From ff4ef37bd6529f26066ca8df8dd9b98fed71577a Mon Sep 17 00:00:00 2001
From: Marc Benz <marc.benz@ost.ch>
Date: Thu, 12 May 2022 15:46:25 +0200
Subject: first changes

---
 buch/papers/transfer/teil2.tex | 38 ++++++++------------------------------
 1 file changed, 8 insertions(+), 30 deletions(-)

(limited to 'buch/papers/transfer/teil2.tex')

diff --git a/buch/papers/transfer/teil2.tex b/buch/papers/transfer/teil2.tex
index ce8f798..d79d80c 100644
--- a/buch/papers/transfer/teil2.tex
+++ b/buch/papers/transfer/teil2.tex
@@ -3,38 +3,16 @@
 %
 % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
-\section{Teil 2 
+\section{MiniMax-Polinom 
 \label{transfer:section:teil2}}
-\rhead{Teil 2}
-Sed ut perspiciatis unde omnis iste natus error sit voluptatem
-accusantium doloremque laudantium, totam rem aperiam, eaque ipsa
-quae ab illo inventore veritatis et quasi architecto beatae vitae
-dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit
-aspernatur aut odit aut fugit, sed quia consequuntur magni dolores
-eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam
-est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci
-velit, sed quia non numquam eius modi tempora incidunt ut labore
-et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima
-veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam,
-nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure
-reprehenderit qui in ea voluptate velit esse quam nihil molestiae
-consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla
-pariatur?
+\rhead{MiniMax-Polinom}
 
-\subsection{De finibus bonorum et malorum
+
+
+\subsection{Problemstellung
 \label{transfer:subsection:bonorum}}
-At vero eos et accusamus et iusto odio dignissimos ducimus qui
-blanditiis praesentium voluptatum deleniti atque corrupti quos
-dolores et quas molestias excepturi sint occaecati cupiditate non
-provident, similique sunt in culpa qui officia deserunt mollitia
-animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis
-est et expedita distinctio. Nam libero tempore, cum soluta nobis
-est eligendi optio cumque nihil impedit quo minus id quod maxime
-placeat facere possimus, omnis voluptas assumenda est, omnis dolor
-repellendus. Temporibus autem quibusdam et aut officiis debitis aut
-rerum necessitatibus saepe eveniet ut et voluptates repudiandae
-sint et molestiae non recusandae. Itaque earum rerum hic tenetur a
-sapiente delectus, ut aut reiciendis voluptatibus maiores alias
-consequatur aut perferendis doloribus asperiores repellat.
 
+\[
+\max _{a \leq x \leq b}|\operatorname{TanH}(x)-P(x)|
+\]
 
-- 
cgit v1.2.1


From d17d215c1fbfa384dd8edcc772df865f5819c1ab Mon Sep 17 00:00:00 2001
From: Marc Benz <marc.benz@ost.ch>
Date: Thu, 18 Aug 2022 20:07:48 +0200
Subject: first commit (without teil4)

---
 buch/papers/transfer/teil2.tex | 66 +++++++++++++++++++++++++++++++++++++-----
 1 file changed, 58 insertions(+), 8 deletions(-)

(limited to 'buch/papers/transfer/teil2.tex')

diff --git a/buch/papers/transfer/teil2.tex b/buch/papers/transfer/teil2.tex
index d79d80c..aae81a7 100644
--- a/buch/papers/transfer/teil2.tex
+++ b/buch/papers/transfer/teil2.tex
@@ -1,18 +1,68 @@
 %
-% teil2.tex -- Beispiel-File für teil2 
+% teil1.tex -- Beispiel-File für das Paper
 %
 % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
 %
-\section{MiniMax-Polinom 
+\section{Padé-Approximation
 \label{transfer:section:teil2}}
-\rhead{MiniMax-Polinom}
+\rhead{}
 
+\subsection{Idee
+	\label{transfer:pade:idee}}
+Die Taylorapproximation ist für den Gebrauch als Ersatz des Tangenshyperbolicus als Transferfunktion nicht brauchbar. Die Padé-Approximation kann die grössten Probleme aber entschärfen und dies mit sehr begrenztem zusätzlichen Rechenaufwand. Dafür wird die Taylorapproximation in einen Bruch von zwei Polynom zerlegt.
 
+\subsection{Definition
+\label{transfer:pade:definition}}
+Sei
+\begin{equation}
+	R(x)=\frac{\sum_{j=0}^{m} a_{j} x^{j}}{1+\sum_{k=1}^{n} b_{k} x^{k}}=\frac{a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{m} x^{m}}{1+b_{1} x+b_{2} x^{2}+\cdots+b_{n} x^{n}} 
+\end{equation}
+und gilt
+\begin{gather*}
+	f(0) =R(0) \\
+	f^{\prime}(0) =R^{\prime}(0) \\
+	f^{\prime \prime}(0) =R^{\prime \prime}(0) \\
+	\vdots \\
+	f^{(m+n)}(0) =R^{(m+n)}(0),
+\end{gather*}
+so ist $R(x)$ die Padé-Approximation von $f(x)$.
+\subsection{Beispiel
+	\label{transfer:pade:beispiel}}
+Sei $f(x) = \tanh (x)$ und $T_{5} \tanh(x ; a) = x-\frac{x^{3}}{3}+\frac{2 x^{5}}{15}$, dann gilt
+$$
+	\begin{gathered}  
+	[3 / 2]_{f}(x) = \frac{A_{0}+A_{1} x+A_{2} x^{2}+A_{3} x^{3}}{B_{0}+B_{1} x+B_{2} x^{2}}=x-\frac{x^{3}}{3}+\frac{2 x^{5}}{15}+O\left(x^{6}\right), B_{0} = 1,\\
+	\Downarrow \\
+	[3 / 2]_{f}(x) = \frac{15x+x^3}{15+6x^2}
+\end{gathered}
+$$
+
+\begin{figure}
+\centering
+\begin{tikzpicture}
+	\begin{axis}[
+		xmin=-3.5, xmax=3.5,
+		ymin=-1.5, ymax=1.5,
+		axis lines=center,
+		axis on top=true,
+		domain=-3.5:3.5,
+		ylabel=$y$,
+		xlabel=$x$,
+		]
+		
+		\addplot [mark=none,draw=red,thick] {tanh(\x)};
+		\node [right, red] at (axis cs: 1.4,0.7) {$\tanh(x)$};
+		\addplot [mark=none,draw=blue,ultra thick, samples=100, smooth] expression{x*(15+x^2)/(15+6*x^2)};
+		\node [right, blue] at (axis cs: -1.8,0.7) {$Padé$};
+		
+		%% Add the asymptotes
+		\draw [blue, dotted, thick] (axis cs:-2.5,-1)-- (axis cs:0,-1);
+		\draw [blue, dotted, thick] (axis cs:+2.5,+1)-- (axis cs:0,+1);
+	\end{axis}
+\end{tikzpicture}
+\caption{$[3 / 2]_{f}(x)$
+\label{motivation:figure:Pade32}}
+\end{figure}
 
-\subsection{Problemstellung
-\label{transfer:subsection:bonorum}}
 
-\[
-\max _{a \leq x \leq b}|\operatorname{TanH}(x)-P(x)|
-\]
 
-- 
cgit v1.2.1

