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PDF statistics: - 1336 PDF objects out of 1440 (max. 8388607) - 1157 compressed objects within 12 object streams - 607 named destinations out of 1000 (max. 500000) - 385 words of extra memory for PDF output out of 10000 (max. 10000000) + 1375 PDF objects out of 1440 (max. 8388607) + 1192 compressed objects within 12 object streams + 615 named destinations out of 1000 (max. 500000) + 417 words of extra memory for PDF output out of 10000 (max. 10000000) diff --git a/ENUME/projectB/Report/report.out b/ENUME/projectB/Report/report.out index 173654f5..3e0ccc62 100644 --- a/ENUME/projectB/Report/report.out +++ b/ENUME/projectB/Report/report.out @@ -13,21 +13,25 @@ \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}{Comparison of results between MM1 and MM2}{section.2.3}% 16 -\BOOKMARK [2][-]{subsection.2.3.2}{Comparison of results between Newton's method and MM2}{section.2.3}% 17 -\BOOKMARK [0][-]{chapter.3}{Find real and complex roots of the polynomial using Laguerre's method}{}% 18 -\BOOKMARK [1][-]{section.3.1}{Problem}{chapter.3}% 19 -\BOOKMARK [1][-]{section.3.2}{Theoretical Introduction}{chapter.3}% 20 -\BOOKMARK [1][-]{section.3.3}{Results}{chapter.3}% 21 -\BOOKMARK [2][-]{subsection.3.3.1}{Comparison of results between MM1 and MM2}{section.3.3}% 22 -\BOOKMARK [0][-]{chapter.4}{Code appendix}{}% 23 -\BOOKMARK [1][-]{section.4.1}{Task 1}{chapter.4}% 24 -\BOOKMARK [2][-]{subsection.4.1.1}{task1Bisection.m}{section.4.1}% 25 -\BOOKMARK [2][-]{subsection.4.1.2}{task1Newton.m}{section.4.1}% 26 -\BOOKMARK [1][-]{section.4.2}{Task 2}{chapter.4}% 27 -\BOOKMARK [2][-]{subsection.4.2.1}{task2MM1.m}{section.4.2}% 28 -\BOOKMARK [2][-]{subsection.4.2.2}{task2MM2.m}{section.4.2}% 29 -\BOOKMARK [1][-]{section.4.3}{Task 3}{chapter.4}% 30 -\BOOKMARK [2][-]{subsection.4.3.1}{rootBrackering.m}{section.4.3}% 31 -\BOOKMARK [2][-]{subsection.4.3.2}{printGraph.m}{section.4.3}% 32 -\BOOKMARK [2][-]{subsection.4.3.3}{printComplexGraph.m}{section.4.3}% 33 +\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 diff --git a/ENUME/projectB/Report/report.pdf b/ENUME/projectB/Report/report.pdf index 1cab059f..f186152e 100644 Binary files a/ENUME/projectB/Report/report.pdf and b/ENUME/projectB/Report/report.pdf differ diff --git a/ENUME/projectB/Report/report.synctex.gz b/ENUME/projectB/Report/report.synctex.gz index 588fc543..06dadc2e 100644 Binary files a/ENUME/projectB/Report/report.synctex.gz and b/ENUME/projectB/Report/report.synctex.gz differ diff --git a/ENUME/projectB/Report/report.tex b/ENUME/projectB/Report/report.tex index 3e0fa79d..a8509aa5 100644 --- a/ENUME/projectB/Report/report.tex +++ b/ENUME/projectB/Report/report.tex @@ -97,6 +97,106 @@ It is superlinearly convergent, globally convergent and length of intervals we g \includegraphics[scale=0.25]{task1falsepositionzommedright.eps} \end{center} +\begin{center} + \begin{tabular}{| c c c |} +\hline +Iteration & root & f(root) \\ +\hline +1 & -0.17673 &-1.9421 \\ +\hline +2 & -0.32943 &-1.7409 \\ +\hline +3 & -0.39578 &-1.6308 \\ +\hline +4 &-0.61812 &-1.1386 \\ +\hline +5 &-0.76152 &-0.69765 \\ +\hline +6 &-0.84516 &-0.38573 \\ +\hline +7 &-0.89015 &-0.19911 \\ +\hline +8 &-0.91304 &-0.098733 \\ +\hline +9 & -0.92431 &-0.047922 \\ +\hline +10 & -0.92976 & -0.02301 \\ +\hline +11 &-0.93237 & -0.01099 \\ +\hline +12 &-0.93362 &-0.005236 \\ +\hline +13 &-0.93421 &-0.0024915 \\ +\hline +14 &-0.9345 &-0.0011848 \\ +\hline +15 &-0.93463 &-0.0005633 \\ +\hline +16 &-0.93469 &-0.00026777 \\ +\hline +17 &-0.93473 &-0.00012728 \\ +\hline +18 &-0.93474 &-6.0499e-05 \\ +\hline +19 &-0.93475 &-2.8756e-05 \\ +\hline +20 &-0.93475 & -1.3668e-05 \\ +\hline +21 &-0.93475 & -6.4964e-06 \\ +\hline +22 &-0.93475 & -3.0878e-06 \\ +\hline +23 &-0.93475 & -1.4676e-06 \\ +\hline +24 &-0.93475 & -6.9758e-07 \\ +\hline +25 &-0.93475 & -3.3157e-07 \\ +\hline +26 &-0.93475 & -1.5759e-07 \\ +\hline +27 &-0.93475 & -7.4906e-08 \\ +\hline +28 &-0.93475 & -3.5603e-08 \\ +\hline +29 &-0.93475 & -1.6922e-08 \\ +\hline +30 &-0.93475 & -8.0433e-09 \\ +\hline +31 &-0.93475 & -3.823e-09 \\ +\hline +32 &-0.93475 & -1.8171e-09 \\ +\hline +33 &-0.93475 & -8.6368e-10 \\ +\hline +34 &-0.93475 & -4.1051e-10 \\ +\hline +35 &-0.93475 & -1.9512e-10 \\ +\hline +36 &-0.93475 & -9.2741e-11 \\ +\hline +\end{tabular} +\end{center} + +\begin{center} + \begin{tabular}{| c c c |} +\hline +Iteration & root & f(root) \\ +\hline +1 & 7.0323 & 0.0034663 \\ +\hline +2 & 7.0212 & 9.0344e-05 \\ +\hline +3 & 7.0211 & 4.6349e-05 \\ +\hline +4 & 7.0208 & -4.3921e-05 \\ +\hline +5 & 7.0209 & 4.8939e-11 \\ +\hline + +\hline +\end{tabular} +\end{center} + \section{b) the Newton's method} @@ -132,6 +232,47 @@ Newton's method is also effective if the function derrivative is far from zero, \includegraphics[scale=0.25]{task1Newtonzommedright.eps} \end{center} +\begin{center} + \begin{tabular}{| c c c |} +\hline +Iteration & root & f(root) \\ +\hline +1 & -1.25 & 1.8879 \\ +\hline +2 & -1.0001 & 0.31855 \\ +\hline +3 & -0.93804 & 0.015256 \\ +\hline +4 & -0.93476 & 4.0314e-05 \\ +\hline +5 & -0.93475 & 2.8356e-10 \\ +\hline +6 & -0.93475 & 0 \\ +\hline +\hline + +\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 & 7.0144 & -0.0019866 \\ +\hline +3 & 7.0209 & -9.517e-08 \\ +\hline +4 &7.0209 & 7.468e-16 \\ +\hline +\hline + +\hline +\end{tabular} +\end{center} \chapter{Find real and complex roots of the polynomial} @@ -214,6 +355,7 @@ This method is locally convergent with order of convergence equal to 1.84. It is \section{Results} +\subsection{Graphs for MM1} \begin{center} \includegraphics[scale=0.25]{task2mm1overall.eps} \end{center} @@ -238,6 +380,7 @@ This method is locally convergent with order of convergence equal to 1.84. It is \includegraphics[scale=0.25]{task2mm1complexzommedup.eps} \end{center} +\subsection{Graphs for MM2} \begin{center} \includegraphics[scale=0.25]{task2mm2overall.eps} \end{center} @@ -262,6 +405,134 @@ This method is locally convergent with order of convergence equal to 1.84. It is \includegraphics[scale=0.25]{task2mm2complexzommedup.eps} \end{center} +\subsection{Tables for MM1} +\begin{center} + \begin{tabular}{| c c c |} +\hline +Iteration & root & f(root) \\ +\hline +1 & -1.25 & -20.32 \\ +\hline +2 & -1.0009 & -8.0351 \\ +\hline +3 & -0.76855 & -1.5512 \\ +\hline +4 & -0.63496 & 0.58064 \\ +\hline +5 & -0.67945 & -0.023155 \\ +\hline +6 & -0.67788 & -0.00010725 \\ +\hline +7 & -0.67787 & -8.6306e-09 \\ +\hline +8 & -0.67787 & 0 \\ +\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 & 6.3257 & 4.4966 \\ +\hline +3 & 6.3341 & 0.033379 \\ +\hline +4 & 6.3341 & -1.6916e-06 \\ +\hline +5 & 6.3341 & -7.1942e-14 \\ +\hline + +\end{tabular} +\end{center} + +\begin{center} + \begin{tabular}{| c c c |} +\hline +Iteration & root & f(root) \\ +\hline +1 & -0.5-0.5i & 6.1847 \\ +\hline +2 & -0.42823-0.15033i & 2.6872 \\ +\hline +3 & -0.59202+0.094367i & 1.6012 \\ +\hline +4 & -0.71249-0.0089336i & 0.55524 \\ +\hline +5 & -0.67654+0.00084164i & 0.022962 \\ +\hline +6 & -0.67787+8.4773e-06i & 0.00012412 \\ +\hline +7 & -0.67787+3.6458e-10i & 8.3668e-09 \\ +\hline +8 & -0.67787+9.1168e-18i & 1.3321e-16 \\ +\hline + +\end{tabular} +\end{center} + +\subsection{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 + +\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 + +\end{tabular} +\end{center} + +\begin{center} + \begin{tabular}{| c c c |} +\hline +Iteration & root & f(root) \\ +\hline +1 & -0.5-0.5i & 6.1847 \\ +\hline +2 & -0.60346-0.076584i & 1.3673 \\ +\hline +3 & -0.67709+0.001186i & 0.020767 \\ +\hline +4 & -0.67787+1.518e-08i & 2.2673e-07 \\ +\hline +5 & -0.67787-2.0556e-17i & 3.0035e-16 \\ +\hline + +\end{tabular} +\end{center} + \subsection{Comparison of results between MM1 and MM2} \subsection{Comparison of results between Newton's method and MM2} @@ -309,6 +580,61 @@ This formula is similar to the one from MM2 but also takes order of the polynomi \includegraphics[scale=0.25]{task3complexup.eps} \end{center} +\begin{center} + \begin{tabular}{| c c c |} +\hline +Iteration & root & f(root) \\ +\hline +1 & -1.25 & -20.32 \\ +\hline +2 & -0.67905 & -0.017268 \\ +\hline +3 & -0.67787 & -3.1489e-07 \\ +\hline +4 & -0.67787 & 0 \\ +\hline +\hline + +\end{tabular} +\end{center} + +\begin{center} + \begin{tabular}{| c c c |} +\hline +Iteration & root & f(root) \\ +\hline +1 & -1.25 & -20.32 \\ +\hline +2 & -0.67905 & -0.017268 \\ +\hline +3 & -0.67787 & -3.1489e-07 \\ +\hline +4 & -0.67787 & 0 \\ +\hline +\hline + +\end{tabular} +\end{center} + + +\begin{center} + \begin{tabular}{| c c c |} +\hline +Iteration & root & f(root) \\ +\hline +1 & -1.25 & -20.32 \\ +\hline +2 & -0.67905 & -0.017268 \\ +\hline +3 & -0.67787 & -3.1489e-07 \\ +\hline +4 & -0.67787 & 0 \\ +\hline +\hline + +\end{tabular} +\end{center} + \subsection{Comparison of results between MM1 and MM2} diff --git a/ENUME/projectB/Report/report.toc b/ENUME/projectB/Report/report.toc index 901445d8..985858f3 100644 --- a/ENUME/projectB/Report/report.toc +++ b/ENUME/projectB/Report/report.toc @@ -5,91 +5,95 @@ \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}{6}{section.1.2}% -\contentsline {subsection}{\numberline {1.2.1}Problem}{6}{subsection.1.2.1}% -\contentsline {subsection}{\numberline {1.2.2}Theoretical Introduction}{6}{subsection.1.2.2}% -\contentsline {subsection}{\numberline {1.2.3}Results}{7}{subsection.1.2.3}% -\contentsline {chapter}{\numberline {2}Find real and complex roots of the polynomial}{9}{chapter.2}% -\contentsline {section}{\numberline {2.1}Problem}{9}{section.2.1}% -\contentsline {section}{\numberline {2.2}Theoretical Introduction}{9}{section.2.2}% -\contentsline {subsection}{\numberline {2.2.1}MM1}{10}{subsection.2.2.1}% -\contentsline {subsection}{\numberline {2.2.2}MM2}{11}{subsection.2.2.2}% -\contentsline {section}{\numberline {2.3}Results}{12}{section.2.3}% -\contentsline {subsection}{\numberline {2.3.1}Comparison of results between MM1 and MM2}{16}{subsection.2.3.1}% -\contentsline {subsection}{\numberline {2.3.2}Comparison of results between Newton's method and MM2}{16}{subsection.2.3.2}% -\contentsline {chapter}{\numberline {3}Find real and complex roots of the polynomial using Laguerre's method}{17}{chapter.3}% -\contentsline {section}{\numberline {3.1}Problem}{17}{section.3.1}% -\contentsline {section}{\numberline {3.2}Theoretical Introduction}{17}{section.3.2}% -\contentsline {section}{\numberline {3.3}Results}{18}{section.3.3}% -\contentsline {subsection}{\numberline {3.3.1}Comparison of results between MM1 and MM2}{20}{subsection.3.3.1}% -\contentsline {chapter}{\numberline {4}Code appendix}{21}{chapter.4}% -\contentsline {section}{\numberline {4.1}Task 1}{21}{section.4.1}% -\contentsline {subsection}{\numberline {4.1.1}task1Bisection.m}{21}{subsection.4.1.1}% -\contentsline {subsubsection}{Top of task1Bisection.m}{21}{section*.4}% -\contentsline {subsubsection}{taskFunction}{21}{section*.5}% -\contentsline {subsubsection}{falsePosition}{22}{section*.6}% -\contentsline {subsubsection}{initialize}{22}{section*.7}% -\contentsline {subsubsection}{taskFunction}{22}{section*.8}% -\contentsline {subsubsection}{firstTwoIterations}{23}{section*.9}% -\contentsline {subsubsection}{falsePositionLoop}{23}{section*.10}% -\contentsline {subsubsection}{insideLoop}{24}{section*.11}% -\contentsline {subsubsection}{changeLastTwoAB}{24}{section*.12}% -\contentsline {subsubsection}{calculateZero}{25}{section*.13}% -\contentsline {subsubsection}{newSubInterval}{25}{section*.14}% -\contentsline {subsection}{\numberline {4.1.2}task1Newton.m}{25}{subsection.4.1.2}% -\contentsline {subsubsection}{Top of task1Newton.m}{25}{section*.15}% -\contentsline {subsubsection}{taskFunction}{26}{section*.16}% -\contentsline {subsubsection}{newtonMethod}{26}{section*.17}% -\contentsline {subsubsection}{initialize}{26}{section*.18}% -\contentsline {subsubsection}{newtonLoop}{26}{section*.19}% -\contentsline {subsubsection}{insideLoop}{27}{section*.20}% -\contentsline {subsubsection}{calculateZeroIterations}{27}{section*.21}% -\contentsline {subsubsection}{checkForDivergence}{27}{section*.22}% -\contentsline {section}{\numberline {4.2}Task 2}{28}{section.4.2}% -\contentsline {subsection}{\numberline {4.2.1}task2MM1.m}{28}{subsection.4.2.1}% -\contentsline {subsubsection}{Top of task2MM1}{28}{section*.23}% -\contentsline {subsubsection}{polynomial}{28}{section*.24}% -\contentsline {subsubsection}{mm1}{28}{section*.25}% -\contentsline {subsubsection}{initialize}{29}{section*.26}% -\contentsline {subsubsection}{mm1Loop}{29}{section*.27}% -\contentsline {subsubsection}{insideLoop}{29}{section*.28}% -\contentsline {subsubsection}{createEquationSystem}{30}{section*.29}% -\contentsline {subsubsection}{rootsOfQuadraticFormula}{30}{section*.30}% -\contentsline {subsubsection}{updateApproximations}{30}{section*.31}% -\contentsline {subsubsection}{initializeEquationSystem}{31}{section*.32}% -\contentsline {subsubsection}{solveEquationSystem}{31}{section*.33}% -\contentsline {subsubsection}{createApproximatedQuadraticFormula}{32}{section*.34}% -\contentsline {subsubsection}{findRootsOfQuadraticFormula}{32}{section*.35}% -\contentsline {subsubsection}{chooseNewRoot}{32}{section*.36}% -\contentsline {subsubsection}{addZeroToIterationVector}{32}{section*.37}% -\contentsline {subsubsection}{getWorstApproximationIndex}{33}{section*.38}% -\contentsline {subsubsection}{deleteWorstApproximation}{33}{section*.39}% -\contentsline {subsection}{\numberline {4.2.2}task2MM2.m}{34}{subsection.4.2.2}% -\contentsline {subsubsection}{top of task2MM2.m}{34}{section*.40}% -\contentsline {subsubsection}{polynomial}{34}{section*.41}% -\contentsline {subsubsection}{mm2}{34}{section*.42}% -\contentsline {subsubsection}{initialize}{34}{section*.43}% -\contentsline {subsubsection}{mm2Loop}{35}{section*.44}% -\contentsline {subsubsection}{insideLoop}{35}{section*.45}% -\contentsline {subsubsection}{getABC}{35}{section*.46}% -\contentsline {subsubsection}{findRoots}{35}{section*.47}% -\contentsline {subsubsection}{chooseNewApproximation}{36}{section*.48}% -\contentsline {subsubsection}{updateApproximations}{36}{section*.49}% -\contentsline {subsubsection}{derivative}{36}{section*.50}% -\contentsline {section}{\numberline {4.3}Task 3}{37}{section.4.3}% -\contentsline {subsubsection}{Top of Task 3}{37}{section*.51}% -\contentsline {subsubsection}{polynomial}{37}{section*.52}% -\contentsline {subsubsection}{laguerre}{37}{section*.53}% -\contentsline {subsubsection}{initialize}{38}{section*.54}% -\contentsline {subsubsection}{laguerreLoop}{38}{section*.55}% -\contentsline {subsubsection}{insideLoop}{38}{section*.56}% -\contentsline {subsubsection}{calculateDerrivatives}{39}{section*.57}% -\contentsline {subsubsection}{calculateZ}{39}{section*.58}% -\contentsline {subsubsection}{chooseNewZero}{39}{section*.59}% -\contentsline {subsubsection}{updateZeros}{40}{section*.60}% -\contentsline {subsection}{\numberline {4.3.1}rootBrackering.m}{40}{subsection.4.3.1}% -\contentsline {subsubsection}{updateZeros}{40}{section*.61}% -\contentsline {subsubsection}{initializeValues}{41}{section*.62}% -\contentsline {subsubsection}{bracketingLoop}{42}{section*.63}% -\contentsline {subsection}{\numberline {4.3.2}printGraph.m}{43}{subsection.4.3.2}% -\contentsline {subsection}{\numberline {4.3.3}printComplexGraph.m}{45}{subsection.4.3.3}% +\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 {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}%