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Movind all code to apendix

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PolishPigeon 2021-11-12 11:29:10 +01:00
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101
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Chapter 1. Chapter 1.
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Chapter 2.
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Chapter 3. Chapter 3.
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Chapter 4. Chapter 4.
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Chapter 5. Chapter 5.
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@ -4,72 +4,75 @@
\BOOKMARK [2][-]{subsection.1.2.1}{Definition of machine epsilion}{section.1.2}% 4 \BOOKMARK [2][-]{subsection.1.2.1}{Definition of machine epsilion}{section.1.2}% 4
\BOOKMARK [2][-]{subsection.1.2.2}{Practical applications of machine epsilion}{section.1.2}% 5 \BOOKMARK [2][-]{subsection.1.2.2}{Practical applications of machine epsilion}{section.1.2}% 5
\BOOKMARK [1][-]{section.1.3}{Solution}{chapter.1}% 6 \BOOKMARK [1][-]{section.1.3}{Solution}{chapter.1}% 6
\BOOKMARK [2][-]{subsection.1.3.1}{Matlab code}{section.1.3}% 7 \BOOKMARK [1][-]{section.1.4}{Results}{chapter.1}% 7
\BOOKMARK [1][-]{section.1.4}{Discussion of the result}{chapter.1}% 8 \BOOKMARK [0][-]{chapter.2}{Problem 2 - Solving a system of n linear equations - indicated method}{}% 8
\BOOKMARK [0][-]{chapter.2}{Problem 2 - Solving a system of n linear equations - indicated method}{}% 9 \BOOKMARK [1][-]{section.2.1}{Problem}{chapter.2}% 9
\BOOKMARK [1][-]{section.2.1}{Problem}{chapter.2}% 10 \BOOKMARK [1][-]{section.2.2}{Theoretical Introduction}{chapter.2}% 10
\BOOKMARK [1][-]{section.2.2}{Theoretical Introduction}{chapter.2}% 11 \BOOKMARK [2][-]{subsection.2.2.1}{Transform matrix into upper-triangular matrix}{section.2.2}% 11
\BOOKMARK [2][-]{subsection.2.2.1}{Transform matrix into upper-triangular matrix}{section.2.2}% 12 \BOOKMARK [2][-]{subsection.2.2.2}{Backward substitution}{section.2.2}% 12
\BOOKMARK [2][-]{subsection.2.2.2}{Backward substitution}{section.2.2}% 13 \BOOKMARK [2][-]{subsection.2.2.3}{Partial Pivoting}{section.2.2}% 13
\BOOKMARK [2][-]{subsection.2.2.3}{Partial Pivoting}{section.2.2}% 14 \BOOKMARK [1][-]{section.2.3}{Results}{chapter.2}% 14
\BOOKMARK [1][-]{section.2.3}{Results}{chapter.2}% 15 \BOOKMARK [2][-]{subsection.2.3.1}{2a\)}{section.2.3}% 15
\BOOKMARK [2][-]{subsection.2.3.1}{2a\)}{section.2.3}% 16 \BOOKMARK [2][-]{subsection.2.3.2}{2b\)}{section.2.3}% 16
\BOOKMARK [2][-]{subsection.2.3.2}{2b\)}{section.2.3}% 17 \BOOKMARK [1][-]{section.2.4}{Discussion of results}{chapter.2}% 17
\BOOKMARK [1][-]{section.2.4}{Discussion of results}{chapter.2}% 18 \BOOKMARK [2][-]{subsection.2.4.1}{Errors in b\)}{section.2.4}% 18
\BOOKMARK [2][-]{subsection.2.4.1}{Errors in b\)}{section.2.4}% 19 \BOOKMARK [0][-]{chapter.3}{Problem 3 - Solving a system of n linear equations - iterative algorithm}{}% 19
\BOOKMARK [0][-]{chapter.3}{Problem 3 - Solving a system of n linear equations - iterative algorithm}{}% 20 \BOOKMARK [1][-]{section.3.1}{Problem}{chapter.3}% 20
\BOOKMARK [1][-]{section.3.1}{Problem}{chapter.3}% 21 \BOOKMARK [1][-]{section.3.2}{Theoretical introduction}{chapter.3}% 21
\BOOKMARK [1][-]{section.3.2}{Theoretical introduction}{chapter.3}% 22 \BOOKMARK [2][-]{subsection.3.2.1}{Procedure}{section.3.2}% 22
\BOOKMARK [2][-]{subsection.3.2.1}{Procedure}{section.3.2}% 23 \BOOKMARK [1][-]{section.3.3}{Results}{chapter.3}% 23
\BOOKMARK [1][-]{section.3.3}{Results}{chapter.3}% 24 \BOOKMARK [2][-]{subsection.3.3.1}{Jacobi method result}{section.3.3}% 24
\BOOKMARK [2][-]{subsection.3.3.1}{Jacobi method result}{section.3.3}% 25 \BOOKMARK [2][-]{subsection.3.3.2}{Gauss-Seidel method result}{section.3.3}% 25
\BOOKMARK [2][-]{subsection.3.3.2}{Gauss-Seidel method result}{section.3.3}% 26 \BOOKMARK [1][-]{section.3.4}{Discussion of results}{chapter.3}% 26
\BOOKMARK [1][-]{section.3.4}{Discussion of results}{chapter.3}% 27 \BOOKMARK [2][-]{subsection.3.4.1}{Comparison based on table}{section.3.4}% 27
\BOOKMARK [2][-]{subsection.3.4.1}{Comparison based on table}{section.3.4}% 28 \BOOKMARK [2][-]{subsection.3.4.2}{Convergence}{section.3.4}% 28
\BOOKMARK [2][-]{subsection.3.4.2}{Convergence}{section.3.4}% 29 \BOOKMARK [0][-]{chapter.4}{Problem 4 - QR method of finding eigenvalues}{}% 29
\BOOKMARK [0][-]{chapter.4}{Problem 4 - QR method of finding eigenvalues}{}% 30 \BOOKMARK [1][-]{section.4.1}{Problem}{chapter.4}% 30
\BOOKMARK [1][-]{section.4.1}{Problem}{chapter.4}% 31 \BOOKMARK [1][-]{section.4.2}{Theoretical introduction}{chapter.4}% 31
\BOOKMARK [1][-]{section.4.2}{Theoretical introduction}{chapter.4}% 32 \BOOKMARK [2][-]{subsection.4.2.1}{Eigenvalues}{section.4.2}% 32
\BOOKMARK [2][-]{subsection.4.2.1}{Eigenvalues}{section.4.2}% 33 \BOOKMARK [2][-]{subsection.4.2.2}{QR method for finding eigenvalues}{section.4.2}% 33
\BOOKMARK [2][-]{subsection.4.2.2}{QR method for finding eigenvalues}{section.4.2}% 34 \BOOKMARK [1][-]{section.4.3}{Results}{chapter.4}% 34
\BOOKMARK [1][-]{section.4.3}{Results}{chapter.4}% 35 \BOOKMARK [2][-]{subsection.4.3.1}{Starting matrix}{section.4.3}% 35
\BOOKMARK [2][-]{subsection.4.3.1}{Starting matrix}{section.4.3}% 36 \BOOKMARK [2][-]{subsection.4.3.2}{QR method with no shifts}{section.4.3}% 36
\BOOKMARK [2][-]{subsection.4.3.2}{QR method with no shifts}{section.4.3}% 37 \BOOKMARK [2][-]{subsection.4.3.3}{QR method with shifts}{section.4.3}% 37
\BOOKMARK [2][-]{subsection.4.3.3}{QR method with shifts}{section.4.3}% 38 \BOOKMARK [1][-]{section.4.4}{Discussion of the result}{chapter.4}% 38
\BOOKMARK [1][-]{section.4.4}{Discussion of the result}{chapter.4}% 39 \BOOKMARK [2][-]{subsection.4.4.1}{Plot}{section.4.4}% 39
\BOOKMARK [2][-]{subsection.4.4.1}{Plot}{section.4.4}% 40 \BOOKMARK [2][-]{subsection.4.4.2}{Shift method superiority}{section.4.4}% 40
\BOOKMARK [2][-]{subsection.4.4.2}{Shift method superiority}{section.4.4}% 41 \BOOKMARK [0][-]{chapter.5}{Code appendix}{}% 41
\BOOKMARK [0][-]{chapter.5}{Code appendix}{}% 42 \BOOKMARK [1][-]{section.5.1}{Task 1 Code}{chapter.5}% 42
\BOOKMARK [1][-]{section.5.1}{Task 2 Code}{chapter.5}% 43 \BOOKMARK [2][-]{subsection.5.1.1}{Find macheps}{section.5.1}% 43
\BOOKMARK [2][-]{subsection.5.1.1}{Main function}{section.5.1}% 44 \BOOKMARK [2][-]{subsection.5.1.2}{Display results}{section.5.1}% 44
\BOOKMARK [2][-]{subsection.5.1.2}{checkIfMatrixIsSquareMatrix}{section.5.1}% 45 \BOOKMARK [1][-]{section.5.2}{Task 2 Code}{chapter.5}% 45
\BOOKMARK [2][-]{subsection.5.1.3}{gaussianEliminationWithPartialPivoting}{section.5.1}% 46 \BOOKMARK [2][-]{subsection.5.2.1}{Main function}{section.5.2}% 46
\BOOKMARK [2][-]{subsection.5.1.4}{partialPivoting}{section.5.1}% 47 \BOOKMARK [2][-]{subsection.5.2.2}{checkIfMatrixIsSquareMatrix}{section.5.2}% 47
\BOOKMARK [2][-]{subsection.5.1.5}{partialPivotingSwapOneRow}{section.5.1}% 48 \BOOKMARK [2][-]{subsection.5.2.3}{gaussianEliminationWithPartialPivoting}{section.5.2}% 48
\BOOKMARK [2][-]{subsection.5.1.6}{swapRowMatrix}{section.5.1}% 49 \BOOKMARK [2][-]{subsection.5.2.4}{partialPivoting}{section.5.2}% 49
\BOOKMARK [2][-]{subsection.5.1.7}{swapValueVector}{section.5.1}% 50 \BOOKMARK [2][-]{subsection.5.2.5}{partialPivotingSwapOneRow}{section.5.2}% 50
\BOOKMARK [2][-]{subsection.5.1.8}{gaussianElimination}{section.5.1}% 51 \BOOKMARK [2][-]{subsection.5.2.6}{swapRowMatrix}{section.5.2}% 51
\BOOKMARK [2][-]{subsection.5.1.9}{substractRows}{section.5.1}% 52 \BOOKMARK [2][-]{subsection.5.2.7}{swapValueVector}{section.5.2}% 52
\BOOKMARK [2][-]{subsection.5.1.10}{backSubstitutionPhase}{section.5.1}% 53 \BOOKMARK [2][-]{subsection.5.2.8}{gaussianElimination}{section.5.2}% 53
\BOOKMARK [2][-]{subsection.5.1.11}{iterativeResidualCorrection}{section.5.1}% 54 \BOOKMARK [2][-]{subsection.5.2.9}{substractRows}{section.5.2}% 54
\BOOKMARK [2][-]{subsection.5.1.12}{improveSolution}{section.5.1}% 55 \BOOKMARK [2][-]{subsection.5.2.10}{backSubstitutionPhase}{section.5.2}% 55
\BOOKMARK [2][-]{subsection.5.1.13}{plotErrorsGaussian}{section.5.1}% 56 \BOOKMARK [2][-]{subsection.5.2.11}{iterativeResidualCorrection}{section.5.2}% 56
\BOOKMARK [1][-]{section.5.2}{Task 3 Code}{chapter.5}% 57 \BOOKMARK [2][-]{subsection.5.2.12}{improveSolution}{section.5.2}% 57
\BOOKMARK [2][-]{subsection.5.2.1}{initializeValues}{section.5.2}% 58 \BOOKMARK [2][-]{subsection.5.2.13}{plotErrorsGaussian}{section.5.2}% 58
\BOOKMARK [2][-]{subsection.5.2.2}{decomposeMatrix}{section.5.2}% 59 \BOOKMARK [1][-]{section.5.3}{Task 3 Code}{chapter.5}% 59
\BOOKMARK [2][-]{subsection.5.2.3}{jacobiLoop}{section.5.2}% 60 \BOOKMARK [2][-]{subsection.5.3.1}{initializeValues}{section.5.3}% 60
\BOOKMARK [2][-]{subsection.5.2.4}{jacobiInsideLoop}{section.5.2}% 61 \BOOKMARK [2][-]{subsection.5.3.2}{decomposeMatrix}{section.5.3}% 61
\BOOKMARK [2][-]{subsection.5.2.5}{jacobiEquation}{section.5.2}% 62 \BOOKMARK [2][-]{subsection.5.3.3}{jacobiLoop}{section.5.3}% 62
\BOOKMARK [2][-]{subsection.5.2.6}{gaussSeidelLoop}{section.5.2}% 63 \BOOKMARK [2][-]{subsection.5.3.4}{jacobiInsideLoop}{section.5.3}% 63
\BOOKMARK [2][-]{subsection.5.2.7}{gaussiInsideLoop}{section.5.2}% 64 \BOOKMARK [2][-]{subsection.5.3.5}{jacobiEquation}{section.5.3}% 64
\BOOKMARK [2][-]{subsection.5.2.8}{gaussSeidelEquation}{section.5.2}% 65 \BOOKMARK [2][-]{subsection.5.3.6}{gaussSeidelLoop}{section.5.3}% 65
\BOOKMARK [2][-]{subsection.5.2.9}{checkError}{section.5.2}% 66 \BOOKMARK [2][-]{subsection.5.3.7}{gaussiInsideLoop}{section.5.3}% 66
\BOOKMARK [2][-]{subsection.5.2.10}{endOfLoop}{section.5.2}% 67 \BOOKMARK [2][-]{subsection.5.3.8}{gaussSeidelEquation}{section.5.3}% 67
\BOOKMARK [2][-]{subsection.5.2.11}{dispFinalResults}{section.5.2}% 68 \BOOKMARK [2][-]{subsection.5.3.9}{checkError}{section.5.3}% 68
\BOOKMARK [2][-]{subsection.5.2.12}{plotIterations}{section.5.2}% 69 \BOOKMARK [2][-]{subsection.5.3.10}{endOfLoop}{section.5.3}% 69
\BOOKMARK [1][-]{section.5.3}{Task 4 Code}{chapter.5}% 70 \BOOKMARK [2][-]{subsection.5.3.11}{dispFinalResults}{section.5.3}% 70
\BOOKMARK [2][-]{subsection.5.3.1}{Gram-Schmid algorithm}{section.5.3}% 71 \BOOKMARK [2][-]{subsection.5.3.12}{plotIterations}{section.5.3}% 71
\BOOKMARK [2][-]{subsection.5.3.2}{task4}{section.5.3}% 72 \BOOKMARK [1][-]{section.5.4}{Task 4 Code}{chapter.5}% 72
\BOOKMARK [2][-]{subsection.5.3.3}{QRNoShifts}{section.5.3}% 73 \BOOKMARK [2][-]{subsection.5.4.1}{Gram-Schmid algorithm}{section.5.4}% 73
\BOOKMARK [2][-]{subsection.5.3.4}{QRShifts}{section.5.3}% 74 \BOOKMARK [2][-]{subsection.5.4.2}{task4}{section.5.4}% 74
\BOOKMARK [2][-]{subsection.5.3.5}{task4Plot}{section.5.3}% 75 \BOOKMARK [2][-]{subsection.5.4.3}{QRNoShifts}{section.5.4}% 75
\BOOKMARK [2][-]{subsection.5.4.4}{QRShifts}{section.5.4}% 76
\BOOKMARK [2][-]{subsection.5.4.5}{task4Plot}{section.5.4}% 77
\BOOKMARK [2][-]{subsection.5.4.6}{Matrix generation}{section.5.4}% 78

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@ -62,33 +62,12 @@ Macheps is also essential when we calculate cumulation of errors of given mathem
\newpage \newpage
\section{Solution} \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. 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.
\newpage \newpage
\section{Discussion of the result} \section{Results}
\begin{simplechar} \hyperlink{function1_macheps}{Code for finding macheps}\\
\begin{lstlisting} \hyperlink{function1_display}{Code for displaying results}\\
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: Display calculated macheps:
\[2.220446049250313\mathrm{e}{-16}\] \[2.220446049250313\mathrm{e}{-16}\]
@ -333,6 +312,7 @@ Error for 'B' system of equations for matlab method: \[ 2.611906929269057e-04 \]
\newpage \newpage
\section{Discussion of results} \section{Discussion of results}
\hyperlink{function2_plot}{Code for generating plots}\\
As we can see error for 'B' system of equations increased after our residual correction. But that's just one matrix and vector pair, after testing more pairs we get following graphs (Tested on systems of equations from a) and b) respectively, with maximum size of matrices and vectors equal to 100): As we can see error for 'B' system of equations increased after our residual correction. But that's just one matrix and vector pair, after testing more pairs we get following graphs (Tested on systems of equations from a) and b) respectively, with maximum size of matrices and vectors equal to 100):
\begin{center} \begin{center}
\includegraphics[scale=0.5]{errorsA.eps} \includegraphics[scale=0.5]{errorsA.eps}
@ -702,7 +682,7 @@ Error:
\[ r = \| \mathbf{A}\mathbf{x} - \mathbf{b}\| = 1.699812218689508e-10 \] \[ r = \| \mathbf{A}\mathbf{x} - \mathbf{b}\| = 1.699812218689508e-10 \]
We managed to do this in \textbf{57} iterations of our loop, and the demanded tolerance changed to $4*10^{-10}$ We managed to do this in \textbf{57} iterations of our loop, and the demanded tolerance changed to $4*10^{-10}$
Compared to matlab $ A \ b $ function Compared to matlab function
\[ x_{matlab} = \left( \begin{array}{cc} \[ x_{matlab} = \left( \begin{array}{cc}
-0.930024655110760 \\ -0.930024655110760 \\
-1.223407298665612 \\ -1.223407298665612 \\
@ -912,26 +892,13 @@ For Gaussian-Seidel method the matrix is convergent if it has strong diagonal do
\subsubsection{2b) task convergence } \subsubsection{2b) task convergence }
Let's check strong diagonal dominance from matrix \textbf{A} from task 2b) in matlab: Let's check strong diagonal dominance from matrix \textbf{A} from task 2b) in matlab:
\begin{simplechar} \begin{simplechar}
\begin{lstlisting}
function d = checkIfDiagonallyDominant(Matrix) \hyperlink{function_3_dominant}{Code for checking if matrix is diagonally dominant}
d = 1; \\
[Rows, ~] = size(Matrix);
for i = 1 : Rows
rowsSum = 0;
for j = 1 : Rows
rowsSum = rowsSum + abs(Matrix(i, j));
end
rowsSum = rowsSum - Matrix(i, i);
if Matrix(i, i) <= rowsSum
d = 0;
break
end
end
end
\end{lstlisting}
\end{simplechar}
checkIfDiagonallyDominant(matrixB(10)) returns '0', therefore matrix from task 2b) will \textbf{not} converge. And as expected it does not work when put in my jacobiLoop function. checkIfDiagonallyDominant(matrixB(10)) returns '0', therefore matrix from task 2b) will \textbf{not} converge. And as expected it does not work when put in my jacobiLoop function.
\subsubsection{Iterations as function of size of Matrix} \subsubsection{Iterations as function of size of Matrix}
\hyperlink{function_3_plot}{Code for creating plot}
I have used matrix from task 2a), demanded tolerance equal to 10e-10, and max size of matrix equal to 500. for both methods I have used matrix from task 2a), demanded tolerance equal to 10e-10, and max size of matrix equal to 500. for both methods
\begin{center} \begin{center}
\includegraphics[scale=0.75]{iterations.eps} \includegraphics[scale=0.75]{iterations.eps}
@ -980,17 +947,7 @@ Wher $p_k$ should be chosen as a best estimate of $\lambda_{i+1}$
\section{Results} \section{Results}
\subsection{Starting matrix} \subsection{Starting matrix}
I decided to generate random symmetric matrix using following code: I decided to generate random symmetric matrix using following code:
\begin{simplechar} \hyperlink{function4_matrix_generation}{code for matrix generation.}
\begin{lstlisting}
function A = matrix4()
A = 10 * rand(5); % rand generates 5x5 matrix filled with random numbers
% we multiply by 10 to get at lest one digit in front of the dot
A = floor(A); % we floor the matrix we got to get nice natural numbers matrix
A = A * A'; % we get symmetric matrix
disp(issymmetric(A)); % we check if matrix is symmetric
end
\end{lstlisting}
\end{simplechar}
I got the following matrix: I got the following matrix:
\[ \[
\begin{bmatrix} \begin{bmatrix}
@ -1094,6 +1051,33 @@ QR method with shifts is much more efficient and reliable.
\addtocontents{toc}{\protect\newpage} \addtocontents{toc}{\protect\newpage}
\chapter{Code appendix} \chapter{Code appendix}
\section{Task 1 Code}
\hypertarget{function1_macheps}{\subsection{Find macheps}}
\begin{lstlisting}
macheps = 1;
while 1.0 + macheps / 2 > 1.0
macheps = macheps/2;
end
\end{lstlisting}
\hypertarget{function1_display}{\subsection{Display results}}
\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}
\section{Task 2 Code} \section{Task 2 Code}
\subsection{Main function} \subsection{Main function}
@ -1270,7 +1254,7 @@ end % end function
\end{simplechar} \end{simplechar}
\newpage \newpage
\subsection{plotErrorsGaussian} \hypertarget{function2_plot}{\subsection{plotErrorsGaussian}}
\begin{simplechar} \begin{simplechar}
\begin{lstlisting} \begin{lstlisting}
function plotErrorsGaussian(maxMatrixSize) function plotErrorsGaussian(maxMatrixSize)
@ -1484,7 +1468,7 @@ end
\end{simplechar} \end{simplechar}
\newpage \newpage
\subsection{plotIterations} \hypertarget{function_3_plot}{\subsection{plotIterations}}
\begin{simplechar} \begin{simplechar}
\begin{lstlisting} \begin{lstlisting}
function plotIterations() function plotIterations()
@ -1511,6 +1495,26 @@ end
\end{lstlisting} \end{lstlisting}
\end{simplechar} \end{simplechar}
\hypertarget{function_3_dominant}{Code for checking if matrix is diagonally dominant}
\begin{lstlisting}
function d = checkIfDiagonallyDominant(Matrix)
d = 1;
[Rows, ~] = size(Matrix);
for i = 1 : Rows
rowsSum = 0;
for j = 1 : Rows
rowsSum = rowsSum + abs(Matrix(i, j));
end
rowsSum = rowsSum - Matrix(i, i);
if Matrix(i, i) <= rowsSum
d = 0;
break
end
end
end
\end{lstlisting}
\end{simplechar}
\newpage \newpage
\section{Task 4 Code} \section{Task 4 Code}
@ -1849,6 +1853,19 @@ end
\end{lstlisting} \end{lstlisting}
\end{simplechar} \end{simplechar}
\hypertarget{function4_matrix_generation}{\subsection{Matrix generation}}
\begin{simplechar}
\begin{lstlisting}
function A = matrix4()
A = 10 * rand(5); % rand generates 5x5 matrix filled with random numbers
% we multiply by 10 to get at lest one digit in front of the dot
A = floor(A); % we floor the matrix we got to get nice natural numbers matrix
A = A * A'; % we get symmetric matrix
disp(issymmetric(A)); % we check if matrix is symmetric
end
\end{lstlisting}
\end{simplechar}
\begin{thebibliography}{9} \begin{thebibliography}{9}
\bibitem{texbook} \bibitem{texbook}

101
ENUME/projectA/projectA.toc
View File

@ -4,8 +4,7 @@
\contentsline {subsection}{\numberline {1.2.1}Definition of machine epsilion}{4}{subsection.1.2.1}% \contentsline {subsection}{\numberline {1.2.1}Definition of machine epsilion}{4}{subsection.1.2.1}%
\contentsline {subsection}{\numberline {1.2.2}Practical applications of machine epsilion}{5}{subsection.1.2.2}% \contentsline {subsection}{\numberline {1.2.2}Practical applications of machine epsilion}{5}{subsection.1.2.2}%
\contentsline {section}{\numberline {1.3}Solution}{6}{section.1.3}% \contentsline {section}{\numberline {1.3}Solution}{6}{section.1.3}%
\contentsline {subsection}{\numberline {1.3.1}Matlab code}{6}{subsection.1.3.1}% \contentsline {section}{\numberline {1.4}Results}{7}{section.1.4}%
\contentsline {section}{\numberline {1.4}Discussion of the result}{7}{section.1.4}%
\contentsline {chapter}{\numberline {2}Problem 2 - Solving a system of n linear equations - indicated method}{8}{chapter.2}% \contentsline {chapter}{\numberline {2}Problem 2 - Solving a system of n linear equations - indicated method}{8}{chapter.2}%
\contentsline {section}{\numberline {2.1}Problem}{8}{section.2.1}% \contentsline {section}{\numberline {2.1}Problem}{8}{section.2.1}%
\contentsline {section}{\numberline {2.2}Theoretical Introduction}{8}{section.2.2}% \contentsline {section}{\numberline {2.2}Theoretical Introduction}{8}{section.2.2}%
@ -47,60 +46,63 @@
\contentsline {subsection}{\numberline {3.4.1}Comparison based on table}{34}{subsection.3.4.1}% \contentsline {subsection}{\numberline {3.4.1}Comparison based on table}{34}{subsection.3.4.1}%
\contentsline {subsection}{\numberline {3.4.2}Convergence}{35}{subsection.3.4.2}% \contentsline {subsection}{\numberline {3.4.2}Convergence}{35}{subsection.3.4.2}%
\contentsline {subsubsection}{2b) task convergence }{35}{section*.20}% \contentsline {subsubsection}{2b) task convergence }{35}{section*.20}%
\contentsline {subsubsection}{Iterations as function of size of Matrix}{36}{section*.21}% \contentsline {subsubsection}{Iterations as function of size of Matrix}{35}{section*.21}%
\contentsline {chapter}{\numberline {4}Problem 4 - QR method of finding eigenvalues}{39}{chapter.4}% \contentsline {chapter}{\numberline {4}Problem 4 - QR method of finding eigenvalues}{38}{chapter.4}%
\contentsline {section}{\numberline {4.1}Problem}{39}{section.4.1}% \contentsline {section}{\numberline {4.1}Problem}{38}{section.4.1}%
\contentsline {section}{\numberline {4.2}Theoretical introduction}{39}{section.4.2}% \contentsline {section}{\numberline {4.2}Theoretical introduction}{38}{section.4.2}%
\contentsline {subsection}{\numberline {4.2.1}Eigenvalues}{39}{subsection.4.2.1}% \contentsline {subsection}{\numberline {4.2.1}Eigenvalues}{38}{subsection.4.2.1}%
\contentsline {subsection}{\numberline {4.2.2}QR method for finding eigenvalues}{40}{subsection.4.2.2}% \contentsline {subsection}{\numberline {4.2.2}QR method for finding eigenvalues}{39}{subsection.4.2.2}%
\contentsline {section}{\numberline {4.3}Results}{41}{section.4.3}% \contentsline {section}{\numberline {4.3}Results}{40}{section.4.3}%
\contentsline {subsection}{\numberline {4.3.1}Starting matrix}{41}{subsection.4.3.1}% \contentsline {subsection}{\numberline {4.3.1}Starting matrix}{40}{subsection.4.3.1}%
\contentsline {subsection}{\numberline {4.3.2}QR method with no shifts}{42}{subsection.4.3.2}% \contentsline {subsection}{\numberline {4.3.2}QR method with no shifts}{41}{subsection.4.3.2}%
\contentsline {subsection}{\numberline {4.3.3}QR method with shifts}{42}{subsection.4.3.3}% \contentsline {subsection}{\numberline {4.3.3}QR method with shifts}{41}{subsection.4.3.3}%
\contentsline {section}{\numberline {4.4}Discussion of the result}{43}{section.4.4}% \contentsline {section}{\numberline {4.4}Discussion of the result}{42}{section.4.4}%
\contentsline {subsection}{\numberline {4.4.1}Plot}{43}{subsection.4.4.1}% \contentsline {subsection}{\numberline {4.4.1}Plot}{42}{subsection.4.4.1}%
\contentsline {subsection}{\numberline {4.4.2}Shift method superiority}{44}{subsection.4.4.2}% \contentsline {subsection}{\numberline {4.4.2}Shift method superiority}{43}{subsection.4.4.2}%
\newpage \newpage
\contentsline {chapter}{\numberline {5}Code appendix}{45}{chapter.5}% \contentsline {chapter}{\numberline {5}Code appendix}{44}{chapter.5}%
\contentsline {section}{\numberline {5.1}Task 2 Code}{45}{section.5.1}% \contentsline {section}{\numberline {5.1}Task 1 Code}{44}{section.5.1}%
\contentsline {subsection}{\numberline {5.1.1}Main function}{45}{subsection.5.1.1}% \contentsline {subsection}{\numberline {5.1.1}Find macheps}{44}{subsection.5.1.1}%
\contentsline {subsection}{\numberline {5.1.2}checkIfMatrixIsSquareMatrix}{46}{subsection.5.1.2}% \contentsline {subsection}{\numberline {5.1.2}Display results}{44}{subsection.5.1.2}%
\contentsline {subsection}{\numberline {5.1.3}gaussianEliminationWithPartialPivoting}{47}{subsection.5.1.3}% \contentsline {section}{\numberline {5.2}Task 2 Code}{45}{section.5.2}%
\contentsline {subsection}{\numberline {5.1.4}partialPivoting}{47}{subsection.5.1.4}% \contentsline {subsection}{\numberline {5.2.1}Main function}{45}{subsection.5.2.1}%
\contentsline {subsection}{\numberline {5.1.5}partialPivotingSwapOneRow}{48}{subsection.5.1.5}% \contentsline {subsection}{\numberline {5.2.2}checkIfMatrixIsSquareMatrix}{45}{subsection.5.2.2}%
\contentsline {subsection}{\numberline {5.1.6}swapRowMatrix}{48}{subsection.5.1.6}% \contentsline {subsection}{\numberline {5.2.3}gaussianEliminationWithPartialPivoting}{46}{subsection.5.2.3}%
\contentsline {subsection}{\numberline {5.1.7}swapValueVector}{48}{subsection.5.1.7}% \contentsline {subsection}{\numberline {5.2.4}partialPivoting}{46}{subsection.5.2.4}%
\contentsline {subsection}{\numberline {5.1.8}gaussianElimination}{49}{subsection.5.1.8}% \contentsline {subsection}{\numberline {5.2.5}partialPivotingSwapOneRow}{47}{subsection.5.2.5}%
\contentsline {subsection}{\numberline {5.1.9}substractRows}{49}{subsection.5.1.9}% \contentsline {subsection}{\numberline {5.2.6}swapRowMatrix}{47}{subsection.5.2.6}%
\contentsline {subsection}{\numberline {5.1.10}backSubstitutionPhase}{50}{subsection.5.1.10}% \contentsline {subsection}{\numberline {5.2.7}swapValueVector}{47}{subsection.5.2.7}%
\contentsline {subsection}{\numberline {5.1.11}iterativeResidualCorrection}{50}{subsection.5.1.11}% \contentsline {subsection}{\numberline {5.2.8}gaussianElimination}{48}{subsection.5.2.8}%
\contentsline {subsection}{\numberline {5.1.12}improveSolution}{51}{subsection.5.1.12}% \contentsline {subsection}{\numberline {5.2.9}substractRows}{48}{subsection.5.2.9}%
\contentsline {subsection}{\numberline {5.1.13}plotErrorsGaussian}{52}{subsection.5.1.13}% \contentsline {subsection}{\numberline {5.2.10}backSubstitutionPhase}{49}{subsection.5.2.10}%
\contentsline {section}{\numberline {5.2}Task 3 Code}{53}{section.5.2}% \contentsline {subsection}{\numberline {5.2.11}iterativeResidualCorrection}{49}{subsection.5.2.11}%
\contentsline {subsection}{\numberline {5.2.1}initializeValues}{54}{subsection.5.2.1}% \contentsline {subsection}{\numberline {5.2.12}improveSolution}{50}{subsection.5.2.12}%
\contentsline {subsection}{\numberline {5.2.2}decomposeMatrix}{54}{subsection.5.2.2}% \contentsline {subsection}{\numberline {5.2.13}plotErrorsGaussian}{51}{subsection.5.2.13}%
\contentsline {subsection}{\numberline {5.2.3}jacobiLoop}{55}{subsection.5.2.3}% \contentsline {section}{\numberline {5.3}Task 3 Code}{52}{section.5.3}%
\contentsline {subsection}{\numberline {5.2.4}jacobiInsideLoop}{55}{subsection.5.2.4}% \contentsline {subsection}{\numberline {5.3.1}initializeValues}{53}{subsection.5.3.1}%
\contentsline {subsection}{\numberline {5.2.5}jacobiEquation}{55}{subsection.5.2.5}% \contentsline {subsection}{\numberline {5.3.2}decomposeMatrix}{53}{subsection.5.3.2}%
\contentsline {subsection}{\numberline {5.2.6}gaussSeidelLoop}{56}{subsection.5.2.6}% \contentsline {subsection}{\numberline {5.3.3}jacobiLoop}{54}{subsection.5.3.3}%
\contentsline {subsection}{\numberline {5.2.7}gaussiInsideLoop}{56}{subsection.5.2.7}% \contentsline {subsection}{\numberline {5.3.4}jacobiInsideLoop}{54}{subsection.5.3.4}%
\contentsline {subsection}{\numberline {5.2.8}gaussSeidelEquation}{57}{subsection.5.2.8}% \contentsline {subsection}{\numberline {5.3.5}jacobiEquation}{54}{subsection.5.3.5}%
\contentsline {subsection}{\numberline {5.2.9}checkError}{57}{subsection.5.2.9}% \contentsline {subsection}{\numberline {5.3.6}gaussSeidelLoop}{55}{subsection.5.3.6}%
\contentsline {subsection}{\numberline {5.2.10}endOfLoop}{58}{subsection.5.2.10}% \contentsline {subsection}{\numberline {5.3.7}gaussiInsideLoop}{55}{subsection.5.3.7}%
\contentsline {subsection}{\numberline {5.2.11}dispFinalResults}{58}{subsection.5.2.11}% \contentsline {subsection}{\numberline {5.3.8}gaussSeidelEquation}{56}{subsection.5.3.8}%
\contentsline {subsection}{\numberline {5.2.12}plotIterations}{59}{subsection.5.2.12}% \contentsline {subsection}{\numberline {5.3.9}checkError}{56}{subsection.5.3.9}%
\contentsline {section}{\numberline {5.3}Task 4 Code}{60}{section.5.3}% \contentsline {subsection}{\numberline {5.3.10}endOfLoop}{57}{subsection.5.3.10}%
\contentsline {subsection}{\numberline {5.3.1}Gram-Schmid algorithm}{60}{subsection.5.3.1}% \contentsline {subsection}{\numberline {5.3.11}dispFinalResults}{57}{subsection.5.3.11}%
\contentsline {subsection}{\numberline {5.3.12}plotIterations}{58}{subsection.5.3.12}%
\contentsline {section}{\numberline {5.4}Task 4 Code}{60}{section.5.4}%
\contentsline {subsection}{\numberline {5.4.1}Gram-Schmid algorithm}{60}{subsection.5.4.1}%
\contentsline {subsubsection}{initializeGramSchmid}{60}{section*.22}% \contentsline {subsubsection}{initializeGramSchmid}{60}{section*.22}%
\contentsline {subsubsection}{factorizeColumnsOfQ}{60}{section*.23}% \contentsline {subsubsection}{factorizeColumnsOfQ}{60}{section*.23}%
\contentsline {subsubsection}{normalizeColumns}{61}{section*.24}% \contentsline {subsubsection}{normalizeColumns}{61}{section*.24}%
\contentsline {subsection}{\numberline {5.3.2}task4}{61}{subsection.5.3.2}% \contentsline {subsection}{\numberline {5.4.2}task4}{61}{subsection.5.4.2}%
\contentsline {subsection}{\numberline {5.3.3}QRNoShifts}{61}{subsection.5.3.3}% \contentsline {subsection}{\numberline {5.4.3}QRNoShifts}{61}{subsection.5.4.3}%
\contentsline {subsubsection}{QRNoShiftsLoop}{62}{section*.25}% \contentsline {subsubsection}{QRNoShiftsLoop}{62}{section*.25}%
\contentsline {subsubsection}{QRNoShiftsInsideLoop}{62}{section*.26}% \contentsline {subsubsection}{QRNoShiftsInsideLoop}{62}{section*.26}%
\contentsline {subsubsection}{displayResults}{63}{section*.27}% \contentsline {subsubsection}{displayResults}{63}{section*.27}%
\contentsline {subsubsection}{initializeValues}{63}{section*.28}% \contentsline {subsubsection}{initializeValues}{63}{section*.28}%
\contentsline {subsection}{\numberline {5.3.4}QRShifts}{64}{subsection.5.3.4}% \contentsline {subsection}{\numberline {5.4.4}QRShifts}{64}{subsection.5.4.4}%
\contentsline {subsubsection}{initiateValues}{64}{section*.29}% \contentsline {subsubsection}{initiateValues}{64}{section*.29}%
\contentsline {subsubsection}{QRShiftLoop}{65}{section*.30}% \contentsline {subsubsection}{QRShiftLoop}{65}{section*.30}%
\contentsline {subsubsection}{findEigenValue}{65}{section*.31}% \contentsline {subsubsection}{findEigenValue}{65}{section*.31}%
@ -111,4 +113,5 @@
\contentsline {subsubsection}{solveCharactersticEquation}{67}{section*.36}% \contentsline {subsubsection}{solveCharactersticEquation}{67}{section*.36}%
\contentsline {subsubsection}{calculateZeros}{68}{section*.37}% \contentsline {subsubsection}{calculateZeros}{68}{section*.37}%
\contentsline {subsubsection}{dispResults}{68}{section*.38}% \contentsline {subsubsection}{dispResults}{68}{section*.38}%
\contentsline {subsection}{\numberline {5.3.5}task4Plot}{69}{subsection.5.3.5}% \contentsline {subsection}{\numberline {5.4.5}task4Plot}{69}{subsection.5.4.5}%
\contentsline {subsection}{\numberline {5.4.6}Matrix generation}{69}{subsection.5.4.6}%