finishing touches on matlab code and transfering code and solutions to latex

This commit is contained in:
PolishPigeon 2021-11-06 06:22:19 +01:00
parent 9dad28a4f7
commit c92984e9f5
11 changed files with 2069 additions and 63 deletions

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@ -1,9 +1,21 @@
function x = indicatedMethod(Matrix, Vector)
[~,Columns] = size(Matrix);
function x = indicatedMethod(Matrix, Vector) % Name of the method as in the textbook
% x stands for obtained result
checkIfMatrixIsSquareMatrix(Matrix);
[Matrix, Vector] = gaussianEliminationWithPartialPivoting(Columns, Matrix, Vector);
[Matrix, Vector, x] = backSubstitutionPhase(Columns, Matrix, Vector);
x = iterativeResidualCorrection(Matrix, x, Vector);
[Matrix, Vector, x] = solveSystem(Matrix, Vector);
norm(x)
x = iterativeResidualCorrection(Matrix, x, Vector); % Improve on the solution
norm(x)
end % end function
function [Matrix, Vector, x] = solveSystem(Matrix, Vector)
[~,Columns] = size(Matrix); % We need to know how big the matrix is in next steps
% notice the '~', since we assume we use square matrix, we do not need
% to have another variable for number of rows since it is the same as
% number of columns
[Matrix, Vector] = gaussianEliminationWithPartialPivoting(Columns, Matrix, Vector);
% Change matrix to upper triangular matrix
[Matrix, Vector, x] = backSubstitutionPhase(Columns, Matrix, Vector);
% Get the solution
end % end function
function checkIfMatrixIsSquareMatrix(Matrix)
@ -14,79 +26,81 @@ function checkIfMatrixIsSquareMatrix(Matrix)
end % end function
function [Matrix, Vector] = gaussianEliminationWithPartialPivoting(Columns, Matrix, Vector)
for i1 = 1 : Columns - 1
m = max(Matrix(i1:Columns,i1));
partialPivoting(Matrix, Vector, i1, m);
[Matrix, Vector] = gaussianElimination(i1, Columns, Matrix, Vector);
for j = 1 : Columns
centralElement = max(Matrix(j:Columns,j));
% we stay in the same row (j) but we change columns, as in the
% textbook
[Matrix, Vector] = partialPivoting(Matrix, Vector, j, centralElement, Columns);
% ensures that a_kk != 0 and reduces errors
[Matrix, Vector] = gaussianElimination(j, Columns, Matrix, Vector);
% change matrix into upper triangular matrix
end % end for
end % end function
function [Matrix, Vector] = partialPivoting(Matrix, Vector, i1, m)
[~,Columns] = size(Matrix);
for i3 = i1 : Columns
partialPivotingSwapOneRow(Matrix, Vector, i1, i3, m);
function [Matrix, Vector] = partialPivoting(Matrix, Vector, j, centralElement, Columns)
for k = j : Columns
partialPivotingSwapOneRow(Matrix, Vector, j, k, centralElement);
end % end for
end % end function
function [Matrix, Vector] = partialPivotingSwapOneRow(Matrix, Vector, i1, i3, m)
if Matrix(i3,i1) == m
swapRowMatrix(Matrix, i1, i3);
swapValueVector(Vector, i1, i3);
function [Matrix, Vector] = partialPivotingSwapOneRow(Matrix, Vector, j, k, centralElement)
if Matrix(k,j) == centralElement
swapRowMatrix(Matrix, j, k); % swap jth row with kth row
swapValueVector(Vector, j, k); % swap jth value with kth value
end % end if
end % end function
function Matrix = swapRowMatrix(Matrix, i1, i3)
tempRow = Matrix(i1 , :);
Matrix(i1 , :) = Matrix(i3, :);
Matrix(i3, :) = tempRow;
function Matrix = swapRowMatrix(Matrix, j, k)
temp = Matrix(j , :); % ' : ' denote "all elements in jth row"
Matrix(j , :) = Matrix(k, :);
Matrix(k, :) = temp; % temp equal to previous value of jth row
end
function Vector = swapValueVector(Vector, i1, i3)
tempVal = Vector(i1);
Vector(i1) = Vector(i3);
Vector(i3) = tempVal;
function Vector = swapValueVector(Vector, j, k)
temp = Vector(j);
Vector(j) = Vector(k);
Vector(k) = temp; % temp equal to previous value of k element of vector
end % end function
function [Matrix, Vector] = gaussianElimination(i1, Columns, Matrix, Vector)
for i2 = i1 + 1 : Columns
l = Matrix(i2,i1) / Matrix(i1,i1);
[Matrix, Vector] = substractRows(Matrix, Vector, i2, l, i1, Columns);
function [Matrix, Vector] = gaussianElimination(j, Columns, Matrix, Vector)
for i = j + 1 : Columns
rowMultiplier = Matrix(i,j) / Matrix(j,j);
[Matrix, Vector] = substractRows(Matrix, Vector, i, rowMultiplier, j, Columns);
end % end for
end % end function
function [Matrix, Vector] = substractRows(Matrix, Vector, i2, l, i1, Columns)
Vector(i2,1) = Vector(i2,1) - l * Vector(i1, 1);
for t = 1 : Columns
Matrix(i2,t) = Matrix(i2,t) - l * Matrix(i1, t);
function [Matrix, Vector] = substractRows(Matrix, Vector, i, rowMultiplier, j, Columns)
Vector(i) = Vector(i) - rowMultiplier * Vector(j);
for curentColumn = 1 : Columns
Matrix(i,curentColumn) = Matrix(i,curentColumn) - rowMultiplier * Matrix(j, curentColumn);
end % end for
end % end function
function [Matrix, Vector, x] = backSubstitutionPhase(Columns, Matrix, Vector)
for i3 = Columns : -1 : 1
E = 0;
for iter = i3+1 : Columns
E = E + Matrix(i3,iter) * x(iter,1);
for k = Columns : -1 : 1
% Start at final column and move by -1 each iteration until we reach 1
equation = 0;
for j = k+1 : Columns
equation = equation + Matrix(k,j) * x(j, 1);
% even though x is a vector we still need to put '1' to ensure
% that number of columns in the first matrix matches number of
% rows in second matrix
end % end for
x(i3, 1) = (Vector(i3,1) - E) / Matrix(i3,i3);
x(k, 1) = (Vector(k,1) - equation) / Matrix(k,k);
% even though x is a vector we still need to put '1' to ensure
% that we do not exceed array bounds
end % end for
end % end function
function x = iterativeResidualCorrection(Matrix, x, Vector)
r = Matrix*x - Vector;
euclideanColumnsormOfR = norm(r);
new_euclideanColumnsormOfR = euclideanColumnsormOfR;
x = improveSolution(x, new_euclideanColumnsormOfR, euclideanColumnsormOfR, Matrix, Vector);
function x = iterativeResidualCorrection(A, x, b)
r = A*x - b;
euclideanNormOfR = norm(r);
new_euclideanNormOfR = euclideanNormOfR;
while new_euclideanNormOfR <= euclideanNormOfR
euclideanNormOfR = new_euclideanNormOfR;
r = A*x - b;
x = x - r;
new_euclideanNormOfR = norm(r);
end
end % end function
function x = improveSolution(x, new_euclideanColumnsormOfR, euclideanColumnsormOfR, Matrix, Vector)
while new_euclideanColumnsormOfR <= euclideanColumnsormOfR
euclideanColumnsormOfR = new_euclideanColumnsormOfR;
r = Matrix*x - Vector;
x = x - r;
new_euclideanColumnsormOfR = norm(r);
end % end while
end % end function

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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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@ -0,0 +1,38 @@
\BOOKMARK [0][-]{chapter.1}{Problem 1 - Finding machine epsilion}{}% 1
\BOOKMARK [1][-]{section.1.1}{Problem}{chapter.1}% 2
\BOOKMARK [1][-]{section.1.2}{Theoretical Introduction}{chapter.1}% 3
\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 [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}{Discussion of the result}{chapter.1}% 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}% 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}% 12
\BOOKMARK [2][-]{subsection.2.2.2}{Backward substitution}{section.2.2}% 13
\BOOKMARK [2][-]{subsection.2.2.3}{Partial Pivoting}{section.2.2}% 14
\BOOKMARK [1][-]{section.2.3}{Solution}{chapter.2}% 15
\BOOKMARK [2][-]{subsection.2.3.1}{Main function}{section.2.3}% 16
\BOOKMARK [2][-]{subsection.2.3.2}{checkIfMatrixIsSquareMatrix}{section.2.3}% 17
\BOOKMARK [2][-]{subsection.2.3.3}{gaussianEliminationWithPartialPivoting}{section.2.3}% 18
\BOOKMARK [2][-]{subsection.2.3.4}{partialPivoting}{section.2.3}% 19
\BOOKMARK [2][-]{subsection.2.3.5}{partialPivotingSwapOneRow}{section.2.3}% 20
\BOOKMARK [2][-]{subsection.2.3.6}{swapRowMatrix}{section.2.3}% 21
\BOOKMARK [2][-]{subsection.2.3.7}{swapValueVector}{section.2.3}% 22
\BOOKMARK [2][-]{subsection.2.3.8}{gaussianElimination}{section.2.3}% 23
\BOOKMARK [2][-]{subsection.2.3.9}{substractRows}{section.2.3}% 24
\BOOKMARK [2][-]{subsection.2.3.10}{backSubstitutionPhase}{section.2.3}% 25
\BOOKMARK [2][-]{subsection.2.3.11}{iterativeResidualCorrection}{section.2.3}% 26
\BOOKMARK [2][-]{subsection.2.3.12}{improveSolution}{section.2.3}% 27
\BOOKMARK [1][-]{section.2.4}{Discussion of the result}{chapter.2}% 28
\BOOKMARK [0][-]{chapter.3}{Problem 3 - Solving a system of n linear equations - iterative algorithm}{}% 29
\BOOKMARK [1][-]{section.3.1}{Problem}{chapter.3}% 30
\BOOKMARK [1][-]{section.3.2}{Theoretical introduction}{chapter.3}% 31
\BOOKMARK [1][-]{section.3.3}{Solution}{chapter.3}% 32
\BOOKMARK [1][-]{section.3.4}{Discussion of the result}{chapter.3}% 33
\BOOKMARK [0][-]{chapter.4}{Problem 4 - QR method of finding eigenvalues}{}% 34
\BOOKMARK [1][-]{section.4.1}{Problem}{chapter.4}% 35
\BOOKMARK [1][-]{section.4.2}{Theoretical introduction}{chapter.4}% 36
\BOOKMARK [1][-]{section.4.3}{Solution}{chapter.4}% 37
\BOOKMARK [1][-]{section.4.4}{Discussion of the result}{chapter.4}% 38

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@ -107,7 +107,6 @@ Display difference between calculated macheps and $2^{-52}$:
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}
@ -115,7 +114,7 @@ Write a program solving a system of \textit{n} linear equations Ax = b using the
\section{Theoretical Introduction}
Gaussian elimination with partial pivoting consists of 3 main steps:
\subsection{Transform system of equation into an upper-triangular matrix}
\subsection{Transform matrix into upper-triangular matrix}
\subsubsection{Starting conditions}
We start with the system of linear equations looking like this:
@ -134,7 +133,9 @@ We start with the system of linear equations looking like this:
\end{matrix}
\]
In order for this method to work all the elements of "diagonal" line - $ a_{11}, a_{22}, \dots, a_{nn} $ must be different from zero since we will be dividing by them.
In order for this method to work all the elements of "diagonal" line:
\[ a_{11}, a_{22}, \dots, a_{nn} \]
Must be different from zero since we will be dividing by them.
We will denote rows as '$w_i$' where 'i' is number of the row.
@ -222,25 +223,269 @@ We do it before each Gaussian elimination step since this will lead to smaller e
Then we swap this row with k-th row. Since the matrix we use is assumed to be nonsingular then $|{a_{ik}^{k}}| \neq 0$ will be always true. After that we continue with the Gaussian elimination method.
\newpage
\section{Solution}
\subsection{Main function}
\begin{simplechar}
\begin{lstlisting}
function x = indicatedMethod(Matrix, Vector)
[~,Columns] = size(Matrix);
function x = indicatedMethod(Matrix, Vector) % Name of the method as in the textbook
% x stands for obtained result
[~,Columns] = size(Matrix); % We need to know how big the matrix is in next steps
% notice the '~', since we assume we use square matrix, we do not need
% to have another variable for number of rows since it is the same as
% number of columns
checkIfMatrixIsSquareMatrix(Matrix);
[Matrix, Vector] = gaussianEliminationWithPartialPivoting(Columns, Matrix, Vector);
% Change matrix to upper triangular matrix
[Matrix, Vector, x] = backSubstitutionPhase(Columns, Matrix, Vector);
x = iterativeResidualCorrection(Matrix, x, Vector);
% Get the solution
x = iterativeResidualCorrection(Matrix, x, Vector); % Improve on the solution
end % end function
\end{lstlisting}
\end{simplechar}
\subsection{checkIfMatrixIsSquareMatrix}
\begin{simplechar}
\begin{lstlisting}
function checkIfMatrixIsSquareMatrix(Matrix)
[Rows,Columns] = size(Matrix);
if Rows ~= Columns
error ('Matrix is not square matrix!');
end % end if
end % end function
\end{lstlisting}
\end{simplechar}
\newpage
\subsection{gaussianEliminationWithPartialPivoting}
\begin{simplechar}
\begin{lstlisting}
function [Matrix, Vector] = gaussianEliminationWithPartialPivoting(Columns, Matrix, Vector)
for j = 1 : Columns
centralElement = max(Matrix(j:Columns,j));
% we stay in the same row (j) but we change columns, as in the
% textbook
[Matrix, Vector] = partialPivoting(Matrix, Vector, j, centralElement, Columns);
% ensures that a_kk != 0 and reduces errors
[Matrix, Vector] = gaussianElimination(j, Columns, Matrix, Vector);
% change matrix into upper triangular matrix
end % end for
end % end function
\end{lstlisting}
\end{simplechar}
\subsection{partialPivoting}
\begin{simplechar}
\begin{lstlisting}
function [Matrix, Vector] = partialPivoting(Matrix, Vector, j, centralElement, Columns)
for k = j : Columns
partialPivotingSwapOneRow(Matrix, Vector, j, k, centralElement);
end % end for
end % end function
\end{lstlisting}
\end{simplechar}
\subsection{partialPivotingSwapOneRow}
\begin{simplechar}
\begin{lstlisting}
function [Matrix, Vector] = partialPivotingSwapOneRow(Matrix, Vector, j, k, centralElement)
if Matrix(k,j) == centralElement
swapRowMatrix(Matrix, j, k); % swap jth row with kth row
swapValueVector(Vector, j, k); % swap jth value with kth value
end % end if
end % end function
\end{lstlisting}
\end{simplechar}
\newpage
\subsection{swapRowMatrix}
\begin{simplechar}
\begin{lstlisting}
function Matrix = swapRowMatrix(Matrix, j, k)
temp = Matrix(j , :); % ' : ' denote "all elements in jth row"
Matrix(j , :) = Matrix(k, :);
Matrix(k, :) = temp; % temp equal to previous value of jth row
end
\end{lstlisting}
\end{simplechar}
\subsection{swapValueVector}
\begin{simplechar}
\begin{lstlisting}
function Vector = swapValueVector(Vector, j, k)
temp = Vector(j);
Vector(j) = Vector(k);
Vector(k) = temp; % temp equal to previous value of k element of vector
end % end function
\end{lstlisting}
\end{simplechar}
\subsection{gaussianElimination}
\begin{simplechar}
\begin{lstlisting}
function [Matrix, Vector] = gaussianElimination(j, Columns, Matrix, Vector)
for i = j + 1 : Columns
rowMultiplier = Matrix(i,j) / Matrix(j,j);
[Matrix, Vector] = substractRows(Matrix, Vector, i, rowMultiplier, j, Columns);
end % end for
end % end function
\end{lstlisting}
\end{simplechar}
\subsection{substractRows}
\begin{simplechar}
\begin{lstlisting}
function [Matrix, Vector] = substractRows(Matrix, Vector, i, rowMultiplier, j, Columns)
Vector(i) = Vector(i) - rowMultiplier * Vector(j);
for curentColumn = 1 : Columns
Matrix(i,curentColumn) = Matrix(i,curentColumn) - rowMultiplier * Matrix(j, curentColumn);
end % end for
end % end function
\end{lstlisting}
\end{simplechar}
\newpage
\subsection{backSubstitutionPhase}
\begin{simplechar}
\begin{lstlisting}
function [Matrix, Vector, x] = backSubstitutionPhase(Columns, Matrix, Vector)
for k = Columns : -1 : 1
% Start at final column and move by -1 each iteration until we reach 1
equation = 0;
for j = k+1 : Columns
equation = equation + Matrix(k,j) * x(j, 1);
% even though x is a vector we still need to put '1' to ensure
% that number of columns in the first matrix matches number of
% rows in second matrix
end % end for
x(k, 1) = (Vector(k,1) - equation) / Matrix(k,k);
% even though x is a vector we still need to put '1' to ensure
% that we do not exceed array bounds
end % end for
end % end function
\end{lstlisting}
\end{simplechar}
\subsection{iterativeResidualCorrection}
\begin{simplechar}
\begin{lstlisting}
function x = iterativeResidualCorrection(Matrix, x, Vector)
residuum = Matrix*x - Vector; % as in the book
newResiduum = residuum;
x = improveSolution(x, newResiduum, residuum, Matrix, Vector);
end % end function
\end{lstlisting}
\end{simplechar}
\subsection{improveSolution}
\begin{simplechar}
\begin{lstlisting}
function x = improveSolution(x, newResiduum, oldResiduum, Matrix, Vector)
while newResiduum <= oldResiduum
oldResiduum = newResiduum;
residuum = Matrix*x - Vector;
x = x - residuum;
newResiduum = residuum;
end % end while
end % end function
\end{lstlisting}
\end{simplechar}
\section{Discussion of the result}
Solutions vectors for matrix A and vector A and n = 16:
\[ x_{algorithm} = \left( \begin{array}{cc}
-0.930056653761514 \\
-1.223503294617857 \\
-1.273786563425385 \\
-1.231189728991652 \\
-1.153115956882885 \\
-1.061491474990357 \\
-0.964691801421511 \\
-0.865917262607523 \\
-0.766393319734375 \\
-0.666596029928932 \\
-0.566728103385765 \\
-0.466921613561691 \\
-0.367370070632649 \\
-0.268521931669581 \\
-0.171529057709426 \\
-0.079398574792031
\end{array} \right)
%
x_{Matlab Method} = \left( = \begin{array}{cc}
-0.930056653761507 \\
-1.223503294617854 \\
-1.273786563425390 \\
-1.231189728991649 \\
-1.153115956882891 \\
-1.061491474990357 \\
-0.964691801421514 \\
-0.865917262607518 \\
-0.766393319734373 \\
-0.666596029928935 \\
-0.566728103385765 \\
-0.466921613561692 \\
-0.367370070632646 \\
-0.268521931669579 \\
-0.171529057709426 \\
-0.079398574792031
\end{array} \right)
\]
Error for 'A' method for algorithm: \[ 3.383918772654241 \]
Error for 'A' method for matlab method: \[ 3.383918772654241 \]
\newpage
Solutions vectors for matrix B and vector B and n = 16 (Both multiplied by $ 1.0e+17 $):
\[ x_{algorithm} = \left( \begin{array}{cc}
0.000001960155675 \\
-0.000102773501571 \\
0.001959454282882 \\
-0.018894079120425 \\
0.104895022735396 \\
-0.352396798852209 \\
0.705545645736628 \\
-0.728747526649489 \\
0.112818247768452 \\
0.261440075930356 \\
0.953713133491034 \\
-3.080986443185790 \\
3.765178552913233 \\
-2.440218622397594 \\
0.834768953546100 \\
-0.118974790826378
\end{array} \right)
%
x_{Matlab Method} = \left( = \begin{array}{cc}
0.000001102587209 \\
-0.000066546298462 \\
0.001476714765758 \\
-0.016859999627589 \\
0.113920718370153 \\
-0.488374161741872 \\
1.368641128884513 \\
-2.495283439985873 \\
2.793405296264694 \\
-1.547642305352008 \\
-0.035332172403445 \\
0.154726025421297 \\
0.807694426359552 \\
-1.133485136703852 \\
0.592810708065954 \\
-0.115632364630554
\end{array} \right)
\]
Error for 'B' method for algorithm: \[ 5.699979882700911e+17 \]
Error for 'B' method for matlab method: \[ 4.569118543317684e+17 \]
\addtocontents{toc}{\protect\newpage}
\chapter{Problem 3 - Solving a system of n linear equations - iterative algorithm}
\section{Problem}

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@ -0,0 +1,43 @@
\contentsline {chapter}{\numberline {1}Problem 1 - Finding machine epsilion}{3}{chapter.1}%
\contentsline {section}{\numberline {1.1}Problem}{3}{section.1.1}%
\contentsline {section}{\numberline {1.2}Theoretical Introduction}{3}{section.1.2}%
\contentsline {subsection}{\numberline {1.2.1}Definition of machine epsilion}{3}{subsection.1.2.1}%
\contentsline {subsection}{\numberline {1.2.2}Practical applications of machine epsilion}{3}{subsection.1.2.2}%
\contentsline {section}{\numberline {1.3}Solution}{4}{section.1.3}%
\contentsline {subsection}{\numberline {1.3.1}Matlab code}{4}{subsection.1.3.1}%
\contentsline {section}{\numberline {1.4}Discussion of the result}{4}{section.1.4}%
\contentsline {chapter}{\numberline {2}Problem 2 - Solving a system of n linear equations - indicated method}{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}Transform matrix into upper-triangular matrix}{6}{subsection.2.2.1}%
\contentsline {subsubsection}{Starting conditions}{6}{section*.2}%
\contentsline {subsubsection}{Zeroing first column}{7}{section*.3}%
\contentsline {subsubsection}{Zeroing second column}{7}{section*.4}%
\contentsline {subsubsection}{Zeroing next columns}{7}{section*.5}%
\contentsline {subsection}{\numberline {2.2.2}Backward substitution}{8}{subsection.2.2.2}%
\contentsline {subsection}{\numberline {2.2.3}Partial Pivoting}{8}{subsection.2.2.3}%
\contentsline {section}{\numberline {2.3}Solution}{9}{section.2.3}%
\contentsline {subsection}{\numberline {2.3.1}Main function}{9}{subsection.2.3.1}%
\contentsline {subsection}{\numberline {2.3.2}checkIfMatrixIsSquareMatrix}{9}{subsection.2.3.2}%
\contentsline {subsection}{\numberline {2.3.3}gaussianEliminationWithPartialPivoting}{10}{subsection.2.3.3}%
\contentsline {subsection}{\numberline {2.3.4}partialPivoting}{10}{subsection.2.3.4}%
\contentsline {subsection}{\numberline {2.3.5}partialPivotingSwapOneRow}{10}{subsection.2.3.5}%
\contentsline {subsection}{\numberline {2.3.6}swapRowMatrix}{11}{subsection.2.3.6}%
\contentsline {subsection}{\numberline {2.3.7}swapValueVector}{11}{subsection.2.3.7}%
\contentsline {subsection}{\numberline {2.3.8}gaussianElimination}{11}{subsection.2.3.8}%
\contentsline {subsection}{\numberline {2.3.9}substractRows}{11}{subsection.2.3.9}%
\contentsline {subsection}{\numberline {2.3.10}backSubstitutionPhase}{12}{subsection.2.3.10}%
\contentsline {subsection}{\numberline {2.3.11}iterativeResidualCorrection}{12}{subsection.2.3.11}%
\contentsline {subsection}{\numberline {2.3.12}improveSolution}{12}{subsection.2.3.12}%
\contentsline {section}{\numberline {2.4}Discussion of the result}{13}{section.2.4}%
\newpage
\contentsline {chapter}{\numberline {3}Problem 3 - Solving a system of n linear equations - iterative algorithm}{15}{chapter.3}%
\contentsline {section}{\numberline {3.1}Problem}{15}{section.3.1}%
\contentsline {section}{\numberline {3.2}Theoretical introduction}{15}{section.3.2}%
\contentsline {section}{\numberline {3.3}Solution}{15}{section.3.3}%
\contentsline {section}{\numberline {3.4}Discussion of the result}{15}{section.3.4}%
\contentsline {chapter}{\numberline {4}Problem 4 - QR method of finding eigenvalues}{16}{chapter.4}%
\contentsline {section}{\numberline {4.1}Problem}{16}{section.4.1}%
\contentsline {section}{\numberline {4.2}Theoretical introduction}{16}{section.4.2}%
\contentsline {section}{\numberline {4.3}Solution}{16}{section.4.3}%
\contentsline {section}{\numberline {4.4}Discussion of the result}{16}{section.4.4}%

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@ -32,4 +32,10 @@ https://tex.stackexchange.com/questions/526025/problem-with-vdotswithin - vdots
https://www.overleaf.com/learn/latex/Brackets_and_Parentheses - Curly brackets latex
https://tex.stackexchange.com/questions/130516/how-to-write-something-vertically-below-another-math-symbol - Symbol vertically below another
https://tex.stackexchange.com/questions/130516/how-to-write-something-vertically-below-another-math-symbol - Symbol vertically below another
https://tex.stackexchange.com/questions/103998/how-to-force-a-table-of-contents-to-start-new-page - new page at the start of chapter in table of contents
https://www.codegrepper.com/code-examples/whatever/column+vector+latex - Vector in latex
https://tex.stackexchange.com/questions/9901/how-to-have-matrices-side-by-side-in-latex - Matrices side by side in LaTeX