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ENUME/projectB/Report/task3complexright.eps diff --git a/ENUME/projectB/Code/task1Newton.m b/ENUME/projectB/Code/task1Newton.m index 472be1cd..a4b69513 100644 --- a/ENUME/projectB/Code/task1Newton.m +++ b/ENUME/projectB/Code/task1Newton.m @@ -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); diff --git a/ENUME/projectB/Code/task2MM1.m b/ENUME/projectB/Code/task2MM1.m index 997ebff4..1be8f388 100644 --- a/ENUME/projectB/Code/task2MM1.m +++ b/ENUME/projectB/Code/task2MM1.m @@ -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; diff --git a/ENUME/projectB/Code/task2MM2.m b/ENUME/projectB/Code/task2MM2.m index 1a0b1038..a2f5fea8 100644 --- a/ENUME/projectB/Code/task2MM2.m +++ b/ENUME/projectB/Code/task2MM2.m @@ -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) diff --git a/ENUME/projectB/Code/task3.m b/ENUME/projectB/Code/task3.m index d837642a..bc3dc3e5 100644 --- a/ENUME/projectB/Code/task3.m +++ b/ENUME/projectB/Code/task3.m @@ -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; diff --git a/ENUME/projectB/Report/report.aux b/ENUME/projectB/Report/report.aux index 22a5f278..2e29b227 100644 --- a/ENUME/projectB/Report/report.aux +++ b/ENUME/projectB/Report/report.aux @@ -29,99 +29,100 @@ \@writefile{toc}{\contentsline {subsection}{\numberline {1.2.1}Problem}{8}{subsection.1.2.1}\protected@file@percent } \@writefile{toc}{\contentsline {subsection}{\numberline {1.2.2}Theoretical Introduction}{8}{subsection.1.2.2}\protected@file@percent } \@writefile{toc}{\contentsline {subsection}{\numberline {1.2.3}Results}{9}{subsection.1.2.3}\protected@file@percent } -\@writefile{toc}{\contentsline {chapter}{\numberline {2}Find real and complex roots of the polynomial}{11}{chapter.2}\protected@file@percent } +\@writefile{toc}{\contentsline {subsection}{\numberline {1.2.4}Discussion of results}{10}{subsection.1.2.4}\protected@file@percent } +\@writefile{toc}{\contentsline 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./task3complexleft.eps +INPUT task3complexleft.eps +INPUT ./task3complexleft.eps +INPUT ./task3complexleft.eps +INPUT ./task3complexleft-eps-converted-to.pdf +INPUT ./task3complexleft-eps-converted-to.pdf +INPUT ./task3complexleft.eps +INPUT ./task3complexleft-eps-converted-to.pdf +INPUT ./task3complexleft-eps-converted-to.pdf +INPUT ./task3complexleft-eps-converted-to.pdf +INPUT ./task3complexright.eps +INPUT ./task3complexright.eps +INPUT task3complexright.eps +INPUT ./task3complexright.eps +INPUT ./task3complexright.eps +INPUT ./task3complexright-eps-converted-to.pdf +INPUT ./task3complexright-eps-converted-to.pdf +INPUT ./task3complexright.eps +INPUT ./task3complexright-eps-converted-to.pdf +INPUT ./task3complexright-eps-converted-to.pdf +INPUT ./task3complexright-eps-converted-to.pdf INPUT ./task3complexup.eps INPUT ./task3complexup.eps INPUT task3complexup.eps diff --git a/ENUME/projectB/Report/report.log b/ENUME/projectB/Report/report.log index 7d0f29cd..3a2c296e 100644 --- a/ENUME/projectB/Report/report.log +++ b/ENUME/projectB/Report/report.log @@ -1,4 +1,4 @@ -This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020/Debian) (preloaded format=pdflatex 2021.10.23) 3 DEC 2021 05:11 +This is pdfTeX, Version 3.14159265-2.6-1.40.21 (TeX Live 2020/Debian) (preloaded format=pdflatex 2021.10.23) 3 DEC 2021 05:55 entering extended mode restricted \write18 enabled. file:line:error style messages enabled. @@ -470,7 +470,7 @@ File: epstopdf-sys.cfg 2010/07/13 v1.3 Configuration of (r)epstopdf for TeX Live {/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}] (./report.tocpdfTeX warning (ext4): destination with the same identifier (name{page.1}) has been already used, duplicate ignored \relax -l.28 ...between MM1 and MM2}{24}{subsection.3.3.1} +l.28 ...\numberline {3.3}Results}{24}{section.3.3} % [1 ]) @@ -486,12 +486,12 @@ Package epstopdf Info: Source file: (epstopdf) date: 2021-12-03 04:34:38 (epstopdf) size: 44376 bytes (epstopdf) Output file: -(epstopdf) 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Package epstopdf Info: Output file is already uptodate. - + File: task1Newtonoverall-eps-converted-to.pdf Graphic file (type pdf) Package pdftex.def Info: task1Newtonoverall-eps-converted-to.pdf used on input line 224. @@ -545,12 +545,12 @@ Package epstopdf Info: Source file: (epstopdf) date: 2021-12-03 04:36:25 (epstopdf) size: 17046 bytes (epstopdf) Output file: -(epstopdf) date: 2021-12-03 04:46:18 +(epstopdf) date: 2021-12-03 05:49:32 (epstopdf) size: 9310 bytes (epstopdf) Command: (epstopdf) \includegraphics on input line 228. 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-\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 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+\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 diff --git a/ENUME/projectB/Report/report.pdf b/ENUME/projectB/Report/report.pdf index 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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} diff --git a/ENUME/projectB/Report/report.toc b/ENUME/projectB/Report/report.toc index 985858f3..57c7bb64 100644 --- a/ENUME/projectB/Report/report.toc +++ b/ENUME/projectB/Report/report.toc @@ -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}% 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