diff --git a/ENUME/projectA/pro1.txt b/ENUME/projectA/pro1.txt index 8d2332d9..cb6d7da6 100644 --- a/ENUME/projectA/pro1.txt +++ b/ENUME/projectA/pro1.txt @@ -15,7 +15,11 @@ Tutorial - Matlab Primer 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)) +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 +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. 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(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: $2^{-52} = 2.220446\mathrm{e}{-16}$ +\subsection{Practical applications of machine epsilion} +Since macheps is connected to IEEE754 standard it is always equal to the same number, which means that we can safely compare results from different machines without worrying about their individual errors. + +Macheps is also essential when we calculate cumulation of errors of given mathematical operation. + +\newpage \section{Solution} \subsection{Matlab code} @@ -61,6 +69,7 @@ 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. +\section{Discussion of the result} \begin{simplechar} \begin{lstlisting} format long @@ -68,12 +77,12 @@ disp("Display calculated macheps:") disp(macheps); disp("Display actual eps:") disp(eps); -disp("Display 2^-52) +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(2^-52 - eps) \ disp("Display difference between calculated macheps and 2^-52:") disp(macheps - 2^-52) \end{lstlisting} @@ -88,21 +97,36 @@ Display $2^{-52}$: \[2.220446049250313\mathrm{e}{-16}\] Display difference between calculated macheps and actual eps: - 0 + \[0\] Display difference between $2^{-52}$ and actual eps: - 0 + \[0\] Display difference between calculated macheps and $2^{-52}$: - 0 + \[0\] -As expected they are all equal to eachother. -\section{Discussion of the result} +As expected they are all equal to eachother. It means that our method of calculating macheps was correct. \chapter{Problem 2 - Solving a system of n linear equations - indicated method} \section{Problem} +Write a program solving a system of \textit{n} linear equations Ax = b using the indicated method (Gaussian elimination with partial pivoting). + +\section{Theoretical Introduction} +Gaussian elimination with partial pivoting consists of 3 main steps: +\subsection{Transform system of equation into an upper-triangular matrix} +We start with the system of linear equations looking like this: +\begin{align*} +a_{11}^{(1)} x_1 + a_{12}^{(1)} x_2 + \cdots + a_{1n}^{(1)} x_n &= b_1^{(1)}, \\ +a_{21}^{(1)} x_1 + a_{22}^{(1)} x_2 + \cdots + a_{2n}^{(1)} x_n &= b_2^{(1)}, \\ +&\vdotswithin{=} \\ +a_{n1}^{(1)} x_1 + a_{n2}^{(1)} x_2 + \cdots + a_{nn}^{(1)} x_n &= b_n^{(1)}, \\ +\end{align*} + +\subsection{Backward substitution} + +\subsection{Partial Pivoting} \section{Solution} @@ -112,6 +136,8 @@ As expected they are all equal to eachother. \section{Problem} +\section{Theoretical introduction} + \section{Solution} \section{Discussion of the result} @@ -119,6 +145,8 @@ As expected they are all equal to eachother. \section{Problem} +\section{Theoretical introduction} + \section{Solution} \section{Discussion of the result} diff --git a/ENUME/projectA/projectA.toc b/ENUME/projectA/projectA.toc index 7f5f7e8d..83cabb0e 100644 --- a/ENUME/projectA/projectA.toc +++ b/ENUME/projectA/projectA.toc @@ -1,19 +1,26 @@ -\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}% +\contentsline {chapter}{\numberline {1}Problem 1 - Finding machine epsilion}{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 {subsection}{\numberline {1.2.1}Definition of machine epsilion}{2}{subsection.1.2.1}% +\contentsline {subsection}{\numberline {1.2.2}Practical applications of machine epsilion}{2}{subsection.1.2.2}% +\contentsline {section}{\numberline {1.3}Solution}{3}{section.1.3}% +\contentsline {subsection}{\numberline {1.3.1}Matlab code}{3}{subsection.1.3.1}% +\contentsline {section}{\numberline {1.4}Discussion of the result}{3}{section.1.4}% +\contentsline {chapter}{\numberline {2}Problem 2 - Solving a system of n linear equations - indicated method}{5}{chapter.2}% +\contentsline {section}{\numberline {2.1}Problem}{5}{section.2.1}% +\contentsline {section}{\numberline {2.2}Theoretical Introduction}{5}{section.2.2}% +\contentsline {subsection}{\numberline {2.2.1}Transform system of equation into an upper-triangular matrix}{5}{subsection.2.2.1}% +\contentsline {subsection}{\numberline {2.2.2}Backward substitution}{6}{subsection.2.2.2}% +\contentsline {subsection}{\numberline {2.2.3}Partial Pivoting}{6}{subsection.2.2.3}% +\contentsline {section}{\numberline {2.3}Solution}{6}{section.2.3}% +\contentsline {section}{\numberline {2.4}Discussion of the result}{6}{section.2.4}% +\contentsline {chapter}{\numberline {3}Problem 3 - Solving a system of n linear equations - iterative algorithm}{7}{chapter.3}% +\contentsline {section}{\numberline {3.1}Problem}{7}{section.3.1}% +\contentsline {section}{\numberline {3.2}Theoretical introduction}{7}{section.3.2}% +\contentsline {section}{\numberline {3.3}Solution}{7}{section.3.3}% +\contentsline {section}{\numberline {3.4}Discussion of the result}{7}{section.3.4}% +\contentsline {chapter}{\numberline {4}Problem 4 - QR method of finding eigenvalues}{8}{chapter.4}% +\contentsline {section}{\numberline {4.1}Problem}{8}{section.4.1}% +\contentsline {section}{\numberline {4.2}Theoretical introduction}{8}{section.4.2}% +\contentsline {section}{\numberline {4.3}Solution}{8}{section.4.3}% +\contentsline {section}{\numberline {4.4}Discussion of the result}{8}{section.4.4}% diff --git a/ENUME/projectA/task-USOS_1153296.pdf b/ENUME/projectA/task-USOS_1153296.pdf new file mode 100644 index 00000000..b948de41 Binary files /dev/null and b/ENUME/projectA/task-USOS_1153296.pdf differ diff --git a/ENUME/projectA/usefullinks.txt b/ENUME/projectA/usefullinks.txt new file mode 100644 index 00000000..1fbeebfd --- /dev/null +++ b/ENUME/projectA/usefullinks.txt @@ -0,0 +1,31 @@ +Useful links: + +Problem 1: +https://stackoverflow.com/questions/27511349/how-to-calculate-machine-epsilon-in-matlab - How to calculate machine epsilion + +LaTeX related: +https://latex-tutorial.com/symbols/text-formatting/ - Italics + +https://www.overleaf.com/learn/latex/Table_of_contents - Table of contents + +https://www.overleaf.com/learn/latex/Sections_and_chapters - Sections and chapters + +https://www.overleaf.com/learn/latex/Bibliography_management_with_bibtex - Bibliography management + +https://en.wikibooks.org/wiki/LaTeX/Title_Creation - Title Creation + +https://tex.stackexchange.com/questions/73862/how-can-i-make-a-clickable-table-of-contents - Clickable table of contents + +https://tex.stackexchange.com/questions/75116/what-can-i-use-to-typeset-matlab-code-in-my-document - Matlab in LaTeX + +https://tex.stackexchange.com/questions/269841/1-5e-10-style-scientific-notation-looks-ugly-in-latex-math-mode-how-can-i-forma - Pretty scientific notation LaTeX + +https://tex.stackexchange.com/questions/422197/latex-environment-to-write-in-plain-text-mode - How to treat special characters as simple characters (like '^') + +https://www.overleaf.com/learn/latex/Subscripts_and_superscripts - Subscripts and superscripts latex + +https://tex.stackexchange.com/questions/32217/3-dots-in-matrix/32221 - 3 dots latex + +https://stackoverflow.com/questions/48240982/how-do-you-align-a-system-of-equations - System of equations latex + +https://tex.stackexchange.com/questions/526025/problem-with-vdotswithin - vdots alligned to equal sign