\section{Results}
\subsection{Jacobi method result}
For system of equations we got in this task we got following results:
\\
Without the change in demanded tolerance:
\[ x = \left( \begin{array}{cc}
  -0.076776098668341 \\
   2.105784262642568 \\
   0.395344797635474 \\
   0.397776619764909
\end{array} \right)
\]
Error:
\[ r = \| \mathbf{A}\mathbf{x} - \mathbf{b}\| = 1.154375287358407e-10 \]
We managed to do this in \textbf{38} iterations of our loop, and the demanded tolerance did not change. (This required small change in code where I ommited the part of code responsible for changing demandedTolerance if $ \| \mathbf{A}x-b \| > \delta_2) $ )

With the change in demanded tolerance:
\[ x = \left( \begin{array}{cc}
  -0.076776098668341 \\
   2.105784262642568 \\
   0.395344797635474 \\
   0.397776619764909
\end{array} \right)
\]
Error:
\[ r = \| \mathbf{A}\mathbf{x} - \mathbf{b}\| = 5.770361548895147e-11 \]
We got this result in \textbf{37} iterations and demanded tolerance was equal to $2*10^{-10}$

Compared to matlab function
\[ x_{matlab} = \left( \begin{array}{cc}
  -0.076776098662498 \\
   2.105784262636790 \\
   0.395344797637659 \\
   0.397776619767240 \\
\end{array} \right)
\]
Matlab error:
\[ r = \| \mathbf{A}\mathbf{x} - \mathbf{b}\| = 4.070144838902081e-15 \]

\newpage
For data from task 2a we got: \\
Without change in demanded tolerance:
\[ x_a = \left( \begin{array}{cc}
-0.930024655108186 \\
-1.223407298660663 \\
-1.273530574212508 \\
-1.230517757317628 \\
-1.151356031082747 \\
-1.056883669273682 \\
-0.952628310081466 \\
-0.834334594312996 \\
-0.683708806198363 \\
-0.450125157620744 \\
\end{array} \right)
\]
Error:
\[ r = \| \mathbf{A}\mathbf{x} - \mathbf{b}\| = 6.955194519943778e-11 \]
We managed to do this in \textbf{59} iterations of our loop, and the demanded tolerance did not change.

With change in demanded tolerance:
\[ x_a = \left( \begin{array}{cc}
-0.930024655104470 \\
-1.223407298653515 \\
-1.273530574202540 \\
-1.230517757305602 \\
-1.151356031069692 \\
-1.056883669260597 \\
-0.952628310069469 \\
-0.834334594303006 \\
-0.683708806191233 \\
-0.450125157617020 \\
\end{array} \right)
\]
Error:
\[ r = \| \mathbf{A}\mathbf{x} - \mathbf{b}\| = 1.699812218689508e-10 \]
We managed to do this in \textbf{57} iterations of our loop, and the demanded tolerance changed to $4*10^{-10}$

Compared to matlab function
\[ x_{matlab} = \left( \begin{array}{cc}
  -0.930024655110760 \\
  -1.223407298665612 \\
  -1.273530574219411 \\
  -1.230517757325956 \\
  -1.151356031091789 \\
  -1.056883669282743 \\
  -0.952628310089775 \\
  -0.834334594319914 \\
  -0.683708806203301 \\
  -0.450125157623323 \\
\end{array} \right)
\]
Matlab error:
\[ r = \| \mathbf{A}\mathbf{x} - \mathbf{b}\| = 3.662053438817790e-15 \]

For Matrix and Vector from task 2b) error of
\[ \| x^{(i+1)} - x^{(i)} \| \]
grew to infinity, therefore we could never achieve demanded tolerance, therefore the program executed infinite loop.

\subsubsection{Minimizing the demanded error}
I tried to minimize the demanded error using this steps:
\begin{enumerate}
\item I copied error from matlab function and pasted it into demanded tolerance.
\item If I did not get infinite loop I copied the newly acquired error and pasted it into demanded tolerance.
\item If I got inifinite loop I used the previous error as "minimal" demanded error.
\end{enumerate}
\newpage
\paragraph{For original system of equations:}
We managed to get results with error as low as $1.776356839400250e-15$ with demanded tolerance = $3.202372833989376e-15$ for lower values program went into infinite loop.
Results for demanded tolerance = $3.202372833989376e-15$
For given matrix:
\[ x = \left( \begin{array}{cc}
  -0.076776098662498 \\
   2.105784262636790 \\
   0.395344797637659 \\
   0.397776619767240
\end{array} \right)
\]
Error:
\[ r = \| \mathbf{A}\mathbf{x} - \mathbf{b}\| = 3.108624468950438e-15 \]
We got this result in \textbf{53} iterations and demanded tolerance did not change.

\newpage
\paragraph{For task 2a) system of equations:}
We managed to get results with error as low as
\[ 3.108624468950438e-15 \] with demanded tolerance:
\[ 3.202372833989376e-15 \]
for lower values program went into infinite loop.

For demanded tolerance = $3.202372833989376e-15$:
Results for 2a) system of equation

\[ x_a = \left( \begin{array}{cc}
-0.930024655110760 \\
-1.223407298665613 \\
-1.273530574219411 \\
-1.230517757325955 \\
-1.151356031091788 \\
-1.056883669282743 \\
-0.952628310089775 \\
-0.834334594319914 \\
-0.683708806203301 \\
-0.450125157623323
\end{array} \right)
\]
Error:
\[ r = \| \mathbf{A}\mathbf{x} - \mathbf{b}\| = 3.108624468950438e-15 \]
We managed to do this in \textbf{84} iterations of our loop, and the demanded tolerance did not change.
We managed to achieve slightly better (as in, the error was smaller) results than Matlab custom function.

\newpage
\subsection{Gauss-Seidel method result}
For system of equations we got in this task we got following results:
\\
\[ x = \left( \begin{array}{cc}
  -0.076776098752996 \\
   2.105784262542701 \\
   0.395344797589652 \\
   0.397776619735316
\end{array} \right)
\]
Error:
\[ r = \| \mathbf{A}\mathbf{x} - \mathbf{b}\| = 6.999102842719934e-10 \]
We managed to do this in \textbf{19} iterations of our loop, and the demanded tolerance did not change.
Error:
\[ r = \| \mathbf{A}\mathbf{x} - \mathbf{b}\| = 6.999102842719934e-10 \]
We got this result in \textbf{19} iterations and demanded tolerance was equal to $2*10^{-10}$

For data from task 2a we got: \\
\[ x_a = \left( \begin{array}{cc}
-0.930024655049330 \\
-1.223407298590161 \\
-1.273530574151901 \\
-1.230517757273947 \\
-1.151356031055568 \\
-1.056883669259564 \\
-0.952628310076145 \\
-0.834334594312669 \\
-0.683708806199986 \\
-0.450125157622217 \\
\end{array} \right)
\]
Error:
\[ r = \| \mathbf{A}\mathbf{x} - \mathbf{b}\| = 5.311669113699570e-10 \]
We managed to do this in \textbf{28} iterations of our loop, and the demanded tolerance did not change.

For Matrix and Vector from task 2b) error of
\[ \| x^{(i+1)} - x^{(i)} \| \]
grew to infinity, therefore we could never achieve demanded tolerance, therefore the program executed infinite loop.

\subsubsection{Minimizing the demanded error}
I tried to minimize the demanded error using procedure from before
\paragraph{For original system of equations:}
We managed to get results with error as low as $1.776356839400250e-15$ with demanded tolerance = $ 1.986027322597818e-15 $ for lower values program went into infinite loop.
Results for demanded tolerance = $1.986027322597818e-15$
For given matrix:
\[ x = \left( \begin{array}{cc}
  -0.076776098662498 \\
   2.105784262636790 \\
   0.395344797637659 \\
   0.397776619767240
\end{array} \right)
\]
Error:
\[ r = \| \mathbf{A}\mathbf{x} - \mathbf{b}\| = 1.776356839400250e-15 \]
We got this result in \textbf{30} iterations and demanded tolerance did not change.

\newpage
\paragraph{For task 2a) system of equations:}
We managed to get results with error as low as
\[ 1.538370149106851e-15 \] with demanded tolerance:
\[ 1.538370149106851e-15 \]
for lower values program went into infinite loop.

For demanded tolerance = $1.538370149106851e-15$:
Results for 2a) system of equation

\[ x_a = \left( \begin{array}{cc}
-0.930024655110760 \\
-1.223407298665613 \\
-1.273530574219411 \\
-1.230517757325955 \\
-1.151356031091788 \\
-1.056883669282743 \\
-0.952628310089775 \\
-0.834334594319914 \\
-0.683708806203301 \\
-0.450125157623323 \\
\end{array} \right)
\]
Error:
\[ r = \| \mathbf{A}\mathbf{x} - \mathbf{b}\| = 1.538370149106851e-15 \]
We managed to do this in \textbf{42} iterations of our loop, and the demanded tolerance did not change.
We managed to achieve slightly better (as in, the error was smaller) results than Matlab custom function.


\newpage