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Introduction}{section.1.1}% 4 -\BOOKMARK [2][-]{subsection.1.1.3}{Results}{section.1.1}% 5 -\BOOKMARK [1][-]{section.1.2}{b\) the Newton's method}{chapter.1}% 6 -\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 [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 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49e50865..70b0929c 100644 --- a/ENUME/projectB/Report/report.tex +++ b/ENUME/projectB/Report/report.tex @@ -566,11 +566,67 @@ Iteration & root & f(root) \\ \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. +As we can see MM2 method is much better, it took on average around 5 iterations for it to find a root with sufficient accuracy compared to around 7 iterations that MM1 needed to find roots. MM1 is slightly more accurate with at least 10 times better accuary than MM2, there is also a big error in MM2 result which incorrectly denotes a number that should only have real part as an imaginary number. MM1 does not have this problem and shows root -0.67787 correctly. We can also observe that MM1 calculated complex root with a '-' between real and imaginary part and MM2 calculated complex root with a '+' between real and imaginary part. We can assume that in order to get all the complex roots we should use both of the methods. 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} +\begin{center} + \includegraphics[scale=0.25]{task2newtonoverall.eps} +\end{center} + +\begin{center} + \includegraphics[scale=0.25]{task2newtonleft.eps} +\end{center} + +\begin{center} + \includegraphics[scale=0.25]{task2newtonright.eps} +\end{center} + + +\begin{center} + \begin{tabular}{| c c c |} +\hline +Iteration & root & f(root) \\ +\hline + 1 & -1.25 & 1.8879 \\ + \hline + 2 & -0.92681 & -5.52 \\ + \hline + 3 & -0.74804 & -1.159 \\ + \hline + 4 & -0.68543 & -0.11186 \\ + \hline + 5 & -0.67797 & -0.0014548 \\ + \hline + 6 & -0.67787 & -2.568e-07 \\ + \hline + 7 & -0.67787 & -7.9936e-15 \\ + \hline +\end{tabular} +\end{center} + +\begin{center} + \begin{tabular}{| c c c |} +\hline +Iteration & root & f(root) \\ +\hline + 1 & 6.25 & -0.23707 \\ + \hline + 2 & 6.3376 & -1.8647 \\ + \hline + 3 & 6.3341 & -0.0029912 \\ + \hline + 4 & 6.3341 & -7.7134e-09 \\ + \hline + 5 & 6.3341 & -7.1942e-14 \\ + \hline +\end{tabular} +\end{center} + +As we can see Newton method calculated real roots in around the same number of iterations as method MM1, it was slightly slower than MM2. Newton method had pretty good accuracy, comparable to MM1 accuracy. + + \chapter{Find real and complex roots of the polynomial using Laguerre's method} \section{Problem} @@ -682,7 +738,83 @@ Iteration & root & f(root) \\ \end{center} -\subsection{Comparison of results between MM1 and MM2} +\subsection{Comparison of results between Laguerre's method and MM2 method} + +\subsubsection{Tables for MM2} + +\begin{center} + \begin{tabular}{| c c c |} +\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 \\ +\hline + \hline + +\end{tabular} +\end{center} + +\begin{center} + \begin{tabular}{| c c c |} +\hline +Iteration & root & f(root) \\ +\hline +1 & 6.25 & 43.43 \\ +\hline +2 & 6.3376 & -1.8647 \\ +\hline +3 & 6.3341 & 0.0029695 \\ +\hline +4 & 6.3341 & 1.6294e-08 \\ +\hline +5 & 6.3341 & -4.6896e-13 \\ +\hline +\hline + +\end{tabular} +\end{center} + +\begin{center} + \begin{tabular}{| c c c |} +\hline +Iteration & root & f(root) \\ +\hline +1 & 0+0i & 3 \\ +\hline +2 & -0.125+0.85696i & 8.0341 \\ +\hline +3 & 0.16892+0.62593i & 0.94053 \\ +\hline +4 & 0.17203+0.56536i & 0.0031354 \\ +\hline +5 & 0.17188+0.56551i & 3.8615e-09 \\ +\hline +6 & 0.17188+0.56551i & 0 \\ +\hline +\hline + +\end{tabular} +\end{center} + +As expected Laguerre's method reigns supreme, it is: +\begin{enumerate} +\item Faster +With 5 iterations on average compared to around 5.3 for MM2 +\item More accurate +For real root -0.67787 matlab calculated the value of this function for Laguerre's method as equal to exactly 0 as opposed to around $10^{-11}$ error for MM2. +\item With no errors +Laguerre's method did not mistakenly added imaginary part to real root -0.67787 as MM2 did. +\end{enumerate} + +Laguerre's method is faster and more accurate than MM2 which means that we can compare it to faster version of MM1, this means that it is the best method we tested in this exercise. \chapter{Code appendix} diff --git a/ENUME/projectB/Report/report.toc b/ENUME/projectB/Report/report.toc deleted file mode 100644 index 57c7bb64..00000000 --- a/ENUME/projectB/Report/report.toc +++ /dev/null @@ -1,100 +0,0 @@ -\contentsline {chapter}{\numberline {1}Find all zeros of function}{3}{chapter.1}% -\contentsline {section}{\numberline {1.1}a) False position method}{3}{section.1.1}% -\contentsline {subsection}{\numberline {1.1.1}Problem}{3}{subsection.1.1.1}% -\contentsline {subsection}{\numberline {1.1.2}Theoretical Introduction}{3}{subsection.1.1.2}% -\contentsline {subsubsection}{Properties of \emph {false position method}}{4}{section*.2}% -\contentsline {paragraph}{Improvement to the method}{4}{section*.3}% -\contentsline {subsection}{\numberline {1.1.3}Results}{5}{subsection.1.1.3}% -\contentsline {section}{\numberline {1.2}b) the Newton's method}{8}{section.1.2}% -\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 {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}% diff --git a/ENUME/projectB/Report/task1Newtonoverall-eps-converted-to.pdf b/ENUME/projectB/Report/task1Newtonoverall-eps-converted-to.pdf deleted file mode 100644 index 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