Merge pull request #1 from AndreasFMueller/master

update

1 files changed, 108 insertions, 0 deletions

diff --git a/vorlesungen/slides/4/euklidmatrix.tex b/vorlesungen/slides/4/euklidmatrix.tex
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+++ b/vorlesungen/slides/4/euklidmatrix.tex
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+%
+% euklidmatrix.tex
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostscheizer Fachhochschule
+%
+\begin{frame}[t]
+\frametitle{Matrixform des euklidischen Algorithmus}
+\setlength{\abovedisplayskip}{5pt}
+\setlength{\belowdisplayskip}{5pt}
+\vspace{-20pt}
+\begin{columns}[t,onlytextwidth]
+\begin{column}{0.52\textwidth}
+\begin{block}{Einzelschritt}
+\vspace{-10pt}
+\[
+a_k = b_kq_k + r_k
+\uncover<2->{
+\;\Rightarrow\;
+\left\{
+\begin{aligned}
+a_{k+1} &= b_k = \phantom{a_k-q_k}\llap{$-\mathstrut$}b_k \\
+b_{k+1} &= \phantom{b_k}\llap{$r_k$} = a_k - q_kb_k 
+\end{aligned}
+\right.}
+\]
+\end{block}
+\end{column}
+\begin{column}{0.44\textwidth}
+\uncover<3->{%
+\begin{block}{Matrixschreibweise}
+\vspace{-10pt}
+\begin{align*}
+\begin{pmatrix}
+a_{k+1}\\
+b_{k+1}
+\end{pmatrix}
+&=
+\begin{pmatrix}
+b_k\\r_k
+\end{pmatrix}
+=
+\uncover<4->{
+\underbrace{\begin{pmatrix}
+\uncover<5->{0&1}\\
+\uncover<6->{1&-q_k}
+\end{pmatrix}}_{\uncover<7->{\displaystyle =Q(q_k)}}
+}
+\begin{pmatrix}
+a_k\\b_k
+\end{pmatrix}
+\end{align*}
+\end{block}}
+\end{column}
+\end{columns}
+\vspace{-10pt}
+\uncover<8->{%
+\begin{block}{Ende des Algorithmus}
+\vspace{-10pt}
+\begin{align*}
+\uncover<9->{
+\begin{pmatrix}
+a_{n+1}\\
+b_{n+1}\\
+\end{pmatrix}
+&=}
+\begin{pmatrix}
+r_{n-1}\\
+r_{n}
+\end{pmatrix}
+=
+\begin{pmatrix}
+\operatorname{ggT}(a,b) \\
+0
+\end{pmatrix}
+\uncover<11->{
+=
+\underbrace{\uncover<15->{Q(q_n)}
+\uncover<14->{\dots}
+\uncover<13->{Q(q_1)}
+\uncover<12->{Q(q_0)}}_{\displaystyle =Q}}
+\uncover<10->{
+\begin{pmatrix} a_0\\ b_0\end{pmatrix}
+\uncover<6->{
+=
+Q\begin{pmatrix}a\\b\end{pmatrix}
+}
+}
+\end{align*}
+\end{block}}
+\uncover<16->{%
+\begin{block}{Konsequenzen}
+\[
+Q=\begin{pmatrix}
+q_{11}&q_{12}\\
+q_{21}&q_{22}
+\end{pmatrix}
+\quad\Rightarrow\quad
+\left\{
+\quad
+\begin{aligned}
+\operatorname{ggT}(a,b) &= q_{11}a + q_{12}b = {\color{red}s}a+{\color{red}t}b\\
+                      0 &= q_{21}a + q_{22}b
+\end{aligned}
+\right.
+\]
+\end{block}}
+
+\end{frame}