english design pattern
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\chapter{Design pattern}
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\chapter{Design patterns}
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\selectlanguage{ngerman}
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Beim Programmieren sind sich viel Codes in ihrer Grundstruktur sehr
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\"ahnlich. Viele Konstrukte kommen in den verschiedensten Kontexten
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immer wieder in \"ahnlicher Weise vor. In diesem Kapitel stellen wir
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einige dieser ``Design pattern'' zusammen.
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Many code fragments are variations of some basic patterns. These
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patterns are used in many different variations in many different
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contexts. In this chapter we summarize a few of these \enterm[design
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pattern]{design patterns}.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{for Schleifen \"uber Vektoren}
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Grundlegend ist das Iterieren \"uber den Inhalt eines Vektors mit einer \code{for}-Schleife:
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\begin{lstlisting}[caption={\varcode{for}-Schleife mit Indexen \"uber einen Vektor}]
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x = [2:3:20]; % irgendein Vektor
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for i=1:length(x) % Mit der for-Schleife "loopen" wir ueber den Vektor
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i % das ist der Index, der die Elemente des Vektors indiziert.
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x(i) % das ist der Wert des i-ten Elements des Vektors x.
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a = x(i); % die Variable a bekommt den Wert des i-ten Elements des Vektors x zugewiesen.
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% Benutze den Wert:
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\section{Looping over vector elements}
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Iterating over vector elements by means of a \mcode{for}-loop is a very commen pattern:
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\begin{lstlisting}[caption={\varcode{for}-loop for accessing vector elements by indices}]
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x = [2:3:20]; % Some vector.
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for i=1:length(x) % For loop over the indices of the vector.
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i % This is the index (an integer number)
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% to be used for accessing vector elements.
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x(i) % This is the value of the i-th element of the vector.
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a = x(i); % The value of the i-th vector element
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% is assigned to the variable a.
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% Use the value of the i-th vector element by passing it
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% as an argument to a function:
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do_something( x(i) );
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end
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\end{lstlisting}
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Wenn in der Schleife das Ergebnis in einen Vektor gespeichert werden soll,
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sollten wir vor der Schleife schon einen Vektor f\"ur die Ergebnisse
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erstellen:
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\begin{lstlisting}[caption={\varcode{for}-Schleife zum Schreiben eines Vektors}]
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x = [1.2 2.3 2.6 3.1]; % irgendein Vektor
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y = zeros(length(x),1); % Platz fuer die Ergebnisse, genauso viele wie Loops der Schleife
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If the result of the computation within the loop are single numbers
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that are to be stored in a vector, you should create this vector with
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the right size before the loop:
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\begin{lstlisting}[caption={\varcode{for}-loop for writing a vector}]
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x = [1.2 2.3 2.6 3.1]; % Some vector.
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% Create a vector for the results, as long as the number of loops:
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y = zeros(length(x),1);
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for i=1:length(x)
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% Schreibe den Rueckgabewert der Funktion get_something an die i-te
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% Stelle von y:
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% Write the result of the computation at
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% the i-th position in the y vector:
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y(i) = get_something( x(i) );
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end
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% jetzt koennen wir den Ergebnisvektor weiter bearbeiten:
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mean(y)
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% Now the result vector can be further processed:
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mean(y);
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\end{lstlisting}
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Die Berechnungen in der Schleife k\"onnen statt einer Zahl auch einen Vektor
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zur\"uckgeben. Wenn die L\"ange diese Vektors bekannt ist, dann kann vorher
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eine entsprechend gro{\ss}e Matrix angelegt werden:
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\begin{lstlisting}[caption={\varcode{for}-Schleife zum Schreiben von Zeilen einer Matrix}]
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x = [2:3:20]; % irgendein Vektor
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y = zeros(length(x),10); % Platz fuer die Ergebnisse
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The computation within the loop could also result in a vector of some
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length and not just a single number. If the length of this vector
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(here 10) is known beforehand, then you should create a matrix of
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appropriate size for storing the results:
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\begin{lstlisting}[caption={\varcode{for}-loop for writing rows of a matrix}]
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x = [2:3:20]; % Some vector.
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% Create space for results -
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% as many rows as loops, as many columns as needed:
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y = zeros(length(x), 10);
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for i=1:length(x)
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% Schreibe den Rueckgabewert der Funktion get_something - jetzt ein
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% Vektor mit 10 Elementen - in die i-te
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% Zeile von y:
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y(i,:) = get_something( x(i) );
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% Write the return value of the function get_something - now a
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% column vector with 10 elements - into the i-th row of the matrix y:
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y(i, :) = get_something(x(i));
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end
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% jetzt koennen wir die Ergebnismatrix weiter bearbeiten:
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% Process the results stored in matrix y:
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mean(y, 1)
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\end{lstlisting}
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Alternativ k\"onnen die in der Schleife erzeugten Vektoren zu einem
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einzigen, durchgehenden Vektor zusammengestellt werden:
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\begin{lstlisting}[caption={\varcode{for}-Schleife zum Aneinanderh\"angen von Vektoren}]
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x = [2:3:20]; % irgendein Vektor
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y = []; % Leerer Vektor fuer die Ergebnisse
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Another possibility is that the result vectors (here of unknown size)
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need to be combined into a single large vector:
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\begin{lstlisting}[caption={\varcode{for}-loop for appending vectors}]
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x = [2:3:20]; % Some vector.
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y = []; % Empty vector for storing the results.
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for i=1:length(x)
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% Die Funktion get_something gibt einen Vektor zurueck:
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z = get_something( x(i) );
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% dessen Inhalt h\"angen wir an den Ergebnissvektor an:
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% The function get_something() returns a vector of unspecified size:
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z = get_something(x(i));
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% The content of z is appended to the result vector y:
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y = [y; z(:)];
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% z(:) stellt sicher, das wir auf jeden Fall einen Spaltenvektoren aneinanderreihen.
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% The z(:) syntax ensures that we append column-vectors.
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end
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% jetzt koennen wir dem Ergebnisvektor weiter bearbeiten:
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% Process the results stored in the vector z:
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mean(y)
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\end{lstlisting}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Skalieren und Verschieben nicht nur von Zufallszahlen}
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Zufallsgeneratoren geben oft nur Zufallszahlen mit festen Mittelwerten
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und Standardabweichungen (auch Skalierungen) zur\"uck. Multiplikation
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mit einem Faktor skaliert die Standardabweichung und Addition einer Zahl
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verschiebt den Mittelwert.
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\begin{lstlisting}[caption={Skalierung von Zufallszahlen}]
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% 100 random numbers draw from a Gaussian distribution with mean 0 and standard deviation 1.
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\section{Scaling and shifting random numbers, zeros, and ones}
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Random number generators usually return random numbers of a given mean
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and standard deviation. Multiply those numbers by a factor to change their standard deviation and add a number to shift the mean.
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\begin{lstlisting}[caption={Scaling and shifting of random numbers}]
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% 100 random numbers drawn from a normal distribution
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% with mean 0 and standard deviation 1:
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x = randn(100, 1);
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% 100 random numbers drawn from a Gaussian distribution with mean 4.8 and standard deviation 2.3.
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% 100 random numbers drawn from a normal distribution
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% with mean 4.8 and standard deviation 2.3:
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mu = 4.8;
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sigma = 2.3;
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y = randn(100, 1)*sigma + mu;
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\end{lstlisting}
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Das gleiche Prinzip ist manchmal auch sinnvoll f\"ur \code{zeros()} oder \code{ones()}:
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\begin{lstlisting}[caption={Skalierung von \varcode{zeros()} und \varcode{ones()}}]
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x = -1:0.01:2; % Vektor mit x-Werten
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The same principle can be useful for in the context of the functions
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\mcode{zeros()} or \mcode{ones()}:
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\begin{lstlisting}[caption={Scaling and shifting of \varcode{zeros()} and \varcode{ones()}}]
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x = -1:0.01:2; % Vector of x-values for plotting
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plot(x, exp(-x.*x));
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% Plotte f\"ur die gleichen x-Werte eine Linie mit y=0.8:
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% Plot for the same x-values a horizontal line with y=0.8:
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plot(x, zeros(size(x))+0.8);
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% ... Linie mit y=0.5:
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% ... or a line with y=0.5:
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plot(x, ones(size(x))*0.5);
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\end{lstlisting}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Plotten einer mathematischen Funktion}
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Eine mathematische Funktion ordnet einem beliebigen $x$-Wert einen
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$y$-Wert zu. Um eine solche Funktion zeichnen zu k\"onnen, m\"ussen
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wir eine Wertetabelle aus vielen $x$-Werten und den
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dazugeh\"origen Funktionswerten $y=f(x)$ erstellen.
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Wir erstellen dazu einen Vektor mit geeigneten $x$-Werten, die von
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dem kleinsten bis zu dem gr\"o{\ss}ten $x$-Wert laufen, den wir
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plotten wollen. Die Schrittweite f\"ur die $x$-Werte w\"ahlen wir
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klein genug, um eine sch\"one glatte Kurve zu bekommen. F\"ur jeden
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Wert $x_i$ dieses Vektors berechnen wir den entsprechenden
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Funktionswert und erzeugen damit einen Vektor mit den $y$-Werten. Die
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Werte des $y$-Vektors k\"onnen dann gegen die Werte des $x$-Vektors
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geplottet werden.
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Folgende Programme berechnen und plotten die Funktion $f(x)=e^{-x^2}$:
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\begin{lstlisting}[caption={Plotten einer mathematischen Funktion --- sehr ausf\"uhrlich}]
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\section{Plotting a mathematical function}
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A mathematical function $y=f(x)$ assigns to each value of $x$ a single
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$y$-value. For drawing a function with the computer we need compute a
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table of values with many $x$-values and the corresponding function
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values $y=f(x)$.
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We first create a vector with useful $x$-values. They range from the
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smallest to the largest value we want to plot. We also need to set a
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step size that is small enough to result in a smooth plot of the
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function. We then compute for each value $x_i$ of this vector the
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corresponding function value $y_i$ and store them in a vector. We then
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can plot the values of the $y$ vector against the ones of the $x$
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vector.
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The following scripts compute and plot the function $f(x)=e^{-x^2}$:
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\begin{lstlisting}[caption={Plotting a mathematical function --- very detailed}]
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xmin = -1.0;
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xmax = 2.0;
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dx = 0.01; % Schrittweite
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x = xmin:dx:xmax; % Vektor mit x-Werten
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y = exp(-x.*x); % keine for Schleife! '.*' fuer elementweises multiplizieren
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dx = 0.01; % Step size
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x = xmin:dx:xmax; % Vector with x-values.
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y = exp(-x.*x); % No for loop! '.*' for multiplying the vector elements.
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plot(x, y);
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\end{lstlisting}
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\begin{lstlisting}[caption={Plotten einer mathematischen Funktion --- k\"urzer}]
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x = -1:0.01:2; % Vektor mit x-Werten
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y = exp(-x.*x); % keine for Schleife! '.*' fuer elementweises multiplizieren
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\begin{lstlisting}[caption={Plotting a mathematical function --- shorter}]
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x = -1:0.01:2;
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y = exp(-x.*x);
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plot(x, y);
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\end{lstlisting}
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\begin{lstlisting}[caption={Plotten einer mathematischen Funktion --- sehr kompakt}]
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x = -1:0.01:2; % Vektor mit x-Werten
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\begin{lstlisting}[caption={Plotting a mathematical function --- compact}]
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x = -1:0.01:2;
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plot(x, exp(-x.*x));
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\end{lstlisting}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Normierung von Histogrammen}
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Meistens sollten Histogramme normiert werden, damit sie vergleichbar
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mit anderen Histogrammen oder mit theoretischen
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Wahrscheinlichkeitsverteilungen werden.
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Die \code{histogram()} Funktion macht das mit den entsprechenden Parametern automatisch:
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\begin{lstlisting}[caption={Probability-density-function mit der \varcode{histogram()}-Funktion}]
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x = randn(100, 1); % irgendwelche reellwertige Daten
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\section{Normalizing histograms}
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For estimating probabilities or probability densities from histograms
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we need to normalize them appropriately.
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The \mcode{histogram()} function does this automatically with the appropriate arguments:
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\begin{lstlisting}[caption={Probability density with the \varcode{histogram()}-function}]
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x = randn(100, 1); % Some real-valued data.
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histogram(x, 'Normalization', 'pdf');
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\end{lstlisting}
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\begin{lstlisting}[caption={Probability mit der \varcode{histogram()}-Funktion}]
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x = randi(6, 100, 1); % irgendwelche integer Daten
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\begin{lstlisting}[caption={Probability mit der \varcode{histogram()}-function}]
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x = randi(6, 100, 1); % Some integer-valued data.
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histogram(x, 'Normalization', 'probability');
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\end{lstlisting}
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So geht es mit der \code{hist()}-Funktion:
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\begin{lstlisting}[caption={Probability-density-function mit der \varcode{hist()}- und \varcode{bar()}-Funktion}]
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x = randn(100, 1); % irgendwelche reellwertige Daten
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[h, b] = hist(x); % Histogram berechnen
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h = h/sum(h)/(b(2)-b(1)); % normieren zu einer Wahrscheinlichkeitsdichte
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bar(b, h); % und plotten.
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Alternatively one can normalize the histogram data as returned by the
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\code{hist()}-function manually:
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\begin{lstlisting}[caption={Probability density with the \varcode{hist()}- and \varcode{bar()}-function}]
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x = randn(100, 1); % Some real-valued data.
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[h, b] = hist(x); % Compute histogram.
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h = h/sum(h)/(b(2)-b(1)); % Normalization to a probability density.
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bar(b, h); % Plot the probability density.
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\end{lstlisting}
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\begin{lstlisting}[caption={Probability mit der \varcode{hist()}- und \varcode{bar()}-Funktion}]
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x = randi(6, 100, 1); % irgendwelche integer Daten
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[h, b] = hist(x); % Histogram berechnen
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h = h/sum(h); % normieren zu Wahrscheinlichkeiten
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bar(b, h); % und plotten.
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\begin{lstlisting}[caption={Probability with the \varcode{hist()}- and \varcode{bar()}-function}]
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x = randi(6, 100, 1); % Some integer-valued data.
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[h, b] = hist(x); % Compute histogram.
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h = h/sum(h); % Normalize to probability.
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bar(b, h); % Plot the probabilities.
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\end{lstlisting}
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