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\header{{\bfseries\large Exercise 6}}{{\bfseries\large Scripts and functions}}{{\bfseries\large 06. November, 2018}}
\firstpagefooter{Prof. Jan Benda}{Phone: 29 74 573}{Email:
jan.benda@uni-tuebingen.de}
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\begin{document}
\vspace*{-6.5ex}
\begin{center}
\textbf{\Large Introduction to Scientific Computing}\\[1ex]
{\large Jan Grewe, Jan Benda}\\[-3ex]
Neuroethology \hfill --- \hfill Institute for Neurobiology \hfill --- \hfill \includegraphics[width=0.28\textwidth]{UT_WBMW_Black_RGB} \\
\end{center}
The exercises are meant for self-monitoring and revision of the
lecture. You should try to solve them on your own. In contrast
to previous exercises, the solutions can not be saved in a single file
but each question needs an individual file. Combine the files into a
single zip archive and submit it via ILIAS. Name the archive according
to the pattern: ``scripts\_functions\_\{surname\}.zip''.
\begin{questions}
\question Calculate the factorial of a given number $n$.
\begin{parts}
\part{}
Version 1: Write a script that calculates the factorial of 5 and
prints out the result.
\begin{solution}
\lstinputlisting{factorialscripta.m}
\end{solution}
\part{}
Version 2: like version 1, but as a function that takes $n$ as
input argument.
\begin{solution}
\lstinputlisting{printfactorial.m}
\lstinputlisting{factorialscriptb.m}
\end{solution}
\part{}
Version 3: like version 2, but the calculated result should not be
printed on the command line but returned by the function. Write a
script that calls the function and prints out the result.
\begin{solution}
\lstinputlisting{myfactorial.m}
\lstinputlisting{factorialscriptc.m}
\end{solution}
\end{parts}
\question Graphical display of a sinewave.
\begin{parts}
\part{}
Implement a function that plots a sine with the amplitude 1 and
the frequency $f=50$\,Hz $\left(\sin(2\pi \cdot f \cdot t)\right)$. Call the
function in a script.
\begin{solution}
\lstinputlisting{plotsine50.m}
\lstinputlisting{plotsinea.m}
\end{solution}
\part{}
Improve the function that it takes the duration of the time axis,
the amplitude, and the frequency as input arguments. The
calculation should use a temporal stepsize that is 0.01 of the
frequency.
\begin{solution}
\lstinputlisting{plotsine.m} \lstinputlisting{plotsineb.m}
\end{solution}
\part{}
Write a script that calls the function and controls the
plotting. Change the function in a way that it returns a proper
time-axis and the calculated sinwave.
\begin{solution}
\lstinputlisting{sinewave.m}
\lstinputlisting{plotsinec.m}
\end{solution}
\part{}
Write a second function that does the plotting. It accepts the
time and the sine as arguments. Make sure, that the plot is
properly labeled. Write a small script that uses both funtions to
plot a sine of $5$\,Hz frequency and an amplitude of 2 for a
duration of 1.5\,s.
\begin{solution}
\lstinputlisting{plotsinewave.m}
\lstinputlisting{plotsined.m}
\end{solution}
\end{parts}
%\question Schreibe eine Funktion, die bin\"are Datens\"atze
%('signal.bin' und 'signal2.bin' vom Montag) liest und die Daten als
%Vektor zur\"uckgibt. Welche Argumente muss die Funktion
%\"ubernehmen?
\question Random Walk. In a 1-D random walk an \emph{agent} walks
randomly either in the one ($+1$) or the other ($-1$)
direction. With each simulation step one direction is chosen and the
position is updated accordingly. \textbf{There are some differences
to the solution discussed in the lecture!}
\begin{parts}
\part{}
Read the exercise completely before starting the implementation
and then come up with a proper program layout of scripts and
functions. What would be a suitable function to solve the task? Which
arguments should it take? Which results should it return?
\begin{solution}
One function that computes one realisation of a random walk.
Scripts for plotting and analysis.
\lstinputlisting{randomwalkthresh.m}
\end{solution}
\part{}
Run the simulation 10 times and plot the time-course of the
positions of the \emph{agents} into the same plot. The probability
of the two directions should be the same. Each \emph{agent} starts
at position $0$ and the simulation should run until the position is
greater than $50$ or less than $-50$.
\begin{solution}
\lstinputlisting{randomwalkscriptb.m}
\end{solution}
\part{}
Now we want to know how the probability of $p_{+1}$ (the
probability to walk into the $+1$ direction) impacts the random
walk. Vary $p_{+1}$ in the range $0.5 \le p_{+1} < 0.8$. Do 10
random walks for the four probabilities (apply the same thresholds for
stopping the simulations as before).
\begin{solution}
\lstinputlisting{randomwalkscriptc.m}
\end{solution}
\part{}
How does $p_{+1}$ affect the number of simulation steps? Plot the
averages and standard deviations as a function of $p_{+1}$.
\begin{solution}
\lstinputlisting{randomwalkscriptd.m}
\end{solution}
\end{parts}
%\question Modellierung des exponentiellen Wachstums einer isolierten
%Population. Das exponentielle Wachstum einer isolierten Population
%wird \"uber folgende Differentialgleichung beschrieben:
%\begin{equation}
% \frac{dN}{dt} = N \cdot r,
%\end{equation}
%mit $N$ der Populationsgr\"o{\ss}e und $r$ der Wachstumsrate.
%\begin{parts}
% \part L\"ose die Gleichung numerisch mit dem Euler Verfahren.
% \part Implementiere eine Funktion, die die Populationsgr\"o{\ss}e
% und die Zeit zur\"uckgibt.
% \part Plotte die Populationsgr\"o{\ss}e als Funktion der Zeit.
%\end{parts}
%\question Etwas realistischer ist das logistische Wachstum einer
%isolierten Population, bei der das Wachstum durch eine Kapazit\"at
%gedeckelt ist. Sie wird mit folgender Differentialgleichung
%beschrieben:
%\begin{equation}
% \frac{dN}{dt} = N \cdot r \cdot \left( 1 - \frac{N}{K} \right)
%\end{equation}
%mit $N$ der Population, der Wachstumsrate $r$ und $K$ der ``tragenden''
%Kapazit\"at.
%\begin{parts}
% \part Implementiere die L\"osung des logistischen Wachstums in
% einer Funktion. Benutze das Euler Verfahren. Die Funktion soll die
% Parameter $r$, $K$ sowie den Startwert von $N$ als Argumente
% \"ubernehmen.
% \part Die Funktion soll die Populationsgr\"o{\ss}e und die Zeit
% zur\"uckgeben.
% \part Simuliere das Wachstum mit einer Anzahl unterschiedlicher
% Startwerte f\"ur $N$.
% \part Stelle die Ergebnisse in einem Plot graphisch dar.
% \part Plotte das Wachstum $dN/dt$ als Funktion der
% Populationsgr\"o{\ss}e $N$.
%\end{parts}
\end{questions}
\end{document}