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https://github.com/kuhyx/WUT_Computer_Science.git
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Task 2 code
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43
ENUME/projectA/jacobiMethod.asv
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43
ENUME/projectA/jacobiMethod.asv
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@ -0,0 +1,43 @@
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function x = jacobiMethod(Matrix, Vector)
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[Rows,~] = size(Matrix);
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D = diag(diag(Matrix));
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inverseD = inv(D);
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U = triu(Matrix, 1); % Generates upper triangular part of matrix
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% where the second variable denotes on which diagonal of matrix should we
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% start
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L = tril(Matrix, -1); % Generates lower triangular part of matrix
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% where the second variable denotes on which diagonal of matrix should we
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% start
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initial_x = ones(Rows, 1);
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initial_x = - inverseD * ( ( L + U ) * initial_x) + inverseD * initial_x; % As per formula
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% We will be using D \ initial_x and D \ () since it is faster and more
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% accurate according to matlab
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whichIterationAreWeOn = 0;
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currentError = inf; % We set it to inf so that it the algorithm will always start
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% (See condition below)
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demandedTolerance = 1e-10;
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while currentError >= demandedTolerance
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x = - inverseD * ( ( L + U ) * initial_x) + inverseD * initial_x; % As per formula
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currentError = norm(x - initial_x);
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disp(currentError);
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if currentError <= demandedTolerance
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currentError = norm(Matrix*x-Vector);
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disp(currentError);
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if currentError <= demandedTolerance
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break;
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else
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demandedTolerance = demandedTolerance * 2;
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end
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end
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initial_x = x;
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whichIterationAreWeOn = whichIterationAreWeOn + 1;
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end
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disp("Final demandedTolerance");
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disp(demandedTolerance);
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disp("Final Iteration: ");
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disp(whichIterationAreWeOn);
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disp("A\b matlab:");
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disp(Matrix / Vector);
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end
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50
ENUME/projectA/jacobiMethod.m
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50
ENUME/projectA/jacobiMethod.m
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@ -0,0 +1,50 @@
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function x = jacobiMethod(Matrix, Vector)
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[Rows,~] = size(Matrix);
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[L, D, U] = decomposeMatrix(Matrix);
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initial_x = ones(Rows, 1);
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whichIterationAreWeOn = 0;
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currentError = inf; % We set it to inf so that the algorithm will always start
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% (See condition below)
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demandedTolerance = 1e-10;
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while currentError >= demandedTolerance
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x = jacobiEquation(D, L, U, initial_x, Vector);
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currentError = norm(x - initial_x);
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%disp(currentError);
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if currentError <= demandedTolerance
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currentError = norm(Matrix*x-Vector);
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%disp(currentError);
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if currentError <= demandedTolerance
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break;
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else
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demandedTolerance = demandedTolerance * 2;
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end
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end
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initial_x = x;
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whichIterationAreWeOn = whichIterationAreWeOn + 1;
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end
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disp("Final demandedTolerance");
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disp(demandedTolerance);
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disp("Final Iteration: ");
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disp(whichIterationAreWeOn);
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disp("A\b matlab:");
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disp(Matrix \ Vector);
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end
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function [L, D, U] = decomposeMatrix(Matrix)
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D = diag(diag(Matrix));
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U = triu(Matrix, 1); % Generates upper triangular part of matrix
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% where the second variable denotes on which diagonal of matrix should we
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% start
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L = tril(Matrix, -1); % Generates lower triangular part of matrix
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% where the second variable denotes on which diagonal of matrix should we
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% start
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end
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function x = jacobiEquation(D, L, U, initial_x, Vector)
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x = - D \ ( L + U ) * initial_x + D \ Vector; % As per formula
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% We will be using D \ Vector and D \ ( ) instead of inverseD since
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% this is faster according to matlab
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end
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4
ENUME/projectA/pdflatex36032.fls
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4
ENUME/projectA/pdflatex36032.fls
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PWD /home/kuchy/Zlew/Studia/SEM_5/enume_done/Project/matlabproject/ENUME_Project_31/projectA
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INPUT /usr/share/texmf/web2c/texmf.cnf
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@ -46,30 +46,38 @@
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@ -1,13 +1,17 @@
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@ -530,65 +530,75 @@ Chapter 3.
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[11
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] [12]
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./projectA.tex:472: Extra }, or forgotten $.
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l.472 l_{n1} & l_{n2} & l_{n3} & \cdots & 0}
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[13] [14] [15] [16]
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Chapter 4.
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Chapter 5.
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@ -17,24 +17,25 @@
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||||
\BOOKMARK [0][-]{chapter.3}{Problem 3 - Solving a system of n linear equations - iterative algorithm}{}% 17
|
||||
\BOOKMARK [1][-]{section.3.1}{Problem}{chapter.3}% 18
|
||||
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|
||||
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||||
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|
||||
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|
||||
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||||
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|
||||
\BOOKMARK [1][-]{section.4.3}{Solution}{chapter.4}% 25
|
||||
\BOOKMARK [1][-]{section.4.4}{Discussion of the result}{chapter.4}% 26
|
||||
\BOOKMARK [0][-]{chapter.5}{Code appendix}{}% 27
|
||||
\BOOKMARK [1][-]{section.5.1}{Task 2 Code}{chapter.5}% 28
|
||||
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|
||||
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||||
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||||
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|
||||
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||||
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||||
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||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
\BOOKMARK [1][-]{section.3.4}{Discussion of the result}{chapter.3}% 22
|
||||
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|
||||
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|
||||
\BOOKMARK [1][-]{section.4.2}{Theoretical introduction}{chapter.4}% 25
|
||||
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|
||||
\BOOKMARK [1][-]{section.4.4}{Discussion of the result}{chapter.4}% 27
|
||||
\BOOKMARK [0][-]{chapter.5}{Code appendix}{}% 28
|
||||
\BOOKMARK [1][-]{section.5.1}{Task 2 Code}{chapter.5}% 29
|
||||
\BOOKMARK [2][-]{subsection.5.1.1}{Main function}{section.5.1}% 30
|
||||
\BOOKMARK [2][-]{subsection.5.1.2}{checkIfMatrixIsSquareMatrix}{section.5.1}% 31
|
||||
\BOOKMARK [2][-]{subsection.5.1.3}{gaussianEliminationWithPartialPivoting}{section.5.1}% 32
|
||||
\BOOKMARK [2][-]{subsection.5.1.4}{partialPivoting}{section.5.1}% 33
|
||||
\BOOKMARK [2][-]{subsection.5.1.5}{partialPivotingSwapOneRow}{section.5.1}% 34
|
||||
\BOOKMARK [2][-]{subsection.5.1.6}{swapRowMatrix}{section.5.1}% 35
|
||||
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|
||||
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|
||||
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|
||||
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||||
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|
||||
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|
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Binary file not shown.
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@ -335,7 +335,9 @@ We should also try to solve the equations from problem 2a) and 2b) for n = 10 us
|
||||
|
||||
\section{Theoretical introduction}
|
||||
|
||||
Itertaive methods differ from the Gauss elimination method since they are iterative, which means that our solution will improve with each iteration. Building on that we can cnclude that the number of iterations will depend on what accuracy we want to achieve.
|
||||
We should also answer the question what happens if the sufficient condition is not fullfiled.
|
||||
|
||||
Itertaive methods differ from the Gauss elimination method since they are iterative, which means that our solution will improve with each iteration. Building on that we can cnclude that the number of iterations will depend on what accuracy we want to achieve. Since we are using iterative method we don't have the guarantee of how many iterations will be neeeded before we reach the solution,
|
||||
|
||||
In general:
|
||||
We start with:
|
||||
@ -346,6 +348,9 @@ And we generate next vectors \textbf{$x^{i+1}$} in such way:
|
||||
|
||||
Where $\mathbf{M}$ is some matrix.
|
||||
|
||||
\subsection{Procedure}
|
||||
|
||||
\subsubsection{Decomposing matrix}
|
||||
For both Jacobi and Gauss-Seidel method we first decompose starting matrix $ \mathbf{A} $ to:
|
||||
\[ \mathbf{A} = \mathbf{L} + \mathbf{D} + \mathbf{U} \]
|
||||
where:
|
||||
@ -376,8 +381,8 @@ We can get:
|
||||
\end{bmatrix}
|
||||
\]
|
||||
\[ \mathbf{U} = \begin{bmatrix}
|
||||
0 & 0 & 0\\
|
||||
0 & 0 & 0\\
|
||||
0 & 3 & 3\\
|
||||
0 & 0 & 3\\
|
||||
0 & 0 & 0
|
||||
\end{bmatrix}
|
||||
\]
|
||||
@ -412,10 +417,153 @@ so:
|
||||
\end{bmatrix}
|
||||
\]
|
||||
|
||||
\subsubsection{Jacobi's method}
|
||||
After decomposing matrix we can write the system of equations
|
||||
\[ \mathbf{A}\mathbf{x} = \mathbf{b} \]
|
||||
in the form:
|
||||
\[ \mathbf{D}\mathbf{x} = -(\mathbf{L}+\mathbf{U})\mathbf{x}+\mathbf{b}
|
||||
\]
|
||||
|
||||
If we assume that diagonal entries of matrix \textbf{A} are nonzero, then matrix \textbf{D} is nonsingular therefore we can propose such an iterative method:
|
||||
\[ \mathbf{x}^{i+1} = -\mathbf{D}^{-1}(\mathbf{L}+\mathbf{U})\mathbf{x}^{(i)}+\mathbf{D}^{-1}\mathbf{x}
|
||||
\]
|
||||
This is the Jacobi's method.
|
||||
We can rewritte this equation in the form of \textit{n} independent scalar equations:
|
||||
|
||||
\[
|
||||
{x}_j^{i+1} = -\frac{1}{d_{jj}}(
|
||||
\sum^n_{k=1} (l_{jk} + u_{jk})x_k^{(i)}+b_j)
|
||||
) \]
|
||||
Where $d_{jj}$; $l_{jk}$; $u_{jk}$ are the elements of the respective matrixes \textbf{D}, \textbf{L}, \textbf{U}
|
||||
|
||||
Thanks to this we can do those computations in paraller, totally or partially if we are using a computer that enables a parallelization of the computations.
|
||||
|
||||
\paragraph{Converging}
|
||||
Jacobi's method is convergent if we have strong diagonal dominance of the matrix \textbf{A}
|
||||
|
||||
|
||||
\subsubsection{Gauss-Seidel method}
|
||||
After decomposing matrix we can write the system of equations
|
||||
\[ \mathbf{A}\mathbf{x} = \mathbf{b} \]
|
||||
in the form:
|
||||
\[ (\mathbf{L}+\mathbf{D})\mathbf{x} = -\mathbf{U}\mathbf{x}+\mathbf{b}
|
||||
\]
|
||||
Again we assume that \textbf{D} is nonsingular, in doing so we propose following iterative method:
|
||||
\[ \mathbf{D}\mathbf{x}^{(i+1)} = -\mathbf{L}\mathbf{x}^{(i+1)}-\mathbf{U}\mathbf{x}^{(i)}+\mathbf{b}
|
||||
\]
|
||||
Since matrix \textbf{L} is subdiagonal, provided that we organse the calculation of elements of the vector $x^{(i_1)}$ in a proper way, it does not hurt that $x^{(i_1)}$ is on the right side of the equation.
|
||||
In order to organise the calculation in the correct way we:
|
||||
First take into account the structure of matrixes \textbf{D} and \textbf{L}:
|
||||
\[
|
||||
\begin{bmatrix}
|
||||
d_{11}x_1^{(i_1)}\\
|
||||
d_{22}x_2^{(i_1)}\\
|
||||
d_{33}x_3^{(i_1)}\\
|
||||
\vdots\\
|
||||
d_{nn}x_n^{(i_1)}
|
||||
\end{bmatrix}
|
||||
=
|
||||
-
|
||||
\begin{bmatrix}
|
||||
0 & 0 & 0 & \cdots & 0\\
|
||||
l_{21} & 0 & 0 & \cdots & 0\\
|
||||
l_{32} & l_{32} & 0 & \cdots & 0\\
|
||||
\vdots & \vdots & \vdots & \vdots & \vdots\\
|
||||
l_{n1} & l_{n2} & l_{n3} & \cdots & 0}
|
||||
\end{bmatrix}
|
||||
\begin{bmatrix}
|
||||
x_1^{(i_1)}\\
|
||||
x_2^{(i_1)}\\
|
||||
x_3^{(i_1)}\\
|
||||
\vdots\\
|
||||
x_n^{(i_1)}
|
||||
\end{bmatrix}
|
||||
-
|
||||
\mathbf{w}^{(i)}
|
||||
\]
|
||||
Where
|
||||
\[ \mathbf{w}^{(i)} = \mathbf{U}\mathbf{x}^{(i)} - \mathbf{b} \]
|
||||
|
||||
\newpage
|
||||
So the order of calculations is as follows:
|
||||
\[ x_1^{(i+1)} = -\frac{w_1^{(i)}}{d_{11}} \]
|
||||
\[ x_2^{(i+1)} = -\frac{-l_{21}x_1^{(i+1)} - w_2^{(i)}}{d_{22}} \]
|
||||
\[ x_3^{(i+1)} = -\frac{-l_{31}x_1^{(i+1)} -l_{32}x_2^{(i+1)} - w_3^{(i)}}{d_{33}} \]
|
||||
|
||||
And so on
|
||||
|
||||
As opposed to Jacobi's method, Gauss-Seidel method computations must be performed sequentially. Every subsequent scalar equations uses results from the computation of the previous equations.
|
||||
\paragraph{Converging}
|
||||
Gauss-Seidel method is convergent if the matrix \textbf{A} is strongly row or column diagonnaly dominant. If the matrix is symmetric, the method is also convergent if the matrix \textbf{A} is positive definite. This method is also usually faster convergent compared to Jacobi's method.
|
||||
|
||||
\subsubsection{Stop tests}
|
||||
There are two ways to check when to terminate iterations of the methods we just discussed:
|
||||
\begin{enumerate}
|
||||
\item Check differences between two subsequent iteration points
|
||||
\[
|
||||
\| \mathbf{x}^{(i+1)} - \mathbf{x}^{(i)} \| \leq \delta
|
||||
\]
|
||||
Where $\delta$ is an assumed tolerance, (in our case $10^{-10}$). What we are really interested in though is whether the solution of the system of equation has required accuracy. If we want to check that we can additionaly check (higher level, more computationally demanding test):
|
||||
\item Check differences between two subsequent iteration points
|
||||
\[
|
||||
\| \mathbf{A}\mathbf{x}^{(i+1)} - \mathbf{b}\| \leq \delta_2
|
||||
\]
|
||||
Where $\delta_2$ is an assumed tolerance. If this test is not passed then we can diminish the value of $\delta_2$ and continue with the iterations. Value of $\delta_2$ can not be too small since we are limited by the numerical errors.
|
||||
\end{enumerate}
|
||||
|
||||
\subsubsection{\textbf{A} and \textbf{b}}
|
||||
We have been given with the following system:
|
||||
\[
|
||||
\systeme{10x_1-4x_2+x_3+2x_4=-8, 2x_1-6x_2+3x_3-x_4=-12, x_1+4x_2-12x_3+x_4=4, 2x_1+3x_2-3x_3-10x_4=1}
|
||||
\]
|
||||
Therefore our matrices will look like this:
|
||||
\[
|
||||
\textbf{A} = \begin{bmatrix}
|
||||
10 & -4 & 1 & 2 \\
|
||||
2 & -6 & 3 & -1 \\
|
||||
1 & 4 & -12 & 1 \\
|
||||
2 & 3 & -3 & -10
|
||||
\end{bmatrix}
|
||||
\textbf{b} = \begin{bmatrix}
|
||||
-8 \\
|
||||
-12 \\
|
||||
4 \\
|
||||
11 \\
|
||||
\end{bmatrix}
|
||||
\]
|
||||
|
||||
So:
|
||||
\[
|
||||
\textbf{L} = \begin{bmatrix}
|
||||
0 & 0 & 0 & 0 \\
|
||||
2 & 0 & 0 & 0 \\
|
||||
1 & 4 & 0 & 0 \\
|
||||
2 & 3 & -3 & 0
|
||||
\end{bmatrix}
|
||||
\textbf{D} = \begin{bmatrix}
|
||||
10 & 0 & 0 & 0 \\
|
||||
0 & -6 & 0 & 0 \\
|
||||
0 & 0 & -12 & 0 \\
|
||||
0 & 0 & 0 & -10
|
||||
\end{bmatrix}
|
||||
\textbf{U} = \begin{bmatrix}
|
||||
0 & -4 & 1 & 2 \\
|
||||
0 & 0 & 3 & -1 \\
|
||||
0 & 0 & 0 & 1 \\
|
||||
0 & 0 & 0 & 0
|
||||
\end{bmatrix}
|
||||
\]
|
||||
\[
|
||||
\textbf{D}^{-1} = \begin{bmatrix}
|
||||
\frac{1}{10} & 0 & 0 & 0 \\
|
||||
0 & - \frac{1}{6} & 0 & 0 \\
|
||||
0 & 0 & - \frac{1}{12} & 0 \\
|
||||
0 & 0 & 0 & - \frac{1}{10}
|
||||
\end{bmatrix}
|
||||
\]
|
||||
|
||||
|
||||
|
||||
Since we are using iterative method we don't have the guarantee of how many iteratiosn will be neeeded before we reach the solution,
|
||||
we also need to resolve the issue of convergence, the method we use may converge or it may not. We need to find what condition the method converges. We know that there is a sufficient condition which is faster but not 100 \% correct, the sufficient and necessary condition is slower but 100 \% correct. We should also answer the question what happens if the sufficient condition is not fullfiled.
|
||||
We also need to desin stop tests, when are we going to stop the process
|
||||
\section{Solution}
|
||||
|
||||
\section{Discussion of the result}
|
||||
@ -505,7 +653,7 @@ end % end function
|
||||
\end{lstlisting}
|
||||
\end{simplechar}
|
||||
|
||||
\newpage
|
||||
|
||||
\subsection{swapRowMatrix}
|
||||
\begin{simplechar}
|
||||
\begin{lstlisting}
|
||||
|
||||
@ -22,24 +22,32 @@
|
||||
\contentsline {chapter}{\numberline {3}Problem 3 - Solving a system of n linear equations - iterative algorithm}{11}{chapter.3}%
|
||||
\contentsline {section}{\numberline {3.1}Problem}{11}{section.3.1}%
|
||||
\contentsline {section}{\numberline {3.2}Theoretical introduction}{11}{section.3.2}%
|
||||
\contentsline {section}{\numberline {3.3}Solution}{12}{section.3.3}%
|
||||
\contentsline {section}{\numberline {3.4}Discussion of the result}{12}{section.3.4}%
|
||||
\contentsline {chapter}{\numberline {4}Problem 4 - QR method of finding eigenvalues}{13}{chapter.4}%
|
||||
\contentsline {section}{\numberline {4.1}Problem}{13}{section.4.1}%
|
||||
\contentsline {section}{\numberline {4.2}Theoretical introduction}{13}{section.4.2}%
|
||||
\contentsline {section}{\numberline {4.3}Solution}{13}{section.4.3}%
|
||||
\contentsline {section}{\numberline {4.4}Discussion of the result}{13}{section.4.4}%
|
||||
\contentsline {chapter}{\numberline {5}Code appendix}{14}{chapter.5}%
|
||||
\contentsline {section}{\numberline {5.1}Task 2 Code}{14}{section.5.1}%
|
||||
\contentsline {subsection}{\numberline {5.1.1}Main function}{14}{subsection.5.1.1}%
|
||||
\contentsline {subsection}{\numberline {5.1.2}checkIfMatrixIsSquareMatrix}{14}{subsection.5.1.2}%
|
||||
\contentsline {subsection}{\numberline {5.1.3}gaussianEliminationWithPartialPivoting}{16}{subsection.5.1.3}%
|
||||
\contentsline {subsection}{\numberline {5.1.4}partialPivoting}{16}{subsection.5.1.4}%
|
||||
\contentsline {subsection}{\numberline {5.1.5}partialPivotingSwapOneRow}{16}{subsection.5.1.5}%
|
||||
\contentsline {subsection}{\numberline {5.1.6}swapRowMatrix}{17}{subsection.5.1.6}%
|
||||
\contentsline {subsection}{\numberline {5.1.7}swapValueVector}{17}{subsection.5.1.7}%
|
||||
\contentsline {subsection}{\numberline {5.1.8}gaussianElimination}{17}{subsection.5.1.8}%
|
||||
\contentsline {subsection}{\numberline {5.1.9}substractRows}{17}{subsection.5.1.9}%
|
||||
\contentsline {subsection}{\numberline {5.1.10}backSubstitutionPhase}{18}{subsection.5.1.10}%
|
||||
\contentsline {subsection}{\numberline {5.1.11}iterativeResidualCorrection}{18}{subsection.5.1.11}%
|
||||
\contentsline {subsection}{\numberline {5.1.12}improveSolution}{18}{subsection.5.1.12}%
|
||||
\contentsline {subsection}{\numberline {3.2.1}Procedure}{12}{subsection.3.2.1}%
|
||||
\contentsline {subsubsection}{Decomposing matrix}{12}{section*.6}%
|
||||
\contentsline {subsubsection}{Jacobi's method}{13}{section*.7}%
|
||||
\contentsline {paragraph}{Converging}{13}{section*.8}%
|
||||
\contentsline {subsubsection}{Gauss-Seidel method}{13}{section*.9}%
|
||||
\contentsline {paragraph}{Converging}{15}{section*.10}%
|
||||
\contentsline {subsubsection}{Stop tests}{15}{section*.11}%
|
||||
\contentsline {subsubsection}{\textbf {A} and \textbf {b}}{15}{section*.12}%
|
||||
\contentsline {section}{\numberline {3.3}Solution}{16}{section.3.3}%
|
||||
\contentsline {section}{\numberline {3.4}Discussion of the result}{16}{section.3.4}%
|
||||
\contentsline {chapter}{\numberline {4}Problem 4 - QR method of finding eigenvalues}{17}{chapter.4}%
|
||||
\contentsline {section}{\numberline {4.1}Problem}{17}{section.4.1}%
|
||||
\contentsline {section}{\numberline {4.2}Theoretical introduction}{17}{section.4.2}%
|
||||
\contentsline {section}{\numberline {4.3}Solution}{17}{section.4.3}%
|
||||
\contentsline {section}{\numberline {4.4}Discussion of the result}{17}{section.4.4}%
|
||||
\contentsline {chapter}{\numberline {5}Code appendix}{18}{chapter.5}%
|
||||
\contentsline {section}{\numberline {5.1}Task 2 Code}{18}{section.5.1}%
|
||||
\contentsline {subsection}{\numberline {5.1.1}Main function}{18}{subsection.5.1.1}%
|
||||
\contentsline {subsection}{\numberline {5.1.2}checkIfMatrixIsSquareMatrix}{18}{subsection.5.1.2}%
|
||||
\contentsline {subsection}{\numberline {5.1.3}gaussianEliminationWithPartialPivoting}{20}{subsection.5.1.3}%
|
||||
\contentsline {subsection}{\numberline {5.1.4}partialPivoting}{20}{subsection.5.1.4}%
|
||||
\contentsline {subsection}{\numberline {5.1.5}partialPivotingSwapOneRow}{20}{subsection.5.1.5}%
|
||||
\contentsline {subsection}{\numberline {5.1.6}swapRowMatrix}{20}{subsection.5.1.6}%
|
||||
\contentsline {subsection}{\numberline {5.1.7}swapValueVector}{21}{subsection.5.1.7}%
|
||||
\contentsline {subsection}{\numberline {5.1.8}gaussianElimination}{21}{subsection.5.1.8}%
|
||||
\contentsline {subsection}{\numberline {5.1.9}substractRows}{21}{subsection.5.1.9}%
|
||||
\contentsline {subsection}{\numberline {5.1.10}backSubstitutionPhase}{22}{subsection.5.1.10}%
|
||||
\contentsline {subsection}{\numberline {5.1.11}iterativeResidualCorrection}{22}{subsection.5.1.11}%
|
||||
\contentsline {subsection}{\numberline {5.1.12}improveSolution}{22}{subsection.5.1.12}%
|
||||
|
||||
2
ENUME/projectA/task2Matrix.m
Normal file
2
ENUME/projectA/task2Matrix.m
Normal file
@ -0,0 +1,2 @@
|
||||
function Matrix = task2Matrix
|
||||
Matrix = [10 -4 1 2; 2 -6 3 -1; 1 4 -12 1; 2 3 -3 -10];
|
||||
4
ENUME/projectA/task2MatrixAndVector.m
Normal file
4
ENUME/projectA/task2MatrixAndVector.m
Normal file
@ -0,0 +1,4 @@
|
||||
function Matrix = task2Matrix
|
||||
Matrix = [10 -4 1 2; 2 -6 3 -1; 1 4 -12 1; 2 3 -3 -10];
|
||||
function Vector = task2Vector
|
||||
Vector = [-8 -12 4 1];
|
||||
6
ENUME/projectA/task2Vector.m
Normal file
6
ENUME/projectA/task2Vector.m
Normal file
@ -0,0 +1,6 @@
|
||||
function Vector = task2Vector
|
||||
Vector = zeros(4, 1);
|
||||
Vector(1, 1) = -8;
|
||||
Vector(2, 1) = -12;
|
||||
Vector(3, 1) = 4;
|
||||
Vector(4, 1) = 1;
|
||||
@ -3,6 +3,13 @@ Useful links:
|
||||
Problem 1:
|
||||
https://stackoverflow.com/questions/27511349/how-to-calculate-machine-epsilon-in-matlab - How to calculate machine epsilion
|
||||
|
||||
Problem 3:
|
||||
https://www.mathworks.com/help/matlab/learn_matlab/matrices-and-arrays.html - Matrices and arrays MATLAB
|
||||
|
||||
https://www.mathworks.com/help/matlab/ref/break.html - Get out of while loop
|
||||
|
||||
mathworks.com/help/matlab/ref/triu.html - upper triangular matrix in matlab
|
||||
|
||||
LaTeX related:
|
||||
https://latex-tutorial.com/symbols/text-formatting/ - Italics
|
||||
|
||||
@ -30,7 +37,7 @@ https://stackoverflow.com/questions/48240982/how-do-you-align-a-system-of-equati
|
||||
|
||||
https://tex.stackexchange.com/questions/526025/problem-with-vdotswithin - vdots alligned to equal sign
|
||||
|
||||
https://www.overleaf.com/learn/latex/Brackets_and_Parentheses - Curly brackets latex
|
||||
https://www.overleaf.com/learn/latex/Brackets_and_Parentheses - Curly brackets, double 'pipes'
|
||||
|
||||
https://tex.stackexchange.com/questions/130516/how-to-write-something-vertically-below-another-math-symbol - Symbol vertically below another
|
||||
|
||||
@ -47,3 +54,5 @@ https://tex.stackexchange.com/questions/508341/writing-a-system-of-linear-equati
|
||||
https://www.overleaf.com/learn/latex/Matrices - matrixes latex
|
||||
|
||||
https://tex.stackexchange.com/questions/294561/using-textbf-vs-mathbf-in-math-mode - mathbf vs textbf
|
||||
|
||||
https://www.math-linux.com/latex-26/faq/latex-faq/article/how-to-get-dots-in-latex-ldots-cdots-vdots-and-ddots - Dots LaTeX
|
||||
|
||||
Loading…
Reference in New Issue
Block a user