From 679ae2635cdc6f6f6ea15d672371abd3fc345c17 Mon Sep 17 00:00:00 2001 From: PolishPigeon Date: Sun, 2 Jan 2022 01:22:56 +0100 Subject: [PATCH] Adding problem description to Project C --- .../ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.aux | 34 + ...RZYSZTOF_RUDNICKI_307585_ENUME.fdb_latexmk | 139 +++ .../ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.fls | 835 ++++++++++++++++++ .../ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.log | 533 +++++++++++ .../ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.out | 10 + .../ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.pdf | Bin 0 -> 185325 bytes ...KRZYSZTOF_RUDNICKI_307585_ENUME.synctex.gz | Bin 0 -> 14533 bytes .../ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.tex | 118 +++ .../ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.toc | 10 + 9 files changed, 1679 insertions(+) create mode 100644 ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.aux create mode 100644 ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.fdb_latexmk create mode 100644 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HcmV?d00001 diff --git a/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.tex b/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.tex new file mode 100644 index 00000000..6da17d44 --- /dev/null +++ b/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.tex @@ -0,0 +1,118 @@ +\documentclass[12pt]{report} + +\usepackage{mathtools} +\usepackage{amsmath} +\usepackage{systeme} +\usepackage{siunitx} +\usepackage[T1]{fontenc} +\usepackage{hyperref} +\usepackage{bigfoot} +\usepackage[numbered, framed]{matlab-prettifier} +\usepackage{filecontents} +\usepackage{graphicx} +\hypersetup{ +colorlinks, +citecolor=black, +filecolor=black, +linkcolor=black, +urlcolor=black +linkto=all, +} + +\newenvironment{simplechar}{% + \catcode`\^=12 +}{} + +\title{Numerical Methods, project C, Number 32} +\author{Krzysztof Rudnicki\\ Student number: 307585 \\ Advisor: dr hab. Piotr Marusak} +\date{\today} + +\let\ph\mlplaceholder % shorter macro +\lstMakeShortInline" + +\lstset{ + style = Matlab-editor, + basicstyle = \mlttfamily, + escapechar = ", + mlshowsectionrules = true, +} + +\begin{document} + + + + +\maketitle +\tableofcontents + +\chapter{Determine polynomial function fitting experimental data} +\section{Problem} +Given following samples: +\begin{center} + \begin{tabular}{| c | c |} +\hline +$x_i$ & $y_i$ \\ +\hline +-5 & -6.5743\\ +\hline +-4 & 0.9765\\ +\hline +-3 & 3.1026\\ +\hline +-2 & 1.8572 \\ +\hline +-1 & 1.3165 \\ +\hline +0 & -0.6144 \\ +\hline +1 & 0.1032 \\ +\hline +2 & 0.3729 \\ +\hline +3 & 2.5327 \\ +\hline +4 & 7.3857 \\ +\hline +5 & 9.4892 \\ +\hline + +\end{tabular} +\end{center} + +We have to determine polynomial function $ y = f(x) $ that best fits this data. +We will use least-square approximation using system of normal equation with QR factorization. + +\section{Theoretical introduction} + + + +\chapter{Determine trajectory of the motion} +\section{a) Runge-Kutta method of $4^{th}$ order and Adams PC} +\subsection{Problem} +We are given following equations: +\[ \frac{dx_1}{dt} = x_2 + x_1(0.5 - x_1^2 - x_2^2) \] +\[ \frac{dx_2}{dt} = -x_1 + x_2(0.5 - x_1^2 - x_2^2) \] + +And we have to determine the trajectory of the motion on interval $[0, 15]$ with following initial conditions: +$ x_1(0) = 8; x_2(0) = 9 $ +In this section we will use Runge-Kutta method of $4^{th}$ order and Adams PC with different step-sizes until we find an optimal constant step size - when the decrease of the step size does not influence the solution significantly. + +\subsection{Theoretical Introdution} + +\section{b) Runge-Kutta method of $4^{th}$ order with variable step size automatically adjusted} +\subsection{Problem} +We are given following equations: +\[ \frac{dx_1}{dt} = x_2 + x_1(0.5 - x_1^2 - x_2^2) \] +\[ \frac{dx_2}{dt} = -x_1 + x_2(0.5 - x_1^2 - x_2^2) \] +And we have to determine the trajectory of the motion on interval $[0, 15]$ with following initial conditions: +$ x_1(0) = 8; x_2(0) = 9 $ +In this section we will use Runge-Kutta method of $4^{th}$ order with step size automatically adjusted by the algorithm, with error estimation made according to the step-doubling rule. +\subsection{Theoretical Introdution} + +\begin{thebibliography}{9} +\bibitem{texbook} +Piotr Tatjewski (2014) \emph{Numerical Methods}, Oficyna Wydawnicza Politechniki Warszawskiej +\end{thebibliography} + + +\end{document} diff --git a/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.toc b/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.toc new file mode 100644 index 00000000..51fc6576 --- /dev/null +++ b/ENUME/projectC/ENUME_KRZYSZTOF_RUDNICKI_307585_ENUME.toc @@ -0,0 +1,10 @@ +\contentsline {chapter}{\numberline {1}Determine polynomial function fitting experimental data}{2}{chapter.1}% +\contentsline {section}{\numberline {1.1}Problem}{2}{section.1.1}% +\contentsline {section}{\numberline {1.2}Theoretical introduction}{2}{section.1.2}% +\contentsline {chapter}{\numberline {2}Determine trajectory of the motion}{3}{chapter.2}% +\contentsline {section}{\numberline {2.1}a) Runge-Kutta method of $4^{th}$ order and Adams PC}{3}{section.2.1}% +\contentsline {subsection}{\numberline {2.1.1}Problem}{3}{subsection.2.1.1}% +\contentsline {subsection}{\numberline {2.1.2}Theoretical Introdution}{4}{subsection.2.1.2}% +\contentsline {section}{\numberline {2.2}b) Runge-Kutta method of $4^{th}$ order with variable step size automatically adjusted}{4}{section.2.2}% +\contentsline {subsection}{\numberline {2.2.1}Problem}{4}{subsection.2.2.1}% +\contentsline {subsection}{\numberline {2.2.2}Theoretical Introdution}{4}{subsection.2.2.2}%