\documentclass{article}
\usepackage[pdftex,active,tightpage]{preview}
\setlength\PreviewBorder{2mm}

\usepackage[utf8]{inputenc} % this is needed for umlauts
\usepackage[ngerman]{babel} % this is needed for umlauts
\usepackage[T1]{fontenc}    % this is needed for correct output of umlauts in pdf
\usepackage{amssymb,amsmath,amsfonts} % nice math rendering
\usepackage{braket} % needed for \Set
\usepackage{algorithm,algpseudocode}

\begin{document}
\begin{preview}
    \begin{algorithm}[H]
        \begin{algorithmic}
            \Function{Gauss}{$A \in \mathbb{R}^{n \times {n+1}}$}
                \For{($i=1$; $\;i \leq n$; $\;i$++)}
                    \State // Search for maximum in this column
                    \State $maxEl = |A_{i,i}|$
                    \State $maxRow = i$
                    \For{($k=i+1$; $\;k \leq n$; $\;k$++)}
                        \If{$|A_{k,i}| > maxEl$}
                            \State $maxEl = A_{k,i}$
                            \State $maxRow = k$
                        \EndIf
                    \EndFor
                    \\
                    \State // Swap maximum row with current row
                    \For{($k=i$; $\;k \leq n$; $\;k$++)}
                        \State $tmp = A_{maxRow,k}$
                        \State $A_{maxRow,k} = A_{i,k}$
                        \State $A_{i,k} = tmp$
                    \EndFor
                    \\
                    \State // Make all rows below this one 0 in current column
                    \For{($k=i+1$; $\;k \leq n$; $\;k$++)}
                        \State $c = -\frac{A_{k,i}}{A_{i,i}}$
                        \For{($j=i$; $\;j \leq n$; $\;j$++)}
                            \If{$i == j$}
                                \State $A_{k,j} = 0$
                            \Else
                                \State $A_{k,j} += c \cdot A_{i,j}$
                            \EndIf
                        \EndFor
                    \EndFor
                \EndFor
                \\
                \State // Solve equation for an upper triangular matrix
                \State $x = \Set{0} \in \mathbb{R^n}$
                \For{($i=n$; $\;i \geq 1$; $\;i$-\--)}
                    \State $x_i = \frac{A_{i,n+1}}{A_{i,i}}$
                    \For{($k=i-1$; $\;k \geq 1$; $\;k$-\--)}
                        \State $A_{k,n+1} -= A_{k,i} \cdot x_{i}$
                    \EndFor
                \EndFor
                \\
                \State \Return $x$
            \EndFunction
        \end{algorithmic}
    \caption{Gaussian elimination}
    \label{alg:seq1}
    \end{algorithm}
\end{preview}
\end{document}
