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Chapter 1.
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Chapter 2.
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Chapter 3.
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Chapter 4.
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@ -85,6 +85,18 @@ This is called \emph{modified regula falsi} or \emph{Illinois algorithm}.
It is superlinearly convergent, globally convergent and length of intervals we get in each iterations converges to zero.
\subsection{Results}
\begin{center}
\includegraphics[scale=0.25]{task1falsepositionoverall.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task1falsepositionzommedleft.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task1falsepositionzommedright.eps}
\end{center}
\section{b) the Newton's method}
@ -108,6 +120,19 @@ This method as opposed to \emph{regula falsi} method is locally convergent, shou
Newton's method is also effective if the function derrivative is far from zero, so the slope of the function is steep, conversely if the derrivative is close to zero the method is not recommended.
\subsection{Results}
\begin{center}
\includegraphics[scale=0.25]{task1Newtonoverall.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task1Newtonzommedleft.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task1Newtonzommedright.eps}
\end{center}
\chapter{Find real and complex roots of the polynomial}
@ -189,6 +214,54 @@ This method is locally convergent with order of convergence equal to 1.84. It is
\section{Results}
\begin{center}
\includegraphics[scale=0.25]{task2mm1overall.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task2mm1zommedleft.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task2mm1zommedright.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task2mm1complexzommedleft.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task2mm1complexzommeddown.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task2mm1complexzommedup.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task2mm2overall.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task2mm2zommedleft.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task2mm2zommedright.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task2mm2complexzommedleft.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task2mm2complexzommedmiddle.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task2mm2complexzommedup.eps}
\end{center}
\subsection{Comparison of results between MM1 and MM2}
\subsection{Comparison of results between Newton's method and MM2}
@ -210,6 +283,33 @@ $n$ - order of the polynomial
This formula is similar to the one from MM2 but also takes order of the polynomial into consideration. In general this method is better. For polynomials with real roots it is globally convergent. It does not have formal analysis for complex roots but it usually shows good numerical properties, although divergence may happen.
\section{Results}
\begin{center}
\includegraphics[scale=0.25]{task3overall.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task3left.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task3right.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task3complexoverall.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task3complexdown.eps}
\end{center}
\begin{center}
\includegraphics[scale=0.25]{task3complexup.eps}
\end{center}
\subsection{Comparison of results between MM1 and MM2}
\chapter{Code appendix}

176
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@ -5,91 +5,91 @@
\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}{5}{section.1.2}%
\contentsline {subsection}{\numberline {1.2.1}Problem}{5}{subsection.1.2.1}%
\contentsline {subsection}{\numberline {1.2.2}Theoretical Introduction}{5}{subsection.1.2.2}%
\contentsline {subsection}{\numberline {1.2.3}Results}{5}{subsection.1.2.3}%
\contentsline {chapter}{\numberline {2}Find real and complex roots of the polynomial}{6}{chapter.2}%
\contentsline {section}{\numberline {2.1}Problem}{6}{section.2.1}%
\contentsline {section}{\numberline {2.2}Theoretical Introduction}{6}{section.2.2}%
\contentsline {subsection}{\numberline {2.2.1}MM1}{7}{subsection.2.2.1}%
\contentsline {subsection}{\numberline {2.2.2}MM2}{8}{subsection.2.2.2}%
\contentsline {section}{\numberline {2.3}Results}{9}{section.2.3}%
\contentsline {subsection}{\numberline {2.3.1}Comparison of results between MM1 and MM2}{9}{subsection.2.3.1}%
\contentsline {subsection}{\numberline {2.3.2}Comparison of results between Newton's method and MM2}{9}{subsection.2.3.2}%
\contentsline {chapter}{\numberline {3}Find real and complex roots of the polynomial using Laguerre's method}{10}{chapter.3}%
\contentsline {section}{\numberline {3.1}Problem}{10}{section.3.1}%
\contentsline {section}{\numberline {3.2}Theoretical Introduction}{10}{section.3.2}%
\contentsline {section}{\numberline {3.3}Results}{11}{section.3.3}%
\contentsline {subsection}{\numberline {3.3.1}Comparison of results between MM1 and MM2}{11}{subsection.3.3.1}%
\contentsline {chapter}{\numberline {4}Code appendix}{12}{chapter.4}%
\contentsline {section}{\numberline {4.1}Task 1}{12}{section.4.1}%
\contentsline {subsection}{\numberline {4.1.1}task1Bisection.m}{12}{subsection.4.1.1}%
\contentsline {subsubsection}{Top of task1Bisection.m}{12}{section*.4}%
\contentsline {subsubsection}{taskFunction}{12}{section*.5}%
\contentsline {subsubsection}{falsePosition}{13}{section*.6}%
\contentsline {subsubsection}{initialize}{13}{section*.7}%
\contentsline {subsubsection}{taskFunction}{13}{section*.8}%
\contentsline {subsubsection}{firstTwoIterations}{14}{section*.9}%
\contentsline {subsubsection}{falsePositionLoop}{14}{section*.10}%
\contentsline {subsubsection}{insideLoop}{15}{section*.11}%
\contentsline {subsubsection}{changeLastTwoAB}{15}{section*.12}%
\contentsline {subsubsection}{calculateZero}{16}{section*.13}%
\contentsline {subsubsection}{newSubInterval}{16}{section*.14}%
\contentsline {subsection}{\numberline {4.1.2}task1Newton.m}{16}{subsection.4.1.2}%
\contentsline {subsubsection}{Top of task1Newton.m}{16}{section*.15}%
\contentsline {subsubsection}{taskFunction}{17}{section*.16}%
\contentsline {subsubsection}{newtonMethod}{17}{section*.17}%
\contentsline {subsubsection}{initialize}{17}{section*.18}%
\contentsline {subsubsection}{newtonLoop}{17}{section*.19}%
\contentsline {subsubsection}{insideLoop}{18}{section*.20}%
\contentsline {subsubsection}{calculateZeroIterations}{18}{section*.21}%
\contentsline {subsubsection}{checkForDivergence}{18}{section*.22}%
\contentsline {section}{\numberline {4.2}Task 2}{19}{section.4.2}%
\contentsline {subsection}{\numberline {4.2.1}task2MM1.m}{19}{subsection.4.2.1}%
\contentsline {subsubsection}{Top of task2MM1}{19}{section*.23}%
\contentsline {subsubsection}{polynomial}{19}{section*.24}%
\contentsline {subsubsection}{mm1}{19}{section*.25}%
\contentsline {subsubsection}{initialize}{20}{section*.26}%
\contentsline {subsubsection}{mm1Loop}{20}{section*.27}%
\contentsline {subsubsection}{insideLoop}{20}{section*.28}%
\contentsline {subsubsection}{createEquationSystem}{21}{section*.29}%
\contentsline {subsubsection}{rootsOfQuadraticFormula}{21}{section*.30}%
\contentsline {subsubsection}{updateApproximations}{21}{section*.31}%
\contentsline {subsubsection}{initializeEquationSystem}{22}{section*.32}%
\contentsline {subsubsection}{solveEquationSystem}{22}{section*.33}%
\contentsline {subsubsection}{createApproximatedQuadraticFormula}{23}{section*.34}%
\contentsline {subsubsection}{findRootsOfQuadraticFormula}{23}{section*.35}%
\contentsline {subsubsection}{chooseNewRoot}{23}{section*.36}%
\contentsline {subsubsection}{addZeroToIterationVector}{23}{section*.37}%
\contentsline {subsubsection}{getWorstApproximationIndex}{24}{section*.38}%
\contentsline {subsubsection}{deleteWorstApproximation}{24}{section*.39}%
\contentsline {subsection}{\numberline {4.2.2}task2MM2.m}{25}{subsection.4.2.2}%
\contentsline {subsubsection}{top of task2MM2.m}{25}{section*.40}%
\contentsline {subsubsection}{polynomial}{25}{section*.41}%
\contentsline {subsubsection}{mm2}{25}{section*.42}%
\contentsline {subsubsection}{initialize}{25}{section*.43}%
\contentsline {subsubsection}{mm2Loop}{26}{section*.44}%
\contentsline {subsubsection}{insideLoop}{26}{section*.45}%
\contentsline {subsubsection}{getABC}{26}{section*.46}%
\contentsline {subsubsection}{findRoots}{26}{section*.47}%
\contentsline {subsubsection}{chooseNewApproximation}{27}{section*.48}%
\contentsline {subsubsection}{updateApproximations}{27}{section*.49}%
\contentsline {subsubsection}{derivative}{27}{section*.50}%
\contentsline {section}{\numberline {4.3}Task 3}{28}{section.4.3}%
\contentsline {subsubsection}{Top of Task 3}{28}{section*.51}%
\contentsline {subsubsection}{polynomial}{28}{section*.52}%
\contentsline {subsubsection}{laguerre}{28}{section*.53}%
\contentsline {subsubsection}{initialize}{29}{section*.54}%
\contentsline {subsubsection}{laguerreLoop}{29}{section*.55}%
\contentsline {subsubsection}{insideLoop}{29}{section*.56}%
\contentsline {subsubsection}{calculateDerrivatives}{30}{section*.57}%
\contentsline {subsubsection}{calculateZ}{30}{section*.58}%
\contentsline {subsubsection}{chooseNewZero}{30}{section*.59}%
\contentsline {subsubsection}{updateZeros}{31}{section*.60}%
\contentsline {subsection}{\numberline {4.3.1}rootBrackering.m}{31}{subsection.4.3.1}%
\contentsline {subsubsection}{updateZeros}{31}{section*.61}%
\contentsline {subsubsection}{initializeValues}{32}{section*.62}%
\contentsline {subsubsection}{bracketingLoop}{33}{section*.63}%
\contentsline {subsection}{\numberline {4.3.2}printGraph.m}{34}{subsection.4.3.2}%
\contentsline {subsection}{\numberline {4.3.3}printComplexGraph.m}{36}{subsection.4.3.3}%
\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}%
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\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}%
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\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}%

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