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\chapter{Design pattern}

Many code fragments are variations of some basic pattern. These
pattern are used in many different variations in many different
contexts. In this chapter we summarize a few of these \enterm[design
pattern]{design pattern}.

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\section{Looping over  vector elements}
Iterating over vector elements by means of a \mcode{for}-loop is a very commen pattern:
\begin{lstlisting}[caption={\varcode{for}-loop for accessing vector elements by indices}]
x = [2:3:20];      % Some vector.
for i=1:length(x)  % For loop over the indices of the vector.
  i                % This is the index (an integer number)
                   % to be used for accessing vector elements.
  x(i)             % This is the value of the i-th element of the vector.
  a = x(i);        % The value of the i-th vector element
                   % is assigned to the variable a.
  % Use the value of the i-th vector element by passing it
  % as an argument to a function:
  do_something(x(i));
end
\end{lstlisting}

If the result of the computation within the loop are single numbers
that are to be stored in a vector, you should create this vector with
the right size before the loop:
\begin{lstlisting}[caption={\varcode{for}-loop for writing a vector}]
x = [1.2 2.3 2.6 3.1];   % Some vector.
% Create a vector for the results, as long as the number of loops:
y = zeros(length(x),1);
for i=1:length(x)
  % Write the result of the computation at 
  % the i-th position in the y vector:
  y(i) = get_something(x(i));
end
% Now the result vector can be further processed:
mean(y);
\end{lstlisting}

The computation within the loop could also result in a vector of some
length and not just a single number. If the length of this vector
(here 10) is known beforehand, then you should create a matrix of
appropriate size for storing the results:
\begin{lstlisting}[caption={\varcode{for}-loop for writing rows of a matrix}]
x = [2:3:20];             % Some vector.
% Create space for results -
% as many rows as loops, as many columns as needed:
y = zeros(length(x), 10);
for i=1:length(x)
  % Write the return value of the function get_something - now a 
  % column vector with 10 elements - into the i-th row of the matrix y:
  y(i, :) = get_some_more(x(i));
end
% Process the results stored in matrix y:
mean(y, 1)
\end{lstlisting}

Another possibility is that the result vectors (here of unknown size)
need to be combined into a single large vector:
\begin{lstlisting}[caption={\varcode{for}-loop for appending vectors}]
x = [2:3:20];        % Some vector.
y = [];              % Empty vector for storing the results.
for i=1:length(x)
  % The function get_something() returns a vector of unspecified size:
  z = get_somehow_more(x(i));
  % The content of z is appended to the result vector y:
  y = [y; z(:)];
  % The z(:) syntax ensures that we append column-vectors.
end
% Process the results stored in the vector z:
mean(y)
\end{lstlisting}


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\section{Scaling and shifting random numbers, zeros, and ones}
Random number generators usually return random numbers of a given mean
and standard deviation. Multiply those numbers by a factor to change their standard deviation and add a number to shift the mean.
\begin{lstlisting}[caption={Scaling and shifting of random numbers}]
% 100 random numbers drawn from a normal distribution 
% with mean 0 and standard deviation 1:
x = randn(100, 1);

% 100 random numbers drawn from a normal distribution 
% with mean 4.8 and standard deviation 2.3:
mu = 4.8;
sigma = 2.3;
y = randn(100, 1)*sigma + mu;
\end{lstlisting}

The same principle can be useful for in the context of the functions
\mcode{zeros()} or \mcode{ones()}:
\begin{lstlisting}[caption={Scaling and shifting of \varcode{zeros()} and \varcode{ones()}}]
x = -1:0.01:2;      % Vector of x-values for plotting
plot(x, exp(-x.*x));
% Plot for the same x-values a horizontal line with y=0.8:
plot(x, zeros(size(x))+0.8);
% ... or a line with y=0.5:
plot(x, ones(size(x))*0.5);
\end{lstlisting}


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\section{Plotting  a mathematical function}
A mathematical function $y=f(x)$ assigns to each value of $x$ a single
$y$-value. For drawing a function with the computer we need compute a
table of values with many $x$-values and the corresponding function
values $y=f(x)$.

We first create a vector with useful $x$-values. They range from the
smallest to the largest value we want to plot. We also need to set a
step size that is small enough to result in a smooth plot of the
function. We then compute for each value $x_i$ of this vector the
corresponding function value $y_i$ and store them in a vector. We then
can plot the values of the $y$ vector against the ones of the $x$
vector.

The following scripts compute and plot the function $f(x)=e^{-x^2}$:
\begin{lstlisting}[caption={Plotting a mathematical function --- very detailed}]
xmin = -1.0;
xmax = 2.0;
dx = 0.01;        % Step size
x = xmin:dx:xmax; % Vector with x-values.
y = exp(-x.*x);   % No for loop! '.*' for multiplying the vector elements.
plot(x, y);
\end{lstlisting}

\begin{lstlisting}[caption={Plotting a mathematical function --- shorter}]
x = -1:0.01:2;
y = exp(-x.*x);
plot(x, y);
\end{lstlisting}

\begin{lstlisting}[caption={Plotting a mathematical function --- compact}]
x = -1:0.01:2;
plot(x, exp(-x.*x));
\end{lstlisting}


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\section{Normalizing histograms}
For estimating probabilities or probability densities from histograms
we need to normalize them appropriately.

The \mcode{histogram()} function does this automatically with the appropriate arguments:
\begin{lstlisting}[caption={Probability density with the \varcode{histogram()}-function}]
x = randn(100, 1);         % Some real-valued data.
histogram(x, 'Normalization', 'pdf');
\end{lstlisting}
\begin{lstlisting}[caption={Probability with the \varcode{histogram()}-function}]
x = randi(6, 100, 1);      % Some integer-valued data.
histogram(x, 'Normalization', 'probability');
\end{lstlisting}
Alternatively one can normalize the histogram data as returned by the
\code{hist()}-function manually:
\begin{lstlisting}[caption={Probability density with the \varcode{hist()}- and \varcode{bar()}-function}]
x = randn(100, 1);         % Some real-valued data.
[h, b] = hist(x);          % Compute histogram.
h = h/sum(h)/(b(2)-b(1));  % Normalization to a probability density.
bar(b, h);                 % Plot the probability density.
\end{lstlisting}
\begin{lstlisting}[caption={Probability with the \varcode{hist()}- and \varcode{bar()}-function}]
x = randi(6, 100, 1);      % Some integer-valued data.
[h, b] = hist(x);          % Compute histogram.
h = h/sum(h);              % Normalize to probability.
bar(b, h);                 % Plot the probabilities.
\end{lstlisting}