%
% euklidtabelle.tex
%
% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
%
\begin{frame}[t]
\setlength{\abovedisplayskip}{5pt}
\setlength{\belowdisplayskip}{5pt}
\frametitle{Durchführung des euklidischen Algorithmus}
Problem: Berechnung der Produkte $Q(q_k)\cdots Q(q_1)Q(q_0)$ für $k=0,1,\dots,n$
\uncover<2->{%
\begin{block}{Multiplikation mit $Q(q_k)$}
\vspace{-12pt}
\begin{align*}
Q(q_k)
\ifthenelse{\boolean{presentation}}{
\only<-3>{
\begin{pmatrix}
u&v\\c&d
\end{pmatrix}
=\begin{pmatrix}
0&1\\1&-q_k
\end{pmatrix}
}}{}
\begin{pmatrix}
u&v\\c&d
\end{pmatrix}
&\uncover<3->{=
\begin{pmatrix}
c&d\\
u-q_kc&v-q_kd
\end{pmatrix}}
&&\uncover<5->{\Rightarrow&
\begin{pmatrix}
c_k&d_k\\c_{k+1}&d_{k+1}
\end{pmatrix}
&=
Q(q_k)
%\begin{pmatrix}
%0&1\\1&-q_k
%\end{pmatrix}
\begin{pmatrix}
c_{k-1}&d_{k-1}\\c_{k}&d_{k}
\end{pmatrix}}
\end{align*}
\end{block}}
\vspace{-10pt}
\uncover<6->{%
\begin{equation*}
\begin{tabular}{|>{\tiny$}r<{$}|>{$}c<{$}|>{$}c<{$}>{$}c<{$}|}
\hline
k  &q_k    &                     c_k    &                     d_k    \\
\hline
-1 &       &                     1      &                     0      \\
 0 &\uncover<7->{q_0    }&                     0      &                     1      \\
 1 &\uncover<9->{q_1    }&\uncover<8->{c_{-1} -q_0    \cdot c_0    &d_{-1} -q_0    \cdot d_0    }\\
 2 &\uncover<11->{q_2    }&\uncover<10->{c_0    -q_1    \cdot c_1    &d_0    -q_1    \cdot d_1    }\\
\vdots&\uncover<12->{\vdots}&\uncover<12->{\vdots                    &\vdots                      }\\
 n &\uncover<14->{q_n    }&\uncover<13->{{\color{red}c_{n-2}-q_{n-1}\cdot c_{n-1}}&{\color{red}d_{n-2}-q_{n-1}\cdot d_{n-1}}}\\
n+1&       &\uncover<15->{c_{n-1}-q_{n}  \cdot c_{n}  &d_{n-1}-q_{n}  \cdot d_{n}  }\\
\hline
\end{tabular}
\uncover<16->{
\Rightarrow
\left\{
\begin{aligned}
\rlap{${\color{red}c_{n}}$}\phantom{c_{n+1}} a + \rlap{${\color{red}d_n}$}\phantom{d_{n+1}}b &= \operatorname{ggT}(a,b)
\\
c_{n+1} a + d_{n+1} b &= 0
\end{aligned}
\right.}
\end{equation*}}
\end{frame}