mirror of
https://github.com/MartinThoma/LaTeX-examples.git
synced 2025-04-26 06:48:04 +02:00
7740f0147f
The commands
find . -type f -name '*.md' -exec sed --in-place 's/[[:space:]]\+$//' {} \+
and
find . -type f -name '*.tex' -exec sed --in-place 's/[[:space:]]\+$//' {} \+
were used to do so.
38 lines
1.5 KiB
TeX
38 lines
1.5 KiB
TeX
\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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Sei $n \in \mathbb{N}_{\geq 1}$, $A \in \mathbb{R}^{n \times n}$ und
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positiv definit sowie symmetrisch.
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Dann existiert eine Zerlegung $A = L \cdot L^T$, wobei $L$ eine
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untere Dreiecksmatrix ist. Diese wird von folgendem Algorithmus
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berechnet:
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\begin{algorithm}[H]
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\begin{algorithmic}
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\Function{Cholesky}{$A \in \mathbb{R}^{n \times n}$}
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\State $L = \Set{0} \in \mathbb{R}^{n \times n}$ \Comment{Initialisiere $L$}
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\For{($k=1$; $\;k \leq n$; $\;k$++)}
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\State $L_{k,k} = \sqrt{A_{k,k} - \sum_{i=1}^{k-1} L_{k,i}^2}$
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\For{($i=k+1$; $\;i \leq n$; $\;i$++)}
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\State $L_{i,k} = \frac{A_{i,k} - \sum_{j=1}^{k-1} L_{i,j} \cdot L_{k,j}}{L_{k,k}}$
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\EndFor
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\EndFor
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\State \Return $L$
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\EndFunction
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\end{algorithmic}
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\caption{Cholesky-Zerlegung}
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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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