From ac88036e1c218e9a8617ad720ed2083ccd3c4a2e Mon Sep 17 00:00:00 2001 From: PolishPigeon Date: Wed, 27 Oct 2021 16:00:43 +0200 Subject: [PATCH] Project so far --- ENUME/projectA/findmacheps.m | 18 + ENUME/projectA/findmacheps.mat | Bin 0 -> 226 bytes ENUME/projectA/pro1.txt | 30 ++ ENUME/projectA/projectA.aux | 49 ++ ENUME/projectA/projectA.fdb_latexmk | 129 ++++++ ENUME/projectA/projectA.fls | 684 ++++++++++++++++++++++++++++ ENUME/projectA/projectA.log | 436 ++++++++++++++++++ ENUME/projectA/projectA.out | 19 + ENUME/projectA/projectA.pdf | Bin 0 -> 204156 bytes ENUME/projectA/projectA.synctex.gz | Bin 0 -> 24786 bytes ENUME/projectA/projectA.tex | 131 ++++++ ENUME/projectA/projectA.toc | 19 + 12 files changed, 1515 insertions(+) create mode 100755 ENUME/projectA/findmacheps.m create mode 100755 ENUME/projectA/findmacheps.mat create mode 100644 ENUME/projectA/pro1.txt create mode 100644 ENUME/projectA/projectA.aux create mode 100644 ENUME/projectA/projectA.fdb_latexmk create mode 100644 ENUME/projectA/projectA.fls create mode 100644 ENUME/projectA/projectA.log create mode 100644 ENUME/projectA/projectA.out create mode 100644 ENUME/projectA/projectA.pdf create mode 100644 ENUME/projectA/projectA.synctex.gz create mode 100755 ENUME/projectA/projectA.tex create mode 100644 ENUME/projectA/projectA.toc diff --git a/ENUME/projectA/findmacheps.m b/ENUME/projectA/findmacheps.m new file mode 100755 index 00000000..e43c8862 --- /dev/null +++ b/ENUME/projectA/findmacheps.m @@ -0,0 +1,18 @@ +macheps = 1; +while 1.0 + macheps / 2 > 1.0; + macheps = macheps/2; +end; + +format long; +disp("Display calculated macheps:") +disp(macheps) +disp("Display actual eps:") +disp(eps) +disp("Display 2^(-52)") +disp(2^(-52)) +disp("Display difference between calculated macheps and actual eps:") +disp(macheps - eps) +disp("Display difference between 2^(-52) and actual eps:") +disp(2^(-52) - eps) +disp("Display difference between calculated macheps and 2^(-52):") +disp(macheps - 2^(-52)) diff --git a/ENUME/projectA/findmacheps.mat b/ENUME/projectA/findmacheps.mat new file mode 100755 index 0000000000000000000000000000000000000000..9b2686939c34f1649dad36e7164b6b07facdfc2e GIT binary patch literal 226 zcmeZu4DoSvQZUssQ1EpO(M`+DN!3vZ$Vn_o%P-2cQgHY2i*PhE(NS*g0+E0L1{ghk`GIV8Ag-A6I5{C<14ELT!n1}; z3J18J8Xqz=c20^jWL6K6VKTeHz+iZeX)Q>(He7iDvT~)wGKF)#KIc4iHPRlmoN_+E LCe09I&JqCtE8RM8 literal 0 HcmV?d00001 diff --git a/ENUME/projectA/pro1.txt b/ENUME/projectA/pro1.txt new file mode 100644 index 00000000..8d2332d9 --- /dev/null +++ b/ENUME/projectA/pro1.txt @@ -0,0 +1,30 @@ +Z tablicy: https://studia.elka.pw.edu.pl +If you want to discuss, come within 15 minutes +classes to make some meetings to consult projects to discuss them +3 projects to be completed, time table available on studio elka +first project November 12 deadline, report should be submitted to the report module on the studia elka, tasks have already been distributed, check report on studia elka, you will get the task on PDF file. today quarter past four. then you will be given task to project assignment B and C, Statham is responsible for first two projects, Marusak last project +first two assignment 15 points each +last one 12 points +assessed by the content, by report and possibly by the interview, up to the guys the handle projects, entire group has been split into 3 dates. If you want to come to the different group just let the guy know +check two files from studia elka how to obtain Matlab, second one contains installation code. No need for full installation only basic module will do. +Tutorial - Matlab Primer +1. Start Matlab +2. Study material from Tatjewski book +3. Get acquainted with Matlab +4. Learn how to debug (Step by fashion) + +Project A +4 Tasks to be completed +1. Calculate the machine epsilon (Deliver a lot of background, theoretical findings necessary to deliver solution to the given task, The Definition of machine epsilon, Practical applications, Theoretical grounds on which you are going to design your algorithm which calculates the epsilon, the value of the epsilon should be verified (Best verification is to calculate manually, use some formula, take the value of machine epsilon from mathematical formula, maybe from some documentation maybe there is some standard, maybe Matlab has formula for it)) + +2.System of linear equations: First method is Gaussian elimination with partial pivoting, second with full. Discuss Gaussian elimination, do we need pivoting, can we apply it without pivoting, if so under what conditions, trade off between partial and full pivoting, maybe there is a gain, maybe there is a cost in accuracy, what kind of trade off we deal with, solution error of system of linear equation, discuss solution error. What is interesting is analysis of solution. This course is about analysis (Not implementing!). Take errors from graph and discuss them, for some it will be acceptable for other it will not be, (10^-8 vs 10^-2) what causes the big error, MOSTLY ASSESSED ANALYSIS NOT THE NUMBERS. Is it guessing or based on theory. + +3. Solving system of linear equations using iterative method. You don't have the guarantee how many iterations will reach the solutions. Issue of convergence, method may converge or not , based on what conditions give the condition, there is a sufficient condition there is necessary and sufficient conditions (which is slower). What happens if the sufficient condition is not fulfilled? Answer that. If the necessary condition is fulfilled what's going to happen. If the sufficient condition is fulfilled we know everything. If just necessary we don't know everything. Stop tests, when you are going to stop the process, well design, clarify in report + +4. Exercise is about calculating Eigen value (wartości własne). With shifts and without shifts. 2 algorithms to be implemented from book. You need to compare those algorithms performance ana accuracy generation and there is command eg and you can use this to produce them and then compare your solution with Matlab + +MOSTLY ABOUT ANALYSIS NOT CODING, scientifically grounded, if something goes wrong explain it. + +Bring preliminary reports before the deadline. (not obligatory but strong encouragement) + +Penalty for delivering after deadline: Lose 1 point per each late day. diff --git a/ENUME/projectA/projectA.aux b/ENUME/projectA/projectA.aux new file mode 100644 index 00000000..003e2d8b --- /dev/null +++ b/ENUME/projectA/projectA.aux @@ -0,0 +1,49 @@ +\relax +\providecommand\hyper@newdestlabel[2]{} +\providecommand\HyperFirstAtBeginDocument{\AtBeginDocument} +\HyperFirstAtBeginDocument{\ifx\hyper@anchor\@undefined +\global\let\oldcontentsline\contentsline +\gdef\contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} +\global\let\oldnewlabel\newlabel +\gdef\newlabel#1#2{\newlabelxx{#1}#2} +\gdef\newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} +\AtEndDocument{\ifx\hyper@anchor\@undefined +\let\contentsline\oldcontentsline +\let\newlabel\oldnewlabel +\fi} +\fi} +\global\let\hyper@last\relax +\gdef\HyperFirstAtBeginDocument#1{#1} +\providecommand\HyField@AuxAddToFields[1]{} +\providecommand\HyField@AuxAddToCoFields[2]{} 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+\usepackage[numbered, framed]{matlab-prettifier} +\usepackage{filecontents} +\hypersetup{ +colorlinks, +citecolor=black, +filecolor=black, +linkcolor=black, +urlcolor=black +linkto=all, +} + +\newenvironment{simplechar}{% + \catcode`\^=12 +}{} + +\title{Numerical Methods, project A, Number 31} +\author{Krzysztof Rudnicki\\ Student number: 307585 \\ Advisor: dr Adam Krzemieniowski} +\date{\today} + +\let\ph\mlplaceholder % shorter macro +\lstMakeShortInline" + +\lstset{ + style = Matlab-editor, + basicstyle = \mlttfamily, + escapechar = ", + mlshowsectionrules = true, +} + +\begin{document} + + +\maketitle +\tableofcontents + +\chapter{Introduction} + +\chapter{Problem 1 - Finding machine epsilion} + +\section{Problem} +Write a program finding macheps in the MATLAB environment +\subsection{Definition of machine epsilion} +Machine epsilion is the maximal possible relative error of the floating-point representation. (Tatjewski, p.14) +Machine epsilion is equal to $2^{-t}$ where t is number of bits in the mantissa. +In our case when we use IEEE Standard 754, mantissa is 53 bits long with first bit omitted as it is always equal to '1', so we technicaly work with 52 bits mantissa which makes the machine epsilion equal to: $2^{-52} = 2.220446\mathrm{e}{-16}$ + +\section{Solution} + +\subsection{Matlab code} +\begin{lstlisting} +macheps = 1; +while 1.0 + macheps / 2 > 1.0 + macheps = macheps/2; +end +\end{lstlisting} +Code above shifts macheps one bit to the right each iteration (by dividing by 2), it ends when we run out of mantissa bits which renders us unable to save smaller number. Due to underflow the value of macheps becomes 0 and therefore 1.0 > (macheps / 2) > 1.0 will become false. + +\begin{simplechar} +\begin{lstlisting} +format long +disp("Display calculated macheps:") +disp(macheps); +disp("Display actual eps:") +disp(eps); +disp("Display 2^-52) +disp(2^-52) +disp("Display difference between calculated macheps and actual eps:") +disp(macheps - eps) +disp("Display difference between 2^-52 and actual eps:") +disp(2^-52 - eps) +disp("Display difference between calculated macheps and 2^-52:") +disp(macheps - 2^-52) +\end{lstlisting} +\end{simplechar} +Display calculated macheps: + \[2.220446049250313\mathrm{e}{-16}\] + +Display actual eps: + \[2.220446049250313\mathrm{e}{-16}\] + +Display $2^{-52}$: + \[2.220446049250313\mathrm{e}{-16}\] + +Display difference between calculated macheps and actual eps: + 0 + +Display difference between $2^{-52}$ and actual eps: + 0 + +Display difference between calculated macheps and $2^{-52}$: + 0 + +As expected they are all equal to eachother. +\section{Discussion of the result} + + +\chapter{Problem 2 - Solving a system of n linear equations - indicated method} + +\section{Problem} + +\section{Solution} + +\section{Discussion of the result} + +\chapter{Problem 3 - Solving a system of n linear equations - iterative algorithm} + +\section{Problem} + +\section{Solution} + +\section{Discussion of the result} +\chapter{Problem 4 - QR method of finding eigenvalues} + +\section{Problem} + +\section{Solution} + +\section{Discussion of the result} + +\begin{thebibliography}{9} +\bibitem{texbook} +Piotr Tatjewski (2014) \emph{Numerical Methods}, Oficyna Wydawnicza Politechniki Warszawskiej +\end{thebibliography} + +\end{document} diff --git a/ENUME/projectA/projectA.toc b/ENUME/projectA/projectA.toc new file mode 100644 index 00000000..7f5f7e8d --- /dev/null +++ b/ENUME/projectA/projectA.toc @@ -0,0 +1,19 @@ +\contentsline {chapter}{\numberline {1}Introduction}{2}{chapter.1}% +\contentsline {chapter}{\numberline {2}Problem 1 - Finding machine epsilion}{3}{chapter.2}% +\contentsline {section}{\numberline {2.1}Problem}{3}{section.2.1}% +\contentsline {subsection}{\numberline {2.1.1}Definition of machine epsilion}{3}{subsection.2.1.1}% +\contentsline {section}{\numberline {2.2}Solution}{3}{section.2.2}% +\contentsline {subsection}{\numberline {2.2.1}Matlab code}{3}{subsection.2.2.1}% +\contentsline {section}{\numberline {2.3}Discussion of the result}{4}{section.2.3}% +\contentsline {chapter}{\numberline {3}Problem 2 - Solving a system of n linear equations - indicated method}{5}{chapter.3}% +\contentsline {section}{\numberline {3.1}Problem}{5}{section.3.1}% +\contentsline {section}{\numberline {3.2}Solution}{5}{section.3.2}% +\contentsline {section}{\numberline {3.3}Discussion of the result}{5}{section.3.3}% +\contentsline {chapter}{\numberline {4}Problem 3 - Solving a system of n linear equations - iterative algorithm}{6}{chapter.4}% +\contentsline {section}{\numberline {4.1}Problem}{6}{section.4.1}% +\contentsline {section}{\numberline {4.2}Solution}{6}{section.4.2}% +\contentsline {section}{\numberline {4.3}Discussion of the result}{6}{section.4.3}% +\contentsline {chapter}{\numberline {5}Problem 4 - QR method of finding eigenvalues}{7}{chapter.5}% +\contentsline {section}{\numberline {5.1}Problem}{7}{section.5.1}% +\contentsline {section}{\numberline {5.2}Solution}{7}{section.5.2}% +\contentsline {section}{\numberline {5.3}Discussion of the result}{7}{section.5.3}%