\section*{Problem 8}

%link to relvant files and show plots
\subsection*{a)}
Increasing number of steps result in a decrease of absolute error.
\begin{figure}[H]
    \centering 
    \includegraphics[width=0.8\linewidth]{images/problem8_a.pdf}
    \caption{Absolute error for different step sizes $n_{steps}$.}
\end{figure}

\subsection*{b)}
Increasing number of steps also result in a decrease of absolute error.
\begin{figure}[H]
    \centering 
    \includegraphics[width=0.8\linewidth]{images/problem8_b.pdf}
    \caption{Relative error for different step sizes $n_{steps}$.}
\end{figure}

\subsection*{c)}
Increasing number of steps result in a decrease of maximum relative error, up to a certain number of steps. At $n_{steps} \approx 10^{5}$ the maximumrelative error increase. 
This can be related to loss of numerical precicion when step size is small.
\begin{figure}[H]
    \centering 
    \includegraphics[width=0.7\linewidth]{images/problem8_c.pdf}
    \caption{Maximum relative error for each step sizes $n_{steps}$.}
\end{figure}