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\documentclass{article}

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\usepackage[pdftex,active,tightpage]{preview}

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\setlength\PreviewBorder{2mm}

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\usepackage[utf8]{inputenc} % this is needed for umlauts

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\usepackage[ngerman]{babel} % this is needed for umlauts

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\usepackage[T1]{fontenc}    % this is needed for correct output of umlauts in pdf

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\usepackage{amssymb,amsmath,amsfonts} % nice math rendering

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\usepackage{braket} % needed for \Set

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\usepackage{algorithm,algpseudocode}

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\begin{document}

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\begin{preview}

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    \begin{algorithm}[H]

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        \begin{algorithmic}

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            \Function{Gauss}{$A \in \mathbb{R}^{n \times {n+1}}$}

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                \For{($i=1$; $\;i \leq n$; $\;i$++)}

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                    \State // Search for maximum in this column

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                    \State $maxEl = |A_{i,i}|$

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                    \State $maxRow = i$

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                    \For{($k=i+1$; $\;k \leq n$; $\;k$++)}

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                        \If{$|A_{k,i}| > maxEl$}

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                            \State $maxEl = A_{k,i}$

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                            \State $maxRow = k$

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                        \EndIf

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                    \EndFor

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                    \\

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                    \State // Swap maximum row with current row

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                    \For{($k=i$; $\;k \leq n$; $\;k$++)}

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                        \State $tmp = A_{maxRow,k}$

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                        \State $A_{maxRow,k} = A_{i,k}$

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                        \State $A_{i,k} = tmp$

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                    \EndFor

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                    \\

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                    \State // Make all rows below this one 0 in current column

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                    \For{($k=i+1$; $\;k \leq n$; $\;k$++)}

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                        \State $c = -\frac{A_{k,i}}{A_{i,i}}$

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                        \For{($j=i$; $\;j \leq n$; $\;j$++)}

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                            \If{$i == j$}

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                                \State $A_{k,j} = 0$

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                            \Else

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                                \State $A_{k,j} += c \cdot A_{i,j}$

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                            \EndIf

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                        \EndFor

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                    \EndFor

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                \EndFor

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                \\

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                \State // Solve equation for an upper triangular matrix

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                \State $x = \Set{0} \in \mathbb{R^n}$

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                \For{($i=n$; $\;i \geq 1$; $\;i$-\--)}

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                    \State $x_i = \frac{A_{i,n+1}}{A_{i,i}}$

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                    \For{($k=i-1$; $\;k \geq 1$; $\;k$-\--)}

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                        \State $A_{k,n+1} -= A_{k,i} \cdot x_{i}$

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                    \EndFor

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                \EndFor

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                \\

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                \State \Return $x$

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            \EndFunction

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        \end{algorithmic}

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    \caption{Gaussian elimination}

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    \label{alg:seq1}

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    \end{algorithm}

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\end{preview}

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\end{document}

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