection.4.3.2}{printGraph.m}{section.4.3}% 37
\BOOKMARK [2][-]{subsection.4.3.3}{printComplexGraph.m}{section.4.3}% 38

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@ -274,6 +274,19 @@ Iteration & root & f(root) \\
\end{tabular}
\end{center}
\subsection{Discussion of results}
As we can see and as we could expect from theory Newton method reigns supreme.
It is:
\begin{enumerate}
\item Faster.
Especially for finding first root. It took 36 iterations for false position method to find first root with accuary up to $10^{-10}$ and it took only 6 iterations for the Newton method.
\item More accurate
Not only that we also got more accurate results with Newton method! With first root being so close to the real answer that the matlab calculated the value of this function at this point as exactly equal to 0. For second root the accuary is $10^{6}$ better than for the false position method.
\item Smaller gaps
Looking at graphs we got from both method we can clearly see that the gaps between the final result and each and every step are much smaller for Newton method. This again proves that it is faster and more accurate compared to false position method.
\end{enumerate}
\chapter{Find real and complex roots of the polynomial}
@ -357,52 +370,60 @@ This method is locally convergent with order of convergence equal to 1.84. It is
\subsection{Graphs for MM1}
\begin{center}
\includegraphics[scale=0.25]{task2mm1overall.eps}
\includegraphics[scale=0.25]{task2mm1realoverall.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task2mm1zommedleft.eps}
\includegraphics[scale=0.25]{task2mm1realleft.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task2mm1zommedright.eps}
\includegraphics[scale=0.25]{task2mm1realright.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task2mm1complexzommedleft.eps}
\includegraphics[scale=0.25]{task2mm1complexoverall.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task2mm1complexzommeddown.eps}
\includegraphics[scale=0.25]{task2mm1complexdown.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task2mm1complexzommedup.eps}
\includegraphics[scale=0.25]{task2mm1complexmiddle.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task2mm1complexup.eps}
\end{center}
\subsection{Graphs for MM2}
\begin{center}
\includegraphics[scale=0.25]{task2mm2overall.eps}
\includegraphics[scale=0.25]{task2mm2realoverall.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task2mm2zommedleft.eps}
\includegraphics[scale=0.25]{task2mm2realleft.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task2mm2zommedright.eps}
\includegraphics[scale=0.25]{task2mm2realright.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task2mm2complexzommedleft.eps}
\includegraphics[scale=0.25]{task2mm2complexoverall.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task2mm2complexzommedmiddle.eps}
\includegraphics[scale=0.25]{task2mm2complexright.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task2mm2complexzommedup.eps}
\includegraphics[scale=0.25]{task2mm2complexleft.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task2mm2complexup.eps}
\end{center}
\subsection{Tables for MM1}
@ -411,21 +432,22 @@ This method is locally convergent with order of convergence equal to 1.84. It is
\hline
Iteration & root & f(root) \\
\hline
1 & -1.25 & -20.32 \\
1 & -1.25 & -20.32 \\
\hline
2 & -1.0009 & -8.0351 \\
2 & -1.0009 & -8.0351 \\
\hline
3 & -0.76855 & -1.5512 \\
3 & -0.76855 & -1.5512 \\
\hline
4 & -0.63496 & 0.58064 \\
4 & -0.63496 & 0.58064 \\
\hline
5 & -0.67945 & -0.023155 \\
5 & -0.67945 & -0.023155 \\
\hline
6 & -0.67788 & -0.00010725 \\
6 & -0.67788 & -0.00010725 \\
\hline
7 & -0.67787 & -8.6306e-09 \\
7 & -0.67787 & -8.6306e-09 \\
\hline
8 & -0.67787 & 0 \\
\hline
8 & -0.67787 & 0 \\
\hline
\end{tabular}
@ -440,11 +462,13 @@ Iteration & root & f(root) \\
\hline
2 & 6.3257 & 4.4966 \\
\hline
3 & 6.3341 & 0.033379 \\
3 & 6.3341 & 0.033379 \\
\hline
4 & 6.3341 & -1.6916e-06 \\
4 & 6.3341 & -1.6916e-06 \\
\hline
5 & 6.3341 & -7.1942e-14 \\
5 & 6.3341 & -7.1942e-14 \\
\hline
\hline
\end{tabular}
@ -455,21 +479,24 @@ Iteration & root & f(root) \\
\hline
Iteration & root & f(root) \\
\hline
1 & -0.5-0.5i & 6.1847 \\
1 & 0+0i & 3 \\
\hline
2 & -0.42823-0.15033i & 2.6872 \\
2 & -0.0072549-0.12193i & 2.9384 \\
\hline
3 & -0.59202+0.094367i & 1.6012 \\
3 & 0.069406-0.45062i & 1.7235 \\
\hline
4 & -0.71249-0.0089336i & 0.55524 \\
4 & 0.19733-0.61224i & 0.82578 \\
\hline
5 & -0.67654+0.00084164i & 0.022962 \\
5 & 0.16937-0.56334i & 0.047116 \\
\hline
6 & -0.67787+8.4773e-06i & 0.00012412 \\
6 & 0.17186-0.56551i & 0.0003142 \\
\hline
7 & -0.67787+3.6458e-10i & 8.3668e-09 \\
7 & 0.17188-0.56551i & 4.4741e-08 \\
\hline
8 & 0.17188-0.56551i & 2.6645e-15 \\
\hline
9 & 0.17188-0.56551i & 0 \\
\hline
8 & -0.67787+9.1168e-18i & 1.3321e-16 \\
\hline
\end{tabular}
@ -481,15 +508,16 @@ Iteration & root & f(root) \\
\hline
Iteration & root & f(root) \\
\hline
1 & -1.25+0i & -20.32+0i \\
\hline
2 & -0.72604-0.25326i & 1.0515-4.0744i \\
\hline
3 & -0.68613-0.01483i & -0.11667-0.22303i \\
\hline
4 & -0.67788+2.5407e-06i & -0.00010163+3.7123e-05i\\
\hline
5 & -0.67787-8.0819e-13i & 1.4105e-11-1.1809e-11i \\
1 & -1.25+0i & -20.32+0i \\
\hline
2 & -0.72604-0.25326i & 1.0515-4.0744i \\
\hline
3 & -0.68613-0.01483i & -0.11667-0.22303i \\
\hline
4 & -0.67788+2.5407e-06i & -0.00010163+3.7123e-05i \\
\hline
5 & -0.67787-8.0819e-13i & 1.4105e-11-1.1809e-11i \\
\hline
\hline
\end{tabular}
@ -500,15 +528,16 @@ Iteration & root & f(root) \\
\hline
Iteration & root & f(root) \\
\hline
1 & 6.25 & 43.43 \\
1 & 6.25 & 43.43 \\
\hline
2 & 6.3376 & -1.8647 \\
2 & 6.3376 & -1.8647 \\
\hline
3 & 6.3341 & 0.0029695 \\
3 & 6.3341 & 0.0029695 \\
\hline
4 & 6.3341 & 1.6294e-08 \\
4 & 6.3341 & 1.6294e-08 \\
\hline
5 & 6.3341 & -4.6896e-13 \\
\hline
5 & 6.3341 & -4.6896e-13 \\
\hline
\end{tabular}
@ -519,21 +548,27 @@ Iteration & root & f(root) \\
\hline
Iteration & root & f(root) \\
\hline
1 & -0.5-0.5i & 6.1847 \\
1 & 0+0i & 3 \\
\hline
2 & -0.60346-0.076584i & 1.3673 \\
2 & -0.125+0.85696i & 8.0341 \\
\hline
3 & -0.67709+0.001186i & 0.020767 \\
3 & 0.16892+0.62593i & 0.94053 \\
\hline
4 & -0.67787+1.518e-08i & 2.2673e-07 \\
4 & 0.17203+0.56536i & 0.0031354 \\
\hline
5 & 0.17188+0.56551i & 3.8615e-09 \\
\hline
6 & 0.17188+0.56551i & 0 \\
\hline
5 & -0.67787-2.0556e-17i & 3.0035e-16 \\
\hline
\end{tabular}
\end{center}
\subsection{Comparison of results between MM1 and MM2}
As we can see MM2 method is much better, it took on average 5 iterations for it to find a root with sufficient accuracy compared to around 7 iterations that MM1 needed to find roots. But MM1 is slightly more accurate. There was also fatal error for one of the complex roots for MM1 where the algorithm failed to calculate the complex part. MM2 correctly calculated complex root while MM1 calculated only real part of the result.
We can also observe that on graphs with MM2 approaching the root faster than MM1 and MM1 being slightly more accurate.
\subsection{Comparison of results between Newton's method and MM2}
\chapter{Find real and complex roots of the polynomial using Laguerre's method}
@ -573,7 +608,11 @@ This formula is similar to the one from MM2 but also takes order of the polynomi
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task3complexdown.eps}
\includegraphics[scale=0.25]{task3complexleft.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task3complexright.eps}
\end{center}
\begin{center}
@ -587,11 +626,12 @@ Iteration & root & f(root) \\
\hline
1 & -1.25 & -20.32 \\
\hline
2 & -0.67905 & -0.017268 \\
2 & -0.67905 & -0.017268 \\
\hline
3 & -0.67787 & -3.1489e-07 \\
\hline
4 & -0.67787 & 0 \\
4 & -0.67787 & 0 \\
\hline
\hline
\hline
@ -603,16 +643,18 @@ Iteration & root & f(root) \\
\hline
Iteration & root & f(root) \\
\hline
1 & -1.25 & -20.32 \\
1 & 6.25 & 43.43 \\
\hline
2 & -0.67905 & -0.017268 \\
2 & 6.3376 & -1.8647 \\
\hline
3 & -0.67787 & -3.1489e-07 \\
3 & 6.3341 & 0.0029631 \\
\hline
4 & -0.67787 & 0 \\
4 & 6.3341 & 7.9942e-09\\
\hline
5 & 6.3341 & -7.1942e-14\\
\hline
\end{tabular}
\end{center}
@ -622,15 +664,19 @@ Iteration & root & f(root) \\
\hline
Iteration & root & f(root) \\
\hline
1 & -1.25 & -20.32 \\
1 & 0+0i & 3 \\
\hline
2 & -0.67905 & -0.017268 \\
2 & -0.042857+0.71586i & 4.1826 \\
\hline
3 & -0.67787 & -3.1489e-07 \\
3 & 0.172+0.56266i & 0.040584 \\
\hline
4 & -0.67787 & 0 \\
4 & 0.17188+0.56551i & 2.8484e-06 \\
\hline
5 & 0.17188+0.56551i & 2.2952e-14 \\
\hline
6 & 0.17188+0.56551i & 0 \\
\hline
\end{tabular}
\end{center}
@ -882,7 +928,7 @@ rootBrackets = rootBracketing(@polynomial, interval(1), interval(2));
printGraph(@polynomial, 'MM1', @mm1, interval, rootBrackets, 'Approximate zeros of function for method of ');
printComplexGraph(@polynomial, 'MM1', @mm1, [-1 + i, 0], 'Aproximate complex roots of polynomial');
printComplexGraph(@polynomial, 'MM1', @mm1, [-1 - i, 1 + i], 'Aproximate complex roots of polynomial');
\end{lstlisting}
\end{simplechar}
@ -1083,7 +1129,7 @@ rootBrackets = rootBracketing(@polynomial, interval(1), interval(2));
printGraph(@polynomial, 'MM2', @mm2, interval, rootBrackets, 'Approximate zeros of function for method of ');
printComplexGraph(@polynomial, 'MM2', @mm2, [-1 + i, 0], 'Aproximate complex roots of polynomial');
printComplexGraph(@polynomial, 'MM2', @mm2, [-1 - i, 1 + i], 'Aproximate complex roots of polynomial');
\end{lstlisting}
\end{simplechar}
@ -1210,7 +1256,7 @@ rootBrackets = rootBracketing(@polynomial, interval(1), interval(2));
printGraph(@polynomial, 'Laguerre', @laguerre, interval, rootBrackets, 'Approximate zeros of function for method of ');
printComplexGraph(@polynomial, 'Laguerre', @laguerre, [-1 + i, 0], 'Aproximate complex roots of polynomial');
printComplexGraph(@polynomial, 'Laguerre', @laguerre, [-1 - i, 1 + i], 'Aproximate complex roots of polynomial');
\end{lstlisting}
\end{simplechar}

177
ENUME/projectB/Report/report.toc
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@ -9,91 +9,92 @@
\contentsline {subsection}{\numberline {1.2.1}Problem}{8}{subsection.1.2.1}%
\contentsline {subsection}{\numberline {1.2.2}Theoretical Introduction}{8}{subsection.1.2.2}%
\contentsline {subsection}{\numberline {1.2.3}Results}{9}{subsection.1.2.3}%
\contentsline {chapter}{\numberline {2}Find real and complex roots of the polynomial}{11}{chapter.2}%
\contentsline {section}{\numberline {2.1}Problem}{11}{section.2.1}%
\contentsline {section}{\numberline {2.2}Theoretical Introduction}{11}{section.2.2}%
\contentsline {subsection}{\numberline {2.2.1}MM1}{12}{subsection.2.2.1}%
\contentsline {subsection}{\numberline {2.2.2}MM2}{13}{subsection.2.2.2}%
\contentsline {section}{\numberline {2.3}Results}{14}{section.2.3}%
\contentsline {subsection}{\numberline {2.3.1}Graphs for MM1}{14}{subsection.2.3.1}%
\contentsline {subsection}{\numberline {2.3.2}Graphs for MM2}{16}{subsection.2.3.2}%
\contentsline {subsection}{\numberline {2.3.3}Tables for MM1}{18}{subsection.2.3.3}%
\contentsline {subsection}{\numberline {2.3.4}Tables for MM2}{19}{subsection.2.3.4}%
\contentsline {subsection}{\numberline {2.3.5}Comparison of results between MM1 and MM2}{20}{subsection.2.3.5}%
\contentsline {subsection}{\numberline {2.3.6}Comparison of results between Newton's method and MM2}{20}{subsection.2.3.6}%
\contentsline {chapter}{\numberline {3}Find real and complex roots of the polynomial using Laguerre's method}{21}{chapter.3}%
\contentsline {section}{\numberline {3.1}Problem}{21}{section.3.1}%
\contentsline {section}{\numberline {3.2}Theoretical Introduction}{21}{section.3.2}%
\contentsline {section}{\numberline {3.3}Results}{22}{section.3.3}%
\contentsline {subsection}{\numberline {3.3.1}Comparison of results between MM1 and MM2}{24}{subsection.3.3.1}%
\contentsline {chapter}{\numberline {4}Code appendix}{25}{chapter.4}%
\contentsline {section}{\numberline {4.1}Task 1}{25}{section.4.1}%
\contentsline {subsection}{\numberline {4.1.1}task1Bisection.m}{25}{subsection.4.1.1}%
\contentsline {subsubsection}{Top of task1Bisection.m}{25}{section*.4}%
\contentsline {subsubsection}{taskFunction}{25}{section*.5}%
\contentsline {subsubsection}{falsePosition}{26}{section*.6}%
\contentsline {subsubsection}{initialize}{26}{section*.7}%
\contentsline {subsubsection}{taskFunction}{26}{section*.8}%
\contentsline {subsubsection}{firstTwoIterations}{27}{section*.9}%
\contentsline {subsubsection}{falsePositionLoop}{27}{section*.10}%
\contentsline {subsubsection}{insideLoop}{28}{section*.11}%
\contentsline {subsubsection}{changeLastTwoAB}{28}{section*.12}%
\contentsline {subsubsection}{calculateZero}{29}{section*.13}%
\contentsline {subsubsection}{newSubInterval}{29}{section*.14}%
\contentsline {subsection}{\numberline {4.1.2}task1Newton.m}{29}{subsection.4.1.2}%
\contentsline {subsubsection}{Top of task1Newton.m}{29}{section*.15}%
\contentsline {subsubsection}{taskFunction}{30}{section*.16}%
\contentsline {subsubsection}{newtonMethod}{30}{section*.17}%
\contentsline {subsubsection}{initialize}{30}{section*.18}%
\contentsline {subsubsection}{newtonLoop}{30}{section*.19}%
\contentsline {subsubsection}{insideLoop}{31}{section*.20}%
\contentsline {subsubsection}{calculateZeroIterations}{31}{section*.21}%
\contentsline {subsubsection}{checkForDivergence}{31}{section*.22}%
\contentsline {section}{\numberline {4.2}Task 2}{32}{section.4.2}%
\contentsline {subsection}{\numberline {4.2.1}task2MM1.m}{32}{subsection.4.2.1}%
\contentsline {subsubsection}{Top of task2MM1}{32}{section*.23}%
\contentsline {subsubsection}{polynomial}{32}{section*.24}%
\contentsline {subsubsection}{mm1}{32}{section*.25}%
\contentsline {subsubsection}{initialize}{33}{section*.26}%
\contentsline {subsubsection}{mm1Loop}{33}{section*.27}%
\contentsline {subsubsection}{insideLoop}{33}{section*.28}%
\contentsline {subsubsection}{createEquationSystem}{34}{section*.29}%
\contentsline {subsubsection}{rootsOfQuadraticFormula}{34}{section*.30}%
\contentsline {subsubsection}{updateApproximations}{34}{section*.31}%
\contentsline {subsubsection}{initializeEquationSystem}{35}{section*.32}%
\contentsline {subsubsection}{solveEquationSystem}{35}{section*.33}%
\contentsline {subsubsection}{createApproximatedQuadraticFormula}{36}{section*.34}%
\contentsline {subsubsection}{findRootsOfQuadraticFormula}{36}{section*.35}%
\contentsline {subsubsection}{chooseNewRoot}{36}{section*.36}%
\contentsline {subsubsection}{addZeroToIterationVector}{36}{section*.37}%
\contentsline {subsubsection}{getWorstApproximationIndex}{37}{section*.38}%
\contentsline {subsubsection}{deleteWorstApproximation}{37}{section*.39}%
\contentsline {subsection}{\numberline {4.2.2}task2MM2.m}{38}{subsection.4.2.2}%
\contentsline {subsubsection}{top of task2MM2.m}{38}{section*.40}%
\contentsline {subsubsection}{polynomial}{38}{section*.41}%
\contentsline {subsubsection}{mm2}{38}{section*.42}%
\contentsline {subsubsection}{initialize}{38}{section*.43}%
\contentsline {subsubsection}{mm2Loop}{39}{section*.44}%
\contentsline {subsubsection}{insideLoop}{39}{section*.45}%
\contentsline {subsubsection}{getABC}{39}{section*.46}%
\contentsline {subsubsection}{findRoots}{39}{section*.47}%
\contentsline {subsubsection}{chooseNewApproximation}{40}{section*.48}%
\contentsline {subsubsection}{updateApproximations}{40}{section*.49}%
\contentsline {subsubsection}{derivative}{40}{section*.50}%
\contentsline {section}{\numberline {4.3}Task 3}{41}{section.4.3}%
\contentsline {subsubsection}{Top of Task 3}{41}{section*.51}%
\contentsline {subsubsection}{polynomial}{41}{section*.52}%
\contentsline {subsubsection}{laguerre}{41}{section*.53}%
\contentsline {subsubsection}{initialize}{42}{section*.54}%
\contentsline {subsubsection}{laguerreLoop}{42}{section*.55}%
\contentsline {subsubsection}{insideLoop}{42}{section*.56}%
\contentsline {subsubsection}{calculateDerrivatives}{43}{section*.57}%
\contentsline {subsubsection}{calculateZ}{43}{section*.58}%
\contentsline {subsubsection}{chooseNewZero}{43}{section*.59}%
\contentsline {subsubsection}{updateZeros}{44}{section*.60}%
\contentsline {subsection}{\numberline {4.3.1}rootBrackering.m}{44}{subsection.4.3.1}%
\contentsline {subsubsection}{updateZeros}{44}{section*.61}%
\contentsline {subsubsection}{initializeValues}{45}{section*.62}%
\contentsline {subsubsection}{bracketingLoop}{46}{section*.63}%
\contentsline {subsection}{\numberline {4.3.2}printGraph.m}{47}{subsection.4.3.2}%
\contentsline {subsection}{\numberline {4.3.3}printComplexGraph.m}{49}{subsection.4.3.3}%
\contentsline {subsection}{\numberline {1.2.4}Discussion of results}{10}{subsection.1.2.4}%
\contentsline {chapter}{\numberline {2}Find real and complex roots of the polynomial}{12}{chapter.2}%
\contentsline {section}{\numberline {2.1}Problem}{12}{section.2.1}%
\contentsline {section}{\numberline {2.2}Theoretical Introduction}{12}{section.2.2}%
\contentsline {subsection}{\numberline {2.2.1}MM1}{13}{subsection.2.2.1}%
\contentsline {subsection}{\numberline {2.2.2}MM2}{14}{subsection.2.2.2}%
\contentsline {section}{\numberline {2.3}Results}{15}{section.2.3}%
\contentsline {subsection}{\numberline {2.3.1}Graphs for MM1}{15}{subsection.2.3.1}%
\contentsline {subsection}{\numberline {2.3.2}Graphs for MM2}{18}{subsection.2.3.2}%
\contentsline {subsection}{\numberline {2.3.3}Tables for MM1}{20}{subsection.2.3.3}%
\contentsline {subsection}{\numberline {2.3.4}Tables for MM2}{21}{subsection.2.3.4}%
\contentsline {subsection}{\numberline {2.3.5}Comparison of results between MM1 and MM2}{22}{subsection.2.3.5}%
\contentsline {subsection}{\numberline {2.3.6}Comparison of results between Newton's method and MM2}{22}{subsection.2.3.6}%
\contentsline {chapter}{\numberline {3}Find real and complex roots of the polynomial using Laguerre's method}{23}{chapter.3}%
\contentsline {section}{\numberline {3.1}Problem}{23}{section.3.1}%
\contentsline {section}{\numberline {3.2}Theoretical Introduction}{23}{section.3.2}%
\contentsline {section}{\numberline {3.3}Results}{24}{section.3.3}%
\contentsline {subsection}{\numberline {3.3.1}Comparison of results between MM1 and MM2}{27}{subsection.3.3.1}%
\contentsline {chapter}{\numberline {4}Code appendix}{28}{chapter.4}%
\contentsline {section}{\numberline {4.1}Task 1}{28}{section.4.1}%
\contentsline {subsection}{\numberline {4.1.1}task1Bisection.m}{28}{subsection.4.1.1}%
\contentsline {subsubsection}{Top of task1Bisection.m}{28}{section*.4}%
\contentsline {subsubsection}{taskFunction}{28}{section*.5}%
\contentsline {subsubsection}{falsePosition}{29}{section*.6}%
\contentsline {subsubsection}{initialize}{29}{section*.7}%
\contentsline {subsubsection}{taskFunction}{29}{section*.8}%
\contentsline {subsubsection}{firstTwoIterations}{30}{section*.9}%
\contentsline {subsubsection}{falsePositionLoop}{30}{section*.10}%
\contentsline {subsubsection}{insideLoop}{31}{section*.11}%
\contentsline {subsubsection}{changeLastTwoAB}{31}{section*.12}%
\contentsline {subsubsection}{calculateZero}{32}{section*.13}%
\contentsline {subsubsection}{newSubInterval}{32}{section*.14}%
\contentsline {subsection}{\numberline {4.1.2}task1Newton.m}{32}{subsection.4.1.2}%
\contentsline {subsubsection}{Top of task1Newton.m}{32}{section*.15}%
\contentsline {subsubsection}{taskFunction}{33}{section*.16}%
\contentsline {subsubsection}{newtonMethod}{33}{section*.17}%
\contentsline {subsubsection}{initialize}{33}{section*.18}%
\contentsline {subsubsection}{newtonLoop}{33}{section*.19}%
\contentsline {subsubsection}{insideLoop}{34}{section*.20}%
\contentsline {subsubsection}{calculateZeroIterations}{34}{section*.21}%
\contentsline {subsubsection}{checkForDivergence}{34}{section*.22}%
\contentsline {section}{\numberline {4.2}Task 2}{35}{section.4.2}%
\contentsline {subsection}{\numberline {4.2.1}task2MM1.m}{35}{subsection.4.2.1}%
\contentsline {subsubsection}{Top of task2MM1}{35}{section*.23}%
\contentsline {subsubsection}{polynomial}{35}{section*.24}%
\contentsline {subsubsection}{mm1}{35}{section*.25}%
\contentsline {subsubsection}{initialize}{36}{section*.26}%
\contentsline {subsubsection}{mm1Loop}{36}{section*.27}%
\contentsline {subsubsection}{insideLoop}{36}{section*.28}%
\contentsline {subsubsection}{createEquationSystem}{37}{section*.29}%
\contentsline {subsubsection}{rootsOfQuadraticFormula}{37}{section*.30}%
\contentsline {subsubsection}{updateApproximations}{37}{section*.31}%
\contentsline {subsubsection}{initializeEquationSystem}{38}{section*.32}%
\contentsline {subsubsection}{solveEquationSystem}{38}{section*.33}%
\contentsline {subsubsection}{createApproximatedQuadraticFormula}{39}{section*.34}%
\contentsline {subsubsection}{findRootsOfQuadraticFormula}{39}{section*.35}%
\contentsline {subsubsection}{chooseNewRoot}{39}{section*.36}%
\contentsline {subsubsection}{addZeroToIterationVector}{39}{section*.37}%
\contentsline {subsubsection}{getWorstApproximationIndex}{40}{section*.38}%
\contentsline {subsubsection}{deleteWorstApproximation}{40}{section*.39}%
\contentsline {subsection}{\numberline {4.2.2}task2MM2.m}{41}{subsection.4.2.2}%
\contentsline {subsubsection}{top of task2MM2.m}{41}{section*.40}%
\contentsline {subsubsection}{polynomial}{41}{section*.41}%
\contentsline {subsubsection}{mm2}{41}{section*.42}%
\contentsline {subsubsection}{initialize}{41}{section*.43}%
\contentsline {subsubsection}{mm2Loop}{42}{section*.44}%
\contentsline {subsubsection}{insideLoop}{42}{section*.45}%
\contentsline {subsubsection}{getABC}{42}{section*.46}%
\contentsline {subsubsection}{findRoots}{42}{section*.47}%
\contentsline {subsubsection}{chooseNewApproximation}{43}{section*.48}%
\contentsline {subsubsection}{updateApproximations}{43}{section*.49}%
\contentsline {subsubsection}{derivative}{43}{section*.50}%
\contentsline {section}{\numberline {4.3}Task 3}{44}{section.4.3}%
\contentsline {subsubsection}{Top of Task 3}{44}{section*.51}%
\contentsline {subsubsection}{polynomial}{44}{section*.52}%
\contentsline {subsubsection}{laguerre}{44}{section*.53}%
\contentsline {subsubsection}{initialize}{45}{section*.54}%
\contentsline {subsubsection}{laguerreLoop}{45}{section*.55}%
\contentsline {subsubsection}{insideLoop}{45}{section*.56}%
\contentsline {subsubsection}{calculateDerrivatives}{46}{section*.57}%
\contentsline {subsubsection}{calculateZ}{46}{section*.58}%
\contentsline {subsubsection}{chooseNewZero}{46}{section*.59}%
\contentsline {subsubsection}{updateZeros}{47}{section*.60}%
\contentsline {subsection}{\numberline {4.3.1}rootBrackering.m}{47}{subsection.4.3.1}%
\contentsline {subsubsection}{updateZeros}{47}{section*.61}%
\contentsline {subsubsection}{initializeValues}{48}{section*.62}%
\contentsline {subsubsection}{bracketingLoop}{49}{section*.63}%
\contentsline {subsection}{\numberline {4.3.2}printGraph.m}{50}{subsection.4.3.2}%
\contentsline {subsection}{\numberline {4.3.3}printComplexGraph.m}{52}{subsection.4.3.3}%

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N
1587.74 878 M
1587.74 74 L
1334.257 878 M
1334.257 74 L
S
GR
GS
@ -481,8 +481,8 @@ GS
1 LJ
0.694 LW
N
1733.998 874.034 M
248.997 874.034 L
1467.589 878 M
1467.589 74 L
S
GR
GS
@ -491,8 +491,8 @@ GS
1 LJ
0.694 LW
N
1733.998 759.451 M
248.997 759.451 L
1600.926 878 M
1600.926 74 L
S
GR
GS
@ -501,8 +501,8 @@ GS
1 LJ
0.694 LW
N
1733.998 644.867 M
248.997 644.867 L
1733.997 855.384 M
248.997 855.384 L
S
GR
GS
@ -511,8 +511,8 @@ GS
1 LJ
0.694 LW
N
1733.998 530.284 M
248.997 530.284 L
1733.997 736.996 M
248.997 736.996 L
S
GR
GS
@ -521,8 +521,8 @@ GS
1 LJ
0.694 LW
N
1733.998 415.701 M
248.997 415.701 L
1733.997 618.608 M
248.997 618.608 L
S
GR
GS
@ -531,8 +531,8 @@ GS
1 LJ
0.694 LW
N
1733.998 301.118 M
248.997 301.118 L
1733.997 500.22 M
248.997 500.22 L
S
GR
GS
@ -541,8 +541,28 @@ GS
1 LJ
0.694 LW
N
1733.998 186.534 M
248.997 186.534 L
1733.997 381.832 M
248.997 381.832 L
S
GR
GS
[0.72 0 0 0.72 0 0.08002] CT
0.873 GC
1 LJ
0.694 LW
N
1733.997 263.444 M
248.997 263.444 L
S
GR
GS
[0.72 0 0 0.72 0 0.08002] CT
0.873 GC
1 LJ
0.694 LW
N
1733.997 145.056 M
248.997 145.056 L
S
GR
GS
@ -552,8 +572,8 @@ GS
1 LJ
0.694 LW
N
248.997 644.867 M
1733.998 644.867 L
248.997 500.22 M
1733.997 500.22 L
S
GR
GS
@ -563,8 +583,8 @@ GS
1 LJ
0.694 LW
N
319.559 644.867 M
319.559 630.017 L
267.575 500.22 M
267.575 485.37 L
S
GR
GS
@ -574,8 +594,8 @@ GS
1 LJ
0.694 LW
N
478.086 644.867 M
478.086 630.017 L
400.913 500.22 M
400.913 485.37 L
S
GR
GS
@ -585,8 +605,8 @@ GS
1 LJ
0.694 LW
N
636.605 644.867 M
636.605 630.017 L
534.25 500.22 M
534.25 485.37 L
S
GR
GS
@ -596,8 +616,8 @@ GS
1 LJ
0.694 LW
N
795.131 644.867 M
795.131 630.017 L
667.582 500.22 M
667.582 485.37 L
S
GR
GS
@ -607,8 +627,8 @@ GS
1 LJ
0.694 LW
N
953.65 644.867 M
953.65 630.017 L
800.919 500.22 M
800.919 485.37 L
S
GR
GS
@ -618,8 +638,8 @@ GS
1 LJ
0.694 LW
N
1112.176 644.867 M
1112.176 630.017 L
934.251 500.22 M
934.251 485.37 L
S
GR
GS
@ -629,8 +649,8 @@ GS
1 LJ
0.694 LW
N
1270.695 644.867 M
1270.695 630.017 L
1067.588 500.22 M
1067.588 485.37 L
S
GR
GS
@ -640,8 +660,8 @@ GS
1 LJ
0.694 LW
N
1429.221 644.867 M
1429.221 630.017 L
1200.92 500.22 M
1200.92 485.37 L
S
GR
GS
@ -651,12 +671,45 @@ GS
1 LJ
0.694 LW
N
1587.74 644.867 M
1587.74 630.017 L
1334.257 500.22 M
1334.257 485.37 L
S
GR
GS
[0.72 0 0 0.72 230.08274 468.38455] CT
[0.72 0 0 0.72 0 0.08002] CT
0.149 GC
2 setlinecap
1 LJ
0.694 LW
N
1467.589 500.22 M
1467.589 485.37 L
S
GR
GS
[0.72 0 0 0.72 0 0.08002] CT
0.149 GC
2 setlinecap
1 LJ
0.694 LW
N
1600.926 500.22 M
1600.926 485.37 L
S
GR
GS
[0.72 0 0 0.72 192.65403 364.2386] CT
0.149 GC
/Helvetica 13.889 F
GS
[1 0 0 1 0 0] CT
-15.5 13 moveto
1 -1 scale
(6.24) t
GR
GR
GS
[0.72 0 0 0.72 288.65716 364.2386] CT
0.149 GC
/Helvetica 13.889 F
GS
@ -667,7 +720,7 @@ GS
GR
GR
GS
[0.72 0 0 0.72 344.22173 468.38455] CT
[0.72 0 0 0.72 384.66031 364.2386] CT
0.149 GC
/Helvetica 13.889 F
GS
@ -678,7 +731,7 @@ GS
GR
GR
GS
[0.72 0 0 0.72 458.35528 468.38455] CT
[0.72 0 0 0.72 480.65885 364.2386] CT
0.149 GC
/Helvetica 13.889 F
GS
@ -689,7 +742,7 @@ GS
GR
GR
GS
[0.72 0 0 0.72 572.49427 468.38455] CT
[0.72 0 0 0.72 576.662 364.2386] CT
0.149 GC
/Helvetica 13.889 F
GS
@ -700,7 +753,7 @@ GS
GR
GR
GS
[0.72 0 0 0.72 686.62781 468.38455] CT
[0.72 0 0 0.72 672.66054 364.2386] CT
0.149 GC
/Helvetica 13.889 F
GS
@ -711,7 +764,7 @@ GS
GR
GR
GS
[0.72 0 0 0.72 800.76675 468.38455] CT
[0.72 0 0 0.72 768.66364 364.2386] CT
0.149 GC
/Helvetica 13.889 F
GS
@ -722,7 +775,7 @@ GS
GR
GR
GS
[0.72 0 0 0.72 914.90029 468.38455] CT
[0.72 0 0 0.72 864.66227 364.2386] CT
0.149 GC
/Helvetica 13.889 F
GS
@ -733,7 +786,7 @@ GS
GR
GR
GS
[0.72 0 0 0.72 1029.03927 468.38455] CT
[0.72 0 0 0.72 960.66537 364.2386] CT
0.149 GC
/Helvetica 13.889 F
GS
@ -744,7 +797,7 @@ GS
GR
GR
GS
[0.72 0 0 0.72 1143.17281 468.38455] CT
[0.72 0 0 0.72 1056.66391 364.2386] CT
0.149 GC
/Helvetica 13.889 F
GS
@ -755,6 +808,17 @@ GS
GR
GR
GS
[0.72 0 0 0.72 1152.66702 364.2386] CT
0.149 GC
/Helvetica 13.889 F
GS
[1 0 0 1 0 0] CT
-15.5 13 moveto
1 -1 scale
(6.34) t
GR
GR
GS
[0.72 0 0 0.72 0 0.08002] CT
0.149 GC
2 setlinecap
@ -772,8 +836,8 @@ GS
1 LJ
0.694 LW
N
248.997 874.034 M
263.85 874.034 L
248.997 855.384 M
263.849 855.384 L
S
GR
GS
@ -783,8 +847,8 @@ GS
1 LJ
0.694 LW
N
248.997 759.451 M
263.85 759.451 L
248.997 736.996 M
263.849 736.996 L
S
GR
GS
@ -794,8 +858,8 @@ GS
1 LJ
0.694 LW
N
248.997 644.867 M
263.85 644.867 L
248.997 618.608 M
263.849 618.608 L
S
GR
GS
@ -805,8 +869,8 @@ GS
1 LJ
0.694 LW
N
248.997 530.284 M
263.85 530.284 L
248.997 500.22 M
263.849 500.22 L
S
GR
GS
@ -816,8 +880,8 @@ GS
1 LJ
0.694 LW
N
248.997 415.701 M
263.85 415.701 L
248.997 381.832 M
263.849 381.832 L
S
GR
GS
@ -827,8 +891,8 @@ GS
1 LJ
0.694 LW
N
248.997 301.118 M
263.85 301.118 L
248.997 263.444 M
263.849 263.444 L
S
GR
GS
@ -838,12 +902,34 @@ GS
1 LJ
0.694 LW
N
248.997 186.534 M
263.85 186.534 L
248.997 145.056 M
263.849 145.056 L
S
GR
GS
[0.72 0 0 0.72 175.27731 629.38448] CT
[0.72 0 0 0.72 175.27716 615.95673] CT
0.149 GC
/Helvetica 13.889 F
GS
[1 0 0 1 0 0] CT
-23 4.5 moveto
1 -1 scale
(-60) t
GR
GR
GS
[0.72 0 0 0.72 175.27716 530.71737] CT
0.149 GC
/Helvetica 13.889 F
GS
[1 0 0 1 0 0] CT
-23 4.5 moveto
1 -1 scale
(-40) t
GR
GR
GS
[0.72 0 0 0.72 175.27716 445.47797] CT
0.149 GC
/Helvetica 13.889 F
GS
@ -854,18 +940,7 @@ GS
GR
GR
GS
[0.72 0 0 0.72 175.27731 546.88454] CT
0.149 GC
/Helvetica 13.889 F
GS
[1 0 0 1 0 0] CT
-23 4.5 moveto
1 -1 scale
(-10) t
GR
GR
GS
[0.72 0 0 0.72 175.27731 464.38456] CT
[0.72 0 0 0.72 175.27716 360.23861] CT
0.149 GC
/Helvetica 13.889 F
GS
@ -876,18 +951,7 @@ GS
GR
GR
GS
[0.72 0 0 0.72 175.27731 381.88457] CT
0.149 GC
/Helvetica 13.889 F
GS
[1 0 0 1 0 0] CT
-18 4.5 moveto
1 -1 scale
(10) t
GR
GR
GS
[0.72 0 0 0.72 175.27731 299.38461] CT
[0.72 0 0 0.72 175.27716 274.99923] CT
0.149 GC
/Helvetica 13.889 F
GS
@ -898,18 +962,7 @@ GS
GR
GR
GS
[0.72 0 0 0.72 175.27731 216.88463] CT
0.149 GC
/Helvetica 13.889 F
GS
[1 0 0 1 0 0] CT
-18 4.5 moveto
1 -1 scale
(30) t
GR
GR
GS
[0.72 0 0 0.72 175.27731 134.38467] CT
[0.72 0 0 0.72 175.27716 189.75987] CT
0.149 GC
/Helvetica 13.889 F
GS
@ -920,7 +973,18 @@ GS
GR
GR
GS
[0.72 0 0 0.72 713.87812 51.37999] CT
[0.72 0 0 0.72 175.27716 104.52049] CT
0.149 GC
/Helvetica 13.889 F
GS
[1 0 0 1 0 0] CT
-18 4.5 moveto
1 -1 scale
(60) t
GR
GR
GS
[0.72 0 0 0.72 713.87799 51.38001] CT
/Helvetica-Bold 15.278 F
GS
[1 0 0 1 0 0] CT
@ -935,21 +999,23 @@ GS
1 LJ
0.694 LW
N
248.819 122.004 M
319.559 147.236 L
478.086 204.33 L
636.605 261.976 L
795.131 320.177 L
953.65 378.935 L
1112.176 438.253 L
1270.695 498.134 L
1429.221 558.58 L
1587.74 619.593 L
1734.175 676.48 L
248.82 209.862 M
267.575 213.931 L
400.913 243.143 L
534.25 272.638 L
667.582 302.418 L
800.919 332.484 L
934.251 362.839 L
1067.588 393.483 L
1200.92 424.417 L
1334.257 455.644 L
1467.589 487.164 L
1600.926 518.978 L
1734.174 551.069 L
S
GR
GS
[0.72 0 0 0.72 230.08274 106.08975] CT
[0.72 0 0 0.72 288.65716 175.14264] CT
1 0 0 RC
N
0 -3.819 M
@ -969,7 +1035,7 @@ cp
f
GR
GS
[0.72 0 0 0.72 230.08274 106.08975] CT
[0.72 0 0 0.72 288.65716 175.14264] CT
/Helvetica 13.889 F
GS
[1 0 0 1 0 0] CT
@ -979,7 +1045,7 @@ GS
GR
GR
GS
[0.72 0 0 0.72 1229.71317 479.7681] CT
[0.72 0 0 0.72 1129.45361 368.18579] CT
0 1 0 RC
N
/f235133452{0 -3.819 M
@ -1000,21 +1066,21 @@ f235133452
f
GR
GS
[0.72 0 0 0.72 1190.08127 464.36012] CT
[0.72 0 0 0.72 1096.11898 360.22597] CT
0 1 0 RC
N
f235133452
f
GR
GS
[0.72 0 0 0.72 1190.14112 464.38456] CT
[0.72 0 0 0.72 1096.16934 360.23861] CT
0 1 0 RC
N
f235133452
f
GR
GS
[0.72 0 0 0.72 1190.14112 464.38456] CT
[0.72 0 0 0.72 1096.16934 360.23861] CT
0 1 0 RC
N
f235133452