Commit 0ccf039b authored by Niklas's avatar Niklas

to Euler

parent b44dd2e7
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......@@ -7,6 +7,8 @@
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{mathtools}
\usepackage[osf]{mathpazo}
\usepackage{eulervm}
\usepackage{array}
\usepackage{graphicx}
\usepackage{float}
......
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......@@ -502,8 +502,10 @@ We first only consider the second stage, which is sufficient if setting $x_1 = \
$r \coloneqq \arg\min_{i \in B} \left\{\frac{\bar{b}_i}{\bar{a}_{is}} \mid \bar{a}_{is} > 0\right\}$.\\
\tcp{Pivoting}
$B \coloneqq B \setminus \{r\} \cup \{s\}$.\\
$N \coloneqq N \setminus \{s\} \cup \{r\}$.
%TODO new A, b, c, z
$N \coloneqq N \setminus \{s\} \cup \{r\}$.\\
$a_{rj} \coloneqq \frac{a_{rj}}{a_{rs}}$ for $j \in \{1, \ldots, n+m\}$.\\
$a_{rs} \coloneqq 1$ and $a_{is} = 0$ for $i \in \{0, \ldots, n\} \setminus \{r\}$.~~~~~~~~~\tcp{$0$ refers to $\bar{c}$ and $\bar{z}$ row}
$a_{ij} \coloneqq a_{ij} - \frac{a_{rj} a_{is}}{a_{rs}}$ for $i \in \{0, \ldots, n\} \setminus \{r\}, j \in \{1, \ldots, n+m\} \setminus \{s\}$.
}
\Return{optimal solution}
\end{algorithm}
......@@ -666,7 +668,12 @@ where $s_i$ denotes the slack variable of that constraint and $y_i$ the artifici
\begin{algorithm}[H]
\caption{Basic Simplex algorithm -- stage 1}
TODO
\For{$i \in \{1, \ldots, m\}$ with $b_i < 0$}{
$b_i \coloneqq -b_i$ and $a_{ij} \coloneqq -a_{ij}$ for $j \in \{1, \ldots, n\}$.\\
$a_{-1j} \coloneqq a_{-1j} + M a_{ij}$.~~~~~~~~~~~~~~~~~~~~~~~~\tcp{row $-1$ refers to $M$-row}
}
Apply stage 2 steps to row $-1$ until no artificial variable in basis.\\
Delete artificial variables and row $-1$.
\end{algorithm}
\begin{example*}
......
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