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diff --git a/buch/papers/clifford/0_ElevatorPitch.tex b/buch/papers/clifford/0_ElevatorPitch.tex index 0db5617..ad9bcc2 100644 --- a/buch/papers/clifford/0_ElevatorPitch.tex +++ b/buch/papers/clifford/0_ElevatorPitch.tex @@ -1,2 +1,6 @@ -TODO... -GA [Geometric Algebra i.a.W. Clifford Algebra] provides a unified language for the whole of physics and for much of mathematics and its applications that is conceptually and computationally superior to alternative mathematical systems in many application domains.
\ No newline at end of file + +Der Nutzen, welche die Clifford Algebra hat, lässt sich am besten mit den Worten des modernen Begründers dieser erläutern. + +"GA [Geometric Algebra i.a.W. Clifford Algebra] provides a unified language for the whole of physics and for much of mathematics and its applications that is conceptually and computationally superior to alternative mathematical systems in many application domains." \cite{clifford:hestenes_GA} + +Im folgenden hoffen wir den Leser von der Nützlichkeit und der geometrischen Schönheit der Clifford Algebra zu überzeugen.
\ No newline at end of file diff --git a/buch/papers/clifford/1_Vektordarstellung.tex b/buch/papers/clifford/1_Vektordarstellung.tex index 88a5789..ac00a33 100644 --- a/buch/papers/clifford/1_Vektordarstellung.tex +++ b/buch/papers/clifford/1_Vektordarstellung.tex @@ -1,7 +1,7 @@ \section{Vektoroperationen\label{clifford:section:Vektoroperationen}} \rhead{Vektoroperationen} \subsection{Vektordarstellung\label{clifford:section:Vektordarstellung}} -Vektoren können neben der üblichen Darstellung, auch als Linearkombination aus Basisvektoren dargestellt werden +Vektoren können neben der üblichen Spaltendarstellung, auch als Linearkombination aus Basisvektoren \begin{equation} \begin{split} \textbf{a} @@ -31,12 +31,14 @@ Vektoren können neben der üblichen Darstellung, auch als Linearkombination aus \sum_{i=1}^{n} a_i \textbf{e}_i \qquad a_i \in \mathbb{R} - , \textbf{e}_i \in \mathbb{R}^n. + , \textbf{e}_i \in \mathbb{R}^n \end{split} \end{equation} -Diese Basisvektoren sollen orthonormal sein und um die Darstellung zu vereinfachen werden sie durch $\textbf{e}_1 , \textbf{e}_2, ...$ ersetzt. +dargestellt werden. +Diese Basisvektoren werden so gewählt, dass sie orthonormal sind. +Um die Darstellung zu vereinfachen werden sie durch $\textbf{e}_1 , \textbf{e}_2, \dots$ ersetzt. \begin{beispiel} -Linearkombination von Basisvektoren in $\mathbb{R}^4$ +Eine Linearkombination von Basisvektoren in $\mathbb{R}^4$ könnte wie folgt aussehen \begin{equation} \begin{pmatrix} 42 \\ 2 \\ 1291 \\ 4 @@ -65,7 +67,7 @@ Linearkombination von Basisvektoren in $\mathbb{R}^4$ + 1291\textbf{e}_3 + - 4\textbf{e}_4 + 4\textbf{e}_4. \end{equation} +Dieses Beispiel ist für einen vier dimensionalen Vektor, dies kann selbstverständlich für beliebig viele Dimensionen nach demselben Schema erweitert werden. \end{beispiel} -Wobei Beispiel für einen vier dimensionalen Vektor ist, dies kann selbstverständlich für beliebig viele Dimensionen nach demselben Schema erweitert werden.
\ No newline at end of file diff --git a/buch/papers/clifford/2_QuadratVektoren.tex b/buch/papers/clifford/2_QuadratVektoren.tex index cfb05d6..6c6fb7d 100644 --- a/buch/papers/clifford/2_QuadratVektoren.tex +++ b/buch/papers/clifford/2_QuadratVektoren.tex @@ -1,54 +1,71 @@ \subsection{Quadrat von Vektoren} -Was eine Addition von Vektoren bedeutet ist sehr intuitiv und auch leicht geometrisch darzustellen, was allerdings das Produkt von Vektoren ergibt mag anfänglich unintuitiv wirken. +\subsubsection{Ziel der Multiplikation} +Was eine Addition von Vektoren bedeutet ist sehr intuitiv und auch leicht geometrisch darzustellen wie in Abbildung \ref{figure:addition}, was allerdings das Produkt von Vektoren ergibt mag anfänglich unintuitiv wirken. +\begin{figure}[htb] + \centering + \begin{tikzpicture} + \draw[thin,gray!40] (0,0) grid (4,4); + \draw[blue,thick,->] (0,0)--(3.5,2) node[midway,above,sloped] {$\textbf{a}$}; + \draw[red,thick,->] (3.5,2)--(1.5,3.8) node[midway,above,sloped] {$\textbf{b}$}; + \draw[black,thick,->] (0,0)--(1.5,3.8)node[midway,above,sloped] {$\textbf{a} +\textbf{b} = \textbf{c} $}; + \end{tikzpicture} + \caption{Addition von zwei Vektoren\label{figure:addition}} +\end{figure} Was soll es schon heissen zwei Vektoren miteinander zu multiplizieren? -\newline Im Folgenden werden wir versuchen diese Operation ähnlich intuitiv darzustellen. -\newline -Um sinnvoll eine neue Operation zwischen zwei Elementen einer Algebra, in diesem Fall Vektoren, zu definieren, muss man überlegen, was das Ziel dieser Operation ist. -Als grundsätzliches Ziel wird definiert, dass das Quadrat eines Vektor dessen Länge im Quadrat ergibt, da dies auch in vielen anderen Bereichen der Mathematik,zum Beispiel bei komplexen Zahlen, auch so definiert ist. -\newline -Zusätzlich wollen wir auch das Assoziativgesetz und das Kommutativgesetz für Skalare beibehalten. Wobei das Kommutativgesetz leider, oder wie man sehen wird zum Glück, in der geometrischen Algebra im generellen nicht mehr gilt. Das heisst wir dürfen ausklammern \ref{eq:assoziativ} und die Position von Skalaren im Produkt ändern \ref{eq:kommSkalar}, allerdings nicht die Position der Vektoren \ref{eq:kommVector}. + +Um sinnvoll eine neue Operation zwischen zwei Elementen einer Algebra, in diesem Fall sind diese Elemente Vektoren, zu definieren, muss man überlegen, was das Ziel dieser Operation sein soll. + +Als grundsätzliches Ziel wird definiert, dass das Quadrat eines Vektor dessen Länge im Quadrat ergibt, da dies auch in vielen anderen Bereichen der Mathematik,zum Beispiel bei komplexen Zahlen,so definiert ist. + +Zusätzlich soll auch das Assoziativgesetz für die Multiplikation von Vektoren gelten, dass heisst wir dürfen ausklammern \begin{equation} \label{eq:assoziativ} \textbf{e}_i(\textbf{e}_j + \textbf{e}_k) = - \textbf{e}_i\textbf{e}_j + \textbf{e}_i\textbf{e}_k + \textbf{e}_i\textbf{e}_j + \textbf{e}_i\textbf{e}_k. \end{equation} +Allerdings gilt das Kommutativgesetz leider, oder wie man sehen wird zum Glück, nur für skalare Elemente \begin{equation} \label{eq:kommSkalar} a\textbf{e}_ib\textbf{e}_j = - ab\textbf{e}_i\textbf{e}_j + ab\textbf{e}_i\textbf{e}_j \qquad a,b \in \mathbb{R} \end{equation} +und nicht für Vektoren \begin{equation} \label{eq:kommVector} \textbf{e}_i\textbf{e}_j \neq - \textbf{e}_j\textbf{e}_i + \textbf{e}_j\textbf{e}_i. +\end{equation} +\subsubsection{Quadrieren eines Vektors} +Betrachten wir nun mit diesen Regeln das Quadrat eines Vektors. Zuerst werden die Vektoren als Linearkombinationen geschrieben +\begin{equation} + \textbf{a}^2 = + \left ( + \sum_{i=1}^{n} a_i \textbf{e}_i + \right ) + \left ( + \sum_{i=1}^{n} a_i \textbf{e}_i + \right ) + \label{eq:quad_a_1}. +\end{equation} +Das Quadrat kann nun in zwei Summen aufgeteilt werden +\begin{equation} + \textbf{a}^2 = + \textcolor{red}{\sum_{i=1}^{n} a_i^2\textbf{e}_i^2} + + + \textcolor{blue}{\sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n a_ia_j\textbf{e}_i\textbf{e}_j } + \label{eq:quad_a_2}, +\end{equation} +wobei die roten Summe die quadrierten Terme und die blaue Summe die Mischterme beinhaltet. Da $\textbf{e}_i^2 = 1$ gilt, weil das zuvor definierte Ziel des Quadrates eines Vektors dessen Länge ergibt und die Basisvektoren Länge 1 haben, wird dies nun eingesetzt +\begin{equation} + \textbf{a}^2 = \textcolor{cyan}{\sum_{i=1}^{n} a_i^2} + \textcolor{orange}{\sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n a_ia_j\textbf{e}_i\textbf{e}_j}. + \label{eq:quad_a_3} \end{equation} -Betrachten wir nun mit diesen Regeln das Quadrat eines Vektors. -\begin{align} - \textbf{a}^2 &= - \left ( - \sum_{i=1}^{n} a_i \textbf{e}_i - \right ) - \left ( - \sum_{i=1}^{n} a_i \textbf{e}_i - \right ) - \label{eq:quad_a_1} - \\ - &= - \textcolor{red}{\sum_{i=1}^{n} a_i^2\textbf{e}_i^2} - + - \textcolor{blue}{\sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n a_ia_j\textbf{e}_i\textbf{e}_j } - \label{eq:quad_a_2} - \\ - &= \textcolor{cyan}{\sum_{i=1}^{n} a_i^2} + \textcolor{orange}{\sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n a_ia_j\textbf{e}_i\textbf{e}_j}. - \label{eq:quad_a_3} -\end{align} - \begin{beispiel} -Quadrat eines Vektors in $\mathbb{R}^2$ +Das Quadrat des Vektor $a$ in $\mathbb{R}^2$ ist \begin{equation} \begin{split} \textbf{a}^2 @@ -56,22 +73,17 @@ Quadrat eines Vektors in $\mathbb{R}^2$ &= \textcolor{red}{a_1^2\textbf{e}_1^2 + a_2^2\textbf{e}_2^2} + \textcolor{blue}{a_1\textbf{e}_1a_2\textbf{e}_2 + a_2\textbf{e}_2a_1\textbf{e}_2} \\\ & = \textcolor{cyan}{a_1^2 + a_2^2} + \textcolor{orange}{a_1b\textbf{e}_1a_2\textbf{e}_2 + a_2\textbf{e}_2a_1\textbf{e}_2} - \end{split} + \end{split}. \end{equation} - \end{beispiel} -Der Vektor wird in \ref{eq:quad_a_1} als Linearkombination geschrieben. -Das Quadrat kann, wie in \ref{eq:quad_a_2} gezeigt, in zwei Summen aufteilen werden , wobei die roten Summe die quadrierten Terme und die blaue Summe die Mischterme beinhaltet. -\newline -Da $\textbf{e}_i^2 = 1$ gilt, da zuvor vorausgesetzt wurde, dass man mit orthonormalen Einheitsvektoren arbeitet, wird dies nun eingesetzt ergibt sich \ref{eq:quad_a_3} -\newline -Die hellblaue Teil ist nun bereits Länge im Quadrat eines Vektors, also das Ziel der Multiplikation. -Daher muss der restliche Teil dieser Gleichung null ergeben. -Aus dieser Erkenntnis leiten wir in \ref{eq:Mischterme_Null} weitere Eigenschaften für die Multiplikation her. + +Die hellblaue Teil ist nun bereits die Länge im Quadrat, also das zuvor definierte Ziel der Multiplikation. +Daraus lässt sich schliessen, dass der restliche Teil dieser Gleichung null ergeben muss \begin{equation} \label{eq:Mischterme_Null} - \sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n a_ia_j\textbf{e}_i\textbf{e}_j = \textcolor{blue}{a_1a_2(\textbf{e}_1\textbf{e}_2 + \textbf{e}_2\textbf{e}_1)} + a_1a_3(\textbf{e}_1\textbf{e}_3 + \textbf{e}_3\textbf{e}_1) + \dots = 0 + \sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n a_ia_j\textbf{e}_i\textbf{e}_j = \textcolor{blue}{a_1a_2(\textbf{e}_1\textbf{e}_2 + \textbf{e}_2\textbf{e}_1)} + a_1a_3(\textbf{e}_1\textbf{e}_3 + \textbf{e}_3\textbf{e}_1) + \dots = 0. \end{equation} +Aus dieser Erkenntnis können weitere Eigenschaften für die Multiplikation hergeleitet werden. Da dies für beliebige $a_i$ gelten muss werden alle Terme bis auf $a_1$ und $a_2$ gleich null gesetzt. Somit fallen alle Terme bis auf den blauen weg. Wird dies weiter vereinfacht ergibt sich \begin{equation} \begin{split} @@ -81,15 +93,13 @@ Da dies für beliebige $a_i$ gelten muss werden alle Terme bis auf $a_1$ und $a_ \end{split} \end{equation} \begin{satz} - Die Multiplikation von Vektoren ist antikommutativ, wenn die multiplizierten Vektoren orthogonal sind. + Die Multiplikation von Vektoren ist antikommutativ, wenn die multiplizierten Vektoren orthogonal sind, es gilt also \begin{equation} - \textbf{e}_i\textbf{e}_j = -\textbf{e}_j\textbf{e}_i \qquad \textbf{e}_i \perp \textbf{e}_j + \textbf{e}_i\textbf{e}_j = -\textbf{e}_j\textbf{e}_i \quad \textrm{für} \quad \textbf{e}_i \perp \textbf{e}_j. \end{equation} \end{satz} -Dieses Wissen reicht nun bereits um alle Produkte der Basisvektoren zu berechnen, was in \ref{tab:multip_vec} gemacht wurde. +Dieses Wissen reicht nun bereits um alle Produkte der Basisvektoren zu berechnen, was in Tabelle \ref{tab:multip_vec} gemacht wurde. \begin{table} -\caption{Multiplikationstabelle für Vektoren} -\label{tab:multip_vec} \begin{center} \begin{tabular}{ |c|c|c|c|c|c| } \hline @@ -107,4 +117,6 @@ Dieses Wissen reicht nun bereits um alle Produkte der Basisvektoren zu berechnen \hline \end{tabular} \end{center} +\caption{Multiplikationstabelle für Vektoren} +\label{tab:multip_vec} \end{table}
\ No newline at end of file diff --git a/buch/papers/clifford/3_MultiplikationVektoren.tex b/buch/papers/clifford/3_MultiplikationVektoren.tex index 841dde4..0969b89 100644 --- a/buch/papers/clifford/3_MultiplikationVektoren.tex +++ b/buch/papers/clifford/3_MultiplikationVektoren.tex @@ -1,11 +1,14 @@ \subsection{Multiplikation von Vektoren} -Was geschieht nun wenn zwei beliebige Vektoren,$u$ und $v$, miteinander multipliziert werden? +Was geschieht nun wenn zwei beliebige Vektoren, $u$ und $v$ \begin{equation} \textbf{u} = \sum_{i=1}^{n} u_i \textbf{e}_i \qquad \textbf{v} = \sum_{i=1}^{n} v_i \textbf{e}_i \end{equation} + miteinander multipliziert werden? + + Wieder werden die Vektoren zuerst als Linearkombinationen darstellen und danach in zwei Summen aufgeteilt, eine Summe mit quadrierten Termen und eine Summe mit Mischtermen \begin{equation} \begin{split} \textbf{u}\textbf{v} @@ -18,12 +21,12 @@ Was geschieht nun wenn zwei beliebige Vektoren,$u$ und $v$, miteinander multipli \right) = \sum_{i=1}^n u_iv_i\underbrace{\textbf{e}_i^2}_{1} - + \sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j + + \sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j, \end{split} \end{equation} +wobei die Summe der quadrierten Termen bereits bekannt vorkommen könnte, es ist nämlich das Skalarprodukt von $u$ und $v$. Die Summe der Mischterme bilden etwas neues, dass wir das äussere Produkt von $u$ und $v$ nennen. \begin{beispiel} Multiplikation von Vektoren in $\mathbb{R}^2$ -\end{beispiel} \begin{equation} \begin{split} \textbf{u}\textbf{v} @@ -44,7 +47,7 @@ Was geschieht nun wenn zwei beliebige Vektoren,$u$ und $v$, miteinander multipli \underbrace{(u_1v_2 - u_2v_1)\textbf{e}_1\textbf{e}_2}_{\text{Äusseres Produkt}} \end{split} \end{equation} -Der linke Teil dieser Multiplikation ergibt das Skalarprodukt der zwei Vektoren, der rechte Term ergibt etwas neues das sich das äussere Produkt der zwei Vektoren nennt. +\end{beispiel} \subsubsection{Äusseres Produkt} Das äussere Produkt von zwei Vektoren wird mit einem $\wedge$ dargestellt \begin{equation} @@ -53,123 +56,118 @@ Das äussere Produkt von zwei Vektoren wird mit einem $\wedge$ dargestellt \sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j \end{equation} \begin{beispiel} -Äusseres Produkt von zwei Vektoren in $\mathbb{R}^3$ -\end{beispiel} +Das äusseres Produkt von zwei Vektoren in $\mathbb{R}^3$ ist \begin{equation} - \begin{split} - u \wedge v - &= - u_1v_2\textbf{e}_1\textbf{e}_2 - + - u_1v_3\textbf{e}_1\textbf{e}_3 - + - u_2v_2\textbf{e}_2\textbf{e}_3 - + - u_2v_1\textbf{e}_2\textbf{e}_1 - + - u_3v_1\textbf{e}_3\textbf{e}_1 - + - u_3v_2\textbf{e}_3\textbf{e}_2 \\\ - &= - (u_1v_2 - u_2v_1)\textbf{e}_1\textbf{e}_2 - + - (u_1v_3 - v_3u_1)\textbf{e}_1\textbf{e}_3 - + - (u_2v_3 - u_3v_2)\textbf{e}_2\textbf{e}_3 - \end{split} + \begin{split} + u \wedge v + &= + u_1v_2\textbf{e}_1\textbf{e}_2 + + + u_1v_3\textbf{e}_1\textbf{e}_3 + + + u_2v_2\textbf{e}_2\textbf{e}_3 + + + u_2v_1\textbf{e}_2\textbf{e}_1 + + + u_3v_1\textbf{e}_3\textbf{e}_1 + + + u_3v_2\textbf{e}_3\textbf{e}_2 \\\ + &= + (u_1v_2 - u_2v_1)\textbf{e}_1\textbf{e}_2 + + + (u_1v_3 - v_3u_1)\textbf{e}_1\textbf{e}_3 + + + (u_2v_3 - u_3v_2)\textbf{e}_2\textbf{e}_3. + \end{split} \end{equation} -Im letzten Schritt des Beispiels wurden nun, mit Hilfe der antikommutativität des Produkts, die Vektorprodukte, welche die gleichen Einheitsvektoren beinhalten, zusammengefasst. Dieses Vorgehen kann man auch allgemein anwenden, wie in den Gleichungen \ref{eq:u_wedge_v}-\ref{eq:u_wedge_v_5} hergeleitet. +\end{beispiel} + +Im letzten Schritt des Beispiels wurden nun, mit Hilfe der antikommutativität des Produkts, die Vektorprodukte, welche die gleichen Einheitsvektoren beinhalten, zusammengefasst. Dieses Vorgehen kann man auch allgemein anwenden, wie in den Gleichungen \eqref{eq:u_wedge_v}-\eqref{eq:u_wedge_v_5} hergeleitet. Die Summe, \begin{align} \textbf{u}\wedge \textbf{v} &= \sum_{\begin{subarray}{l}i,j=1\\i \neq j\end{subarray}}^n - u_iv_j\textbf{e}_i\textbf{e}_j + u_iv_j\textbf{e}_i\textbf{e}_j, \label{eq:u_wedge_v} - \\ + \intertext{wird in zwei verschiedene Summen aufgeteilt. + Wobei die linke Summe jeweils den Basisvektor mit dem höheren Index an erster Stelle und die rechte Summe diesen jeweils an zweiter Stelle hat} \label{eq:u_wedge_v_1} &= \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j + - \sum_{\begin{subarray}{l}i,j=1\\j < i\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j - \\ + \sum_{\begin{subarray}{l}i,j=1\\j < i\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j. + \intertext{Nun werden die Indexe der zweiten Summe vertauscht} \label{eq:u_wedge_v_2} &= \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j + - \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_jv_i\textbf{e}_j\textbf{e}_i - \\ - \label{eq:u_wedge_v_3} + \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_jv_i\textbf{e}_j\textbf{e}_i, + \intertext{und diese wird nun mit Hilfe der Antikommutativität umgeformt zu} &= \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_iv_j\textbf{e}_i\textbf{e}_j - - \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_jv_i\textbf{e}_i\textbf{e}_j - \\ + \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n u_jv_i\textbf{e}_i\textbf{e}_j. + \intertext{Nun können die zwei Summen wieder zusammengefasst werden} \label{eq:u_wedge_v_4} &= - \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n (u_iv_j -u_jv_i)\textbf{e}_i\textbf{e}_j - \\ - \label{eq:u_wedge_v_5} + \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n (u_iv_j -u_jv_i)\textbf{e}_i\textbf{e}_j. + \intertext{Der Term in der Summe könnte einem bereits bekannt vorkommen, es ist nämlich die Determinante einer Matrix mit $u$ und $v$ als ihre Spalten} &= + \label{eq:u_wedge_v_5} \sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n \begin{vmatrix} u_i & v_i \\ u_j & v_j - \end{vmatrix}\textbf{e}_i\textbf{e}_j + \end{vmatrix}\textbf{e}_i\textbf{e}_j. \end{align} -Die Summe aus \ref{eq:u_wedge_v_1} wird in \ref{eq:u_wedge_v} in zwei verschiedene Summen aufgeteilt. -Wobei die linke Summe jeweils den Basisvektor mit dem höheren Index an erster Stelle und die rechte Summe diesen jeweils an zweiter Stelle hat. -\newline -Bei \ref{eq:u_wedge_v_2} werden die Indexe der zweiten Summe vertauscht, damit man nun bei beiden Teilen die gleiche Summe hat. -Danach werden in \ref{eq:u_wedge_v_3}, mit Hilfe der Antikommutativität, die Einheitsvektoren der zweiten Summe vertauscht. -\newline -Nun können die Summen, wie in \ref{eq:u_wedge_v_4} wieder in eine Summe zusammengefasst werden. -\newline -Der Term in der Klammer in \ref{eq:u_wedge_v_4} kann auch als Determinante einer 2x2 Matrix dargestellt werden, was in \ref{eq:u_wedge_v_5} gemacht wird. -\newline -Die Determinante einer Matrix beschreibt welche von den Spaltenvektoren aufgespannt wird, wie in Abbildung \ref{figure:det} dargestellt. -\begin{figure} -\centering -\begin{tikzpicture} - \draw[thin,gray!40] (0,0) grid (4,4); - \draw[<->] (0,0)--(4,0) ; - \draw[<->] (0,0)--(0,4) ; - \draw[line width=0,fill=gray!40] (0,0)--(3,1)--(4,3)--(1,2); - \draw[line width=2pt,blue,-stealth](0,0)--(3,1) node[anchor=north - west]{$\boldsymbol{u}$}; - \draw[line width=2pt,red,-stealth](0,0)--(1,2) node[anchor=south east]{$\boldsymbol{v}$}; - \draw[black] (2,1.5)--(-0.5,2.5) node[anchor = east]{$\begin{vmatrix} - u_i & v_i \\ - u_j & v_j - \end{vmatrix} = u_iv_j - v_iu_j$}; -\end{tikzpicture} -\caption{Geometrische Interpretation der Determinante einer 2x2 Matrix\label{figure:det}} +Die Determinante einer Matrix beschreibt die Fläche, welche von den Spaltenvektoren aufgespannt wird, wie in Abbildung \ref{figure:det} dargestellt. +\begin{figure}[htb] + \centering + \begin{minipage}[t]{.45\linewidth} + \centering + \begin{tikzpicture} + \draw[thin,gray!40] (0,0) grid (4,4); + \draw[<->] (0,0)--(4,0) ; + \draw[<->] (0,0)--(0,4) ; + \draw[line width=0,fill=gray!40] (0,0)--(3,1)--(4,3)--(1,2); + \draw[line width=2pt,blue,-stealth](0,0)--(3,1) node[anchor=north + west]{$\boldsymbol{u}$}; + \draw[line width=2pt,red,-stealth](0,0)--(1,2) node[anchor=south east]{$\boldsymbol{v}$}; + \draw[black] (2,1.5)--(1.8,3.2) node[anchor = south]{$\begin{vmatrix} + u_i & v_i \\ + u_j & v_j + \end{vmatrix} = u_iv_j - v_iu_j$}; + \end{tikzpicture} + \caption{Geometrische Interpretation der Determinante einer $2 \times 2$ Matrix\label{figure:det}} + \end{minipage}% + \hfill% + \begin{minipage}[t]{.45\linewidth} + \centering + \begin{tikzpicture} + \draw[thin,gray!40] (0,0) grid (4,4); + \draw[<->] (0,0)--(4,0) node[right]{$x$}; + \draw[<->] (0,0)--(0,4) node[above]{$y$}; + \draw[line width=0,fill=gray!40] (0,0)--(3,1)--(4,3)--(1,2); + \draw[line width=2pt,blue,-stealth](0,0)--(3,1) node[anchor=north + west]{$\boldsymbol{u}$}; + \draw[line width=2pt,red,-stealth](0,0)--(1,2) node[anchor=south east]{$\boldsymbol{v}$}; + \draw[->] (2.15,1.5) arc (0:310:0.3); + \draw[black] (2,1.5)--(2.5,3.2) node[anchor = south]{$u\wedge v = \begin{vmatrix} + u_i & v_i \\ + u_j & v_j + \end{vmatrix} e_1e_2 = (u_iv_j - v_iu_j)\textbf{e}_1\textbf{e}_2$}; + \end{tikzpicture} + \caption{Geometrische Interpretation des äusseren Produktes \label{figure:wedge}} + \end{minipage} \end{figure} -\newline Das äussere Produkt besteht nun also aus der Summe - $\sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n$ + \(\sum_{\begin{subarray}{l}i,j=1\\i < j\end{subarray}}^n\) von Flächen - $\begin{vmatrix} - u_i & v_i \\ - u_j & v_j - \end{vmatrix}$, welche in $\textbf{e}_i\textbf{e}_j$ aufgespannt sind, wie man in \ref{eq:u_wedge_v_5} sieht. + \(\begin{vmatrix} + u_i & v_i \\ + u_j & v_j + \end{vmatrix}\) +, welche in $\textbf{e}_i\textbf{e}_j$ aufgespannt sind, wie man in \ref{eq:u_wedge_v_5} sieht. Dieses Produkt $\textbf{e}_i\textbf{e}_j$ der Basisvektoren interpretiert man als Umlaufrichtung. Wobei die gebildete Fläche in Richtung des ersten Vektors umschritten wird. -Dies ist in \ref{figure:wedge} dargestellt, wobei bei diesem Beispiel die Umlaufrichtung im Gegenuhrzeigersinn ist, da die Fläche in Richtung u umschritten wird. +Dies ist in Abbildung \ref{figure:wedge} dargestellt, wobei bei diesem Beispiel die Umlaufrichtung im Gegenuhrzeigersinn ist, da die Fläche in Richtung u umschritten wird. Diese Fläche mit einer Richtung nennt man in der geometrischen Algebra einen Bivektor, da er eine Art zwei dimensionaler Vektor ist. -\begin{figure} -\centering -\begin{tikzpicture} - \draw[thin,gray!40] (0,0) grid (4,4); - \draw[<->] (0,0)--(4,0) node[right]{$x$}; - \draw[<->] (0,0)--(0,4) node[above]{$y$}; - \draw[line width=0,fill=gray!40] (0,0)--(3,1)--(4,3)--(1,2); - \draw[line width=2pt,blue,-stealth](0,0)--(3,1) node[anchor=north - west]{$\boldsymbol{u}$}; - \draw[line width=2pt,red,-stealth](0,0)--(1,2) node[anchor=south east]{$\boldsymbol{v}$}; - \draw[->] (2.15,1.5) arc (0:310:0.3); - \draw[black] (2,1.5)--(-0.5,2.5) node[anchor = east]{$u\wedge v = \begin{vmatrix} - u_i & v_i \\ - u_j & v_j - \end{vmatrix} e_1e_2 = (u_iv_j - v_iu_j)\textbf{e}_1\textbf{e}_2$}; -\end{tikzpicture} -\caption{Geometrische Interpretation des äusseren Produkt in $\mathbb{R}^2$\label{figure:wedge}} -\end{figure}
\ No newline at end of file diff --git a/buch/papers/clifford/4_GeometrischesProdukt.tex b/buch/papers/clifford/4_GeometrischesProdukt.tex index a19e983..f18b90d 100644 --- a/buch/papers/clifford/4_GeometrischesProdukt.tex +++ b/buch/papers/clifford/4_GeometrischesProdukt.tex @@ -12,9 +12,9 @@ Ein Multivektor besteht aus den verschiedenen Bauteilen, wie zum Beispiel Vektor M = \sum \left ( \prod a_i\textbf{e}_j \right) \end{equation} \end{definition} -Besteht eine Clifford Algebra aus n Basisvektoren so hat sie n Dimensionen, dies wird nicht wie in der linearen Algebra mit $\mathbb{R}^n$ sondern mit $\mathbb{G}^n$ beschrieben. +Besteht eine Clifford Algebra aus n Basisvektoren so hat sie n Dimensionen, dies wird nicht wie in der linearen Algebra mit $\mathbb{R}^n$ sondern mit $G_n(\mathbb{R})$ beschrieben. Dies wird so geschrieben da man eine neue Algebrastruktur um die Vektoren einführt. \begin{beispiel} -Allgemeiner Multivektor in $\mathbb{G}^3$ +Allgemeiner Multivektor in $G_3(\mathbb{R})$ \begin{equation} M = a + @@ -26,34 +26,30 @@ Allgemeiner Multivektor in $\mathbb{G}^3$ \end{equation} \end{beispiel} \begin{definition} -Um das Produkt von Basisvektoren in Zukunft darzustellen wird folgende Notation definiert +Für das Produkt von Basisvektoren wird folgende Notation definiert \begin{equation} - e_ie_j = e_{ij} + e_ie_j = e_{ij}. \end{equation} \end{definition} -Nun da das geometrische Produkt vollständig definiert wurde können Multiplikationstabellen für verschiedene Dimensionen $\mathbb{G}^n$ erstellt werden. In \ref{tab:multip} ist dies für $\mathbb{G}^3$ gemacht. +Nun da das geometrische Produkt vollständig definiert wurde können Multiplikationstabellen für verschiedene Dimensionen $G_n(\mathbb{R})$ erstellt werden. In Tabelle \ref{tab:multip} ist dies für $G_3(\mathbb{R})$ gemacht. \begin{table} - \caption{Multiplikationstabelle für $\mathbb{G^3}$} \label{tab:multip} \begin{center} - \begin{tabular}{ |c|c|c|c|c|c|c|c| } + \begin{tabular}{ |c|ccc|ccc|c| } \hline 1 & $\textbf{e}_1$ & $\textbf{e}_2$ &$\textbf{e}_3$ & $\textbf{e}_{12}$ & $\textbf{e}_{13}$ & $\textbf{e}_{23}$ & $\textbf{e}_{123}$\\ \hline $\textbf{e}_1$ & 1 & $\textbf{e}_{12}$ & $\textbf{e}_{12}$ & $\textbf{e}_2$ & $\textbf{e}_3$ & $\textbf{e}_{123}$ & $\textbf{e}_{23}$\\ - \hline $\textbf{e}_2$ & $-\textbf{e}_{12}$ & 1 & $\textbf{e}_{23}$ & $-\textbf{e}_1$ & $-\textbf{e}_{123}$ & $\textbf{e}_3$ & $-\textbf{e}_{13}$\\ - \hline $\textbf{e}_3$ & $-\textbf{e}_{13}$ & $-\textbf{e}_{23}$ & 1 & $\textbf{e}_{123}$ & $-\textbf{e}_1$ & $-\textbf{e}_2$ & $\textbf{e}_{12}$\\ \hline $\textbf{e}_{12}$ & -$\textbf{e}_2$ & $\textbf{e}_1$& $\textbf{e}_{123}$ & -1 & $-\textbf{e}_{23}$ & $\textbf{e}_{13}$ & $-\textbf{e}_{3}$\\ - \hline $\textbf{e}_{13}$ & $-\textbf{e}_{3}$ & $-\textbf{e}_{123}$ & $\textbf{e}_{1}$ & $\textbf{e}_{23}$ & -1 & $-\textbf{e}_{12}$ & $\textbf{e}_{2}$\\ - \hline $\textbf{e}_{23}$ & $\textbf{e}_{123}$ & $-\textbf{e}_{3}$ & $\textbf{e}_{2}$ & $-\textbf{e}_{13}$ & $\textbf{e}_{12}$ & -1 & $-\textbf{e}_{1}$ \\ \hline $\textbf{e}_{123}$ & $\textbf{e}_{23}$ & $-\textbf{e}_{13}$ & $\textbf{e}_{12}$ & $-\textbf{e}_{3}$& $\textbf{e}_{2}$ & $-\textbf{e}_{1}$ & -1 \\ \hline \end{tabular} \end{center} + \caption{Multiplikationstabelle für $G_3(\mathbb{R})$} \end{table} diff --git a/buch/papers/clifford/references.bib b/buch/papers/clifford/references.bib index ff829d6..9090005 100644 --- a/buch/papers/clifford/references.bib +++ b/buch/papers/clifford/references.bib @@ -4,32 +4,13 @@ % (c) 2020 Autor, Hochschule Rapperswil % -@online{clifford:bibtex, - title = {BibTeX}, - url = {https://de.wikipedia.org/wiki/BibTeX}, - date = {2020-02-06}, - year = {2020}, - month = {2}, - day = {6} -} - -@book{clifford:numerical-analysis, - title = {Numerical Analysis}, - author = {David Kincaid and Ward Cheney}, - publisher = {American Mathematical Society}, - year = {2002}, - isbn = {978-8-8218-4788-6}, - inseries = {Pure and applied undegraduate texts}, - volume = {2} -} - -@article{clifford:mendezmueller, - author = { Tabea Méndez and Andreas Müller }, - title = { Noncommutative harmonic analysis and image registration }, - journal = { Appl. Comput. Harmon. Anal.}, - year = 2019, - volume = 47, - pages = {607--627}, - url = {https://doi.org/10.1016/j.acha.2017.11.004} +@article{clifford:hestenes_GA, + author = { David Hestenes, Garret Eugene Sobczyk and James S. Marsh }, + title = { Clifford Algebra to Geometric Calculus. A Unified Language for Mathematics and Physics }, + journal = { American Journal of Physics }, + year = 1985, + volume = 53, + pages = {24}, + url = {https://www.researchgate.net/publication/258944244_Clifford_Algebra_to_Geometric_Calculus_A_Unified_Language_for_Mathematics_and_Physics} } diff --git a/buch/papers/multiplikation/Makefile b/buch/papers/multiplikation/Makefile index 8f04c2c..8f04c2c 100644..100755 --- a/buch/papers/multiplikation/Makefile +++ b/buch/papers/multiplikation/Makefile diff --git a/buch/papers/multiplikation/Makefile.inc b/buch/papers/multiplikation/Makefile.inc index b78d67e..074020f 100644..100755 --- a/buch/papers/multiplikation/Makefile.inc +++ b/buch/papers/multiplikation/Makefile.inc @@ -7,8 +7,7 @@ dependencies-multiplikation = \ papers/multiplikation/packages.tex \ papers/multiplikation/main.tex \ papers/multiplikation/references.bib \ - papers/multiplikation/teil0.tex \ - papers/multiplikation/teil1.tex \ - papers/multiplikation/teil2.tex \ - papers/multiplikation/teil3.tex + papers/multiplikation/einlteung.tex \ + papers/multiplikation/loesungsmethoden.tex \ + papers/multiplikation/problemstellung.tex diff --git a/buch/papers/multiplikation/code/Figure_1.png b/buch/papers/multiplikation/code/Figure_1.png Binary files differnew file mode 100755 index 0000000..9def15a --- /dev/null +++ b/buch/papers/multiplikation/code/Figure_1.png diff --git a/buch/papers/multiplikation/code/MM b/buch/papers/multiplikation/code/MM Binary files differnew file mode 100755 index 0000000..f07985f --- /dev/null +++ b/buch/papers/multiplikation/code/MM diff --git a/buch/papers/multiplikation/code/MM.c b/buch/papers/multiplikation/code/MM.c new file mode 100755 index 0000000..04c4dab --- /dev/null +++ b/buch/papers/multiplikation/code/MM.c @@ -0,0 +1,465 @@ +#include <stdio.h>
+#include <stdint.h>
+#include <stdlib.h>
+#include <time.h>
+#include <omp.h>
+#include "c_matrix.h"
+#include <gsl/gsl_cblas.h>
+#include <string.h>
+
+void MM(int *A, int *B, int *C, int n);
+void openMP_MM(int *A, int *B, int *C, int n);
+void winograd(int *A, int *B, int *C, int n);
+int winograd_inner(int *a, int *b, int n);
+void run_algo(void (*algo)(), char alog_name[], int print);
+void run_algo_cblas(int print);
+void MM_dc(int *A, int *B, int *C, int n);
+void strassen(int *A, int *B, int *C, int n);
+void printMatrix(int *C, int n);
+void printMatrix_double(double *C, int n);
+void split(int *in, int *out, int n, int col, int row);
+void join(int *in, int *out, int n, int col, int row);
+void add(int *A, int *B, int *C, int n);
+void sub(int *A, int *B, int *C, int n);
+void multiply(int *A, int *B, int *C, int n);
+
+int main() {
+ // omp_set_dynamic(0);
+ // omp_set_num_threads(4);
+// run_algo(openMP_MM, "openMP_MM",0);
+ run_algo(MM_dc, "MM_dc",0);
+ run_algo(strassen, "strassen",0);
+
+ run_algo(MM, "MM", 0);
+ // run_algo(winograd, "winograd", 0);
+ run_algo_cblas(0);
+
+ return 0;
+}
+
+void MM(int *A, int *B, int *C, int n) {
+ for (int i = 0; i < n; ++i) {
+ for (int j = 0; j < n; ++j) {
+ int sum = 0;
+ for (int k = 0; k < n; ++k) {
+ sum += (*((A + i * n) + k)) * (*((B + k * n) + j));
+ }
+ *((C + i * n) + j) = sum;
+ }
+ }
+}
+
+int winograd_inner(int *a, int *b, int n){
+ int ab = 0;
+ if(n%2==0)
+ {
+ int xi = 0;
+ int etha = 0;
+ for(int i = 0; i<n/2;++i)
+ {
+ xi += a[2*i]*a[2*i+1];
+ etha += b[2*i]*b[2*i+1];
+ ab += (a[2*i]+b[2*i+1])*(a[2*i+1]+b[2*i]);
+ }
+ ab = ab-etha-xi;
+ }
+ return ab;
+ }
+
+ void winograd(int *A, int *B, int *C, int n) {
+
+ int xi_array[n];
+ int etha_array[n];
+ int xi = 0;
+ int etha = 0;
+ int ab = 0;
+
+ for (int i = 0; i < n; ++i) {
+ xi = 0;
+ etha = 0;
+ for(int k = 0;k<n/2;++k)
+ {
+ xi += (*((A + i * n) + 2*k))*(*((A + i * n) + (2*k+1)));
+ etha += (*((B + 2*k * n) + i))*(*((B + (2*k+1) * n) + i));
+ }
+ xi_array[i] = xi;
+ etha_array[i] = etha;
+ }
+
+ for (int i = 0; i < n; ++i) {
+ for (int j = 0; j < n; ++j) {
+ ab = 0;
+ for(int k = 0;k<n/2;++k)
+ {
+ ab += ((*((A + i * n) + 2*k))+(*((B + (2*k+1) * n) + j)))*((*((A + i * n) + (2*k+1)))+(*((B + 2*k * n) + j)));
+ }
+ *((C + i * n) + j) = ab-etha_array[j]-xi_array[i];
+ }
+ }
+
+
+
+
+ // for (int i = 0; i < n; ++i) {
+ // int *a = (int*) malloc(n * sizeof(int));
+ // for(int k = 0; k<n; ++k)
+ // {
+ // a[k] = (*((A + i * n) + k));
+ // }
+ //
+ // for (int j = 0; j < n; ++j) {
+ // int *b = (int*) malloc(n * sizeof(int));
+ // for(int k = 0; k<n; ++k)
+ // {
+ // b[k] =(*((B + k * n) + j));
+ // }
+ // *((C + i * n) + j) = winograd_inner(a,b,n);
+ // }
+ // }
+ }
+
+
+void openMP_MM(int *A, int *B, int *C, int n) {
+
+ #pragma omp parallel for
+ for (int i = 0; i < n; ++i) {
+ for (int j = 0; j < n; ++j) {
+ int sum = 0;
+ for (int k = 0; k < n; ++k) {
+ sum += (*((A + i * n) + k)) * (*((B + k * n) + j));
+ }
+ *((C + i * n) + j) = sum;
+ }
+ }
+}
+
+void MM_dc(int *A, int *B, int *C, int n) {
+ if (n <= 2) {
+ MM((int*) A, (int*) B, (int*) C, n);
+ } else {
+ int *A11 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *A12 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *A21 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *A22 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *B11 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *B12 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *B21 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *B22 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+
+ split((int*) A, (int*) A11, n / 2, 0, 0);
+ split((int*) A, (int*) A12, n / 2, 0, n / 2);
+ split((int*) A, (int*) A21, n / 2, n / 2, 0);
+ split((int*) A, (int*) A22, n / 2, n / 2, n / 2);
+ split((int*) B, (int*) B11, n / 2, 0, 0);
+ split((int*) B, (int*) B12, n / 2, 0, n / 2);
+ split((int*) B, (int*) B21, n / 2, n / 2, 0);
+ split((int*) B, (int*) B22, n / 2, n / 2, n / 2);
+
+ int *tmp1 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *tmp2 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *tmp3 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *tmp4 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *tmp5 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *tmp6 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *tmp7 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *tmp8 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+
+ MM_dc((int*) A11, (int*) B11, (int*) tmp1, n / 2);
+ MM_dc((int*) A12, (int*) B21, (int*) tmp2, n / 2);
+ MM_dc((int*) A11, (int*) B12, (int*) tmp3, n / 2);
+ MM_dc((int*) A12, (int*) B22, (int*) tmp4, n / 2);
+ MM_dc((int*) A21, (int*) B11, (int*) tmp5, n / 2);
+ MM_dc((int*) A22, (int*) B21, (int*) tmp6, n / 2);
+ MM_dc((int*) A21, (int*) B12, (int*) tmp7, n / 2);
+ MM_dc((int*) A22, (int*) B22, (int*) tmp8, n / 2);
+
+ free(A11);
+ free(A12);
+ free(A21);
+ free(A22);
+ free(B11);
+ free(B12);
+ free(B21);
+ free(B22);
+
+ int *C11 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *C12 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *C21 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *C22 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+
+ add((int*) tmp1, (int*) tmp2, (int*) C11, n / 2);
+ add((int*) tmp3, (int*) tmp4, (int*) C12, n / 2);
+ add((int*) tmp5, (int*) tmp6, (int*) C21, n / 2);
+ add((int*) tmp7, (int*) tmp8, (int*) C22, n / 2);
+
+ free(tmp1);
+ free(tmp2);
+ free(tmp3);
+ free(tmp4);
+ free(tmp5);
+ free(tmp6);
+ free(tmp7);
+ free(tmp8);
+
+ join((int*) C11, (int*) C, n / 2, 0, 0);
+ join((int*) C12, (int*) C, n / 2, 0, n / 2);
+ join((int*) C21, (int*) C, n / 2, n / 2, 0);
+ join((int*) C22, (int*) C, n / 2, n / 2, n / 2);
+
+ free(C11);
+ free(C12);
+ free(C21);
+ free(C22);
+
+ }
+}
+
+void strassen(int *A, int *B, int *C, int n) {
+ if (n <= 2) {
+
+ int P, Q, R, S, T, U, V;
+ P = ((*((A + 0 * n) + 0)) + (*((A + 1 * n) + 1)))
+ * ((*((B + 0 * n) + 0)) + (*((B + 1 * n) + 1)));
+ Q = ((*((A + 1 * n) + 0)) + (*((A + 1 * n) + 1)))
+ * ((*((B + 0 * n) + 0)));
+ R = ((*((A + 0 * n) + 0)))
+ * ((*((B + 0 * n) + 1)) - (*((B + 1 * n) + 1)));
+ S = ((*((A + 1 * n) + 1)))
+ * ((*((B + 1 * n) + 0)) - (*((B + 0 * n) + 0)));
+ T = ((*((A + 0 * n) + 0)) + (*((A + 0 * n) + 1)))
+ * ((*((B + 1 * n) + 1)));
+ U = ((*((A + 1 * n) + 0)) - (*((A + 0 * n) + 0)))
+ * ((*((B + 0 * n) + 0)) + (*((B + 0 * n) + 1)));
+ V = ((*((A + 0 * n) + 1)) - (*((A + 1 * n) + 1)))
+ * ((*((B + 1 * n) + 0)) + (*((B + 1 * n) + 1)));
+ (*((C + 0 * n) + 0)) = P + S - T + V;
+ (*((C + 0 * n) + 1)) = R + T;
+ (*((C + 1 * n) + 0)) = Q + S;
+ (*((C + 1 * n) + 1)) = P + R - Q + U;
+
+ } else {
+ int *A11 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *A12 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *A21 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *A22 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *B11 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *B12 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *B21 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *B22 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+
+ split((int*) A, (int*) A11, n / 2, 0, 0);
+ split((int*) A, (int*) A12, n / 2, 0, n / 2);
+ split((int*) A, (int*) A21, n / 2, n / 2, 0);
+ split((int*) A, (int*) A22, n / 2, n / 2, n / 2);
+ split((int*) B, (int*) B11, n / 2, 0, 0);
+ split((int*) B, (int*) B12, n / 2, 0, n / 2);
+ split((int*) B, (int*) B21, n / 2, n / 2, 0);
+ split((int*) B, (int*) B22, n / 2, n / 2, n / 2);
+
+ int *P = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *Q = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *R = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *S = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *T = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *U = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *V = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+
+ int *addA = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *addB = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+
+ add((int*) A11, (int*) A22, (int*) addA, n / 2);
+ add((int*) B11, (int*) B22, (int*) addB, n / 2);
+ strassen((int*) addA, (int*) addB, (int*) P, n / 2);
+
+ add((int*) A21, (int*) A22, (int*) addA, n / 2);
+ strassen((int*) addA, (int*) B11, (int*) Q, n / 2);
+
+ sub((int*) B12, (int*) B22, (int*) addB, n / 2);
+ strassen((int*) A11, (int*) addB, (int*) R, n / 2);
+
+ sub((int*) B21, (int*) B11, (int*) addB, n / 2);
+ strassen((int*) A22, (int*) addB, (int*) S, n / 2);
+
+ add((int*) A11, (int*) A12, (int*) addA, n / 2);
+ strassen((int*) addA, (int*) B22, (int*) T, n / 2);
+
+ sub((int*) A21, (int*) A11, (int*) addA, n / 2);
+ add((int*) B11, (int*) B12, (int*) addB, n / 2);
+ strassen((int*) addA, (int*) addB, (int*) U, n / 2);
+
+ sub((int*) A12, (int*) A22, (int*) addA, n / 2);
+ add((int*) B21, (int*) B22, (int*) addB, n / 2);
+ strassen((int*) addA, (int*) addB, (int*) V, n / 2);
+
+ free(A11);
+ free(A12);
+ free(A21);
+ free(A22);
+ free(B11);
+ free(B12);
+ free(B21);
+ free(B22);
+ free(addA);
+ free(addB);
+
+ int *C11 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *C12 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *C21 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *C22 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+
+ int *resAdd1 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *resAdd2 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+
+ add((int*) R, (int*) T, (int*) C12, n / 2);
+ add((int*) Q, (int*) S, (int*) C21, n / 2);
+
+ add((int*) P, (int*) S, (int*) resAdd1, n / 2);
+ add((int*) resAdd1, (int*) V, (int*) resAdd2, n / 2);
+ sub((int*) resAdd2, (int*) T, (int*) C11, n / 2);
+
+ add((int*) P, (int*) R, (int*) resAdd1, n / 2);
+ add((int*) resAdd1, (int*) U, (int*) resAdd2, n / 2);
+ sub((int*) resAdd2, (int*) Q, (int*) C22, n / 2);
+
+ free(P);
+ free(Q);
+ free(R);
+ free(S);
+ free(T);
+ free(U);
+ free(V);
+ free(resAdd1);
+ free(resAdd2);
+
+ join((int*) C11, (int*) C, n / 2, 0, 0);
+ join((int*) C12, (int*) C, n / 2, 0, n / 2);
+ join((int*) C21, (int*) C, n / 2, n / 2, 0);
+ join((int*) C22, (int*) C, n / 2, n / 2, n / 2);
+
+ free(C11);
+ free(C12);
+ free(C21);
+ free(C22);
+ }
+}
+
+void add(int *A, int *B, int *C, int n) {
+ for (int i = 0; i < n; i++) {
+ for (int j = 0; j < n; j++) {
+ *((C + i * n) + j) = *((A + i * n) + j) + *((B + i * n) + j);
+ }
+ }
+}
+
+void sub(int *A, int *B, int *C, int n) {
+ for (int i = 0; i < n; i++) {
+ for (int j = 0; j < n; j++) {
+ *((C + i * n) + j) = *((A + i * n) + j) - *((B + i * n) + j);
+ }
+ }
+}
+
+void multiply(int *A, int *B, int *C, int n) {
+ int mul;
+
+ for (int i = 0; i < n; ++i) {
+ for (int j = 0; j < n; ++j) {
+ mul = (*((A + i * n) + j)) * (*((B + i * n) + j));
+ *((C + i * n) + j) = mul;
+ }
+ }
+}
+
+void split(int *in, int *out, int n, int col, int row) {
+ for (int i1 = 0, i2 = col; i1 < n; i1++, i2++)
+ for (int j1 = 0, j2 = row; j1 < n; j1++, j2++) {
+ *((out + i1 * n) + j1) = *((in + i2 * n * 2) + j2);
+
+ }
+}
+
+void join(int *in, int *out, int n, int col, int row) {
+ for (int i1 = 0, i2 = col; i1 < n; i1++, i2++)
+ for (int j1 = 0, j2 = row; j1 < n; j1++, j2++)
+ *((out + i2 * n * 2) + j2) = *((in + i1 * n) + j1);
+}
+
+void printMatrix(int *C, int n) {
+ for (int i = 0; i < n; ++i) {
+ for (int j = 0; j < n; ++j) {
+ printf("%d ", *((C + i * n) + j));
+ }
+ printf("\n");
+ }
+}
+
+void printMatrix_double(double *C, int n) {
+ for (int i = 0; i < n; ++i) {
+ for (int j = 0; j < n; ++j) {
+ printf("%.0f ", *((C + i * n) + j));
+ }
+ printf("\n");
+ }
+}
+
+void run_algo(void (*algo)(), char alog_name[], int print)
+{
+ FILE *fptr;
+
+ char fileName[40] = "meas/";
+ strcat(fileName, alog_name);
+ strcat(fileName, ".txt");
+ fptr = fopen(fileName, "w");
+
+
+ for(int i=0; i<n_arrays; ++i)
+ {
+ for(int j = 0; j<1; ++j)
+ {
+ int *C = (int*) malloc(n[i] * n[i] * sizeof(int));
+ double dtime = omp_get_wtime();
+ algo(Ap[i], Bp[i], (int*) C, n[i]);
+ dtime = omp_get_wtime() - dtime;
+ // printf("The %s program took %f seconds to execute \n", alog_name, dtime);
+ fprintf(fptr, "%f,%d\n", dtime, n[i]);
+
+ if(print==1)
+ {
+ printMatrix((int*)C, n[i]);
+ }
+ free(C);
+ }
+ }
+ fclose(fptr);
+
+}
+
+void run_algo_cblas(int print)
+{
+
+ FILE *fptr;
+
+ fptr = fopen("meas/blas.txt", "w");
+ for(int i=0; i<n_arrays; ++i)
+ {
+ for(int j = 0; j<1; ++j)
+ {
+ double *dC = (double*) malloc(n[i] * n[i] * sizeof(double));
+ double dtime = omp_get_wtime();
+ cblas_dgemm(CblasRowMajor, CblasNoTrans, CblasNoTrans, n[i], n[i], n[i], 1.0, dAp[i], n[i],
+ dBp[i], n[i], 0.0, dC, n[i]);
+ dtime = omp_get_wtime() - dtime;
+ // printf("The cblas program took %f seconds to execute \n", dtime);
+ fprintf(fptr, "%f,%d\n",dtime, n[i]);
+
+ if(print==1)
+ {
+ printMatrix_double( (double*)dC, n[i]);
+ }
+
+ free(dC);
+ }
+ }
+ fclose(fptr);
+
+}
diff --git a/buch/papers/multiplikation/code/MM.py b/buch/papers/multiplikation/code/MM.py new file mode 100644 index 0000000..626b82d --- /dev/null +++ b/buch/papers/multiplikation/code/MM.py @@ -0,0 +1,311 @@ +#!/usr/bin/env python3 +# -*- coding: utf-8 -*- +""" +Created on Fri Mar 19 07:31:29 2021 + +@author: nunigan +""" +import numpy as np +import time +import matplotlib.pyplot as plt +from scipy.optimize import curve_fit +import tikzplotlib +def MM(A, B): + n = np.shape(A)[0] + C = np.zeros((n, n)) + for i in range(n): + for j in range(n): + C[i, j] = 0 + for k in range(n): + C[i, j] += A[i, k]*B[k, j] + return C + + +def MM_dc(A, B): + n = np.shape(A)[0] + if(n <= 2): + C = np.zeros((n, n)) + C[0, 0] = A[0, 0]*B[0, 0]+A[0, 1]*B[1, 0] + C[0, 1] = A[0, 0]*B[0, 1]+A[0, 1]*B[1, 1] + C[1, 0] = A[1, 0]*B[0, 0]+A[1, 1]*B[1, 0] + C[1, 1] = A[1, 0]*B[0, 1]+A[1, 1]*B[1, 1] + return C + else: + A11, A12, A21, A22 = A[:n//2, :n//2], A[:n//2, n//2:], A[n//2:, :n//2], A[n//2:, n//2:] + B11, B12, B21, B22 = B[:n//2, :n//2], B[:n//2, n//2:], B[n//2:, :n//2], B[n//2:, n//2:] + C11 = MM_dc(A11, B11) + MM_dc(A12, B21) + C12 = MM_dc(A11, B12) + MM_dc(A12, B22) + C21 = MM_dc(A21, B11) + MM_dc(A22, B21) + C22 = MM_dc(A21, B12) + MM_dc(A22, B22) + C = np.vstack((np.hstack((C11, C12)), np.hstack((C21, C22)))) + return C + + +def strassen(A, B): + n = np.shape(A)[0] + if(n <= 2): + C = np.zeros((n, n)) + P = (A[0, 0]+A[1, 1])*(B[0, 0]+B[1, 1]) + Q = (A[1, 0]+A[1, 1])*B[0, 0] + R = A[0, 0]*(B[0, 1]-B[1, 1]) + S = A[1, 1]*(B[1, 0]-B[0, 0]) + T = (A[0, 0]+A[0, 1])*B[1, 1] + U = (A[1, 0]-A[0, 0])*(B[0, 0]+B[0, 1]) + V = (A[0, 1]-A[1, 1])*(B[1, 0]+B[1, 1]) + C[0, 0] = P+S-T+V + C[0, 1] = R+T + C[1, 0] = Q+S + C[1, 1] = P+R-Q+U + return C + else: + m = n//2 + A11, A12, A21, A22 = A[:m, :m], A[:m, m:], A[m:, :m], A[m:, m:] + B11, B12, B21, B22 = B[:m, :m], B[:m, m:], B[m:, :m], B[m:, m:] + P = strassen((A11+A22),(B11+B22)) + Q = strassen((A21+A22),B11) + R = strassen(A11,(B12-B22)) + S = strassen(A22,(B21-B11)) + T = strassen((A11+A12),B22) + U = strassen((A21-A11),(B11+B12)) + V = strassen((A12-A22),(B21+B22)) + + C11 = P+S-T+V + C12 = R+T + C21 = Q+S + C22 = P+R-Q+U + + C = np.vstack((np.hstack((C11, C12)), np.hstack((C21, C22)))) + return C + +def winograd_inner(a, b): + n = np.shape(a)[0] + if n%2 == 0: + xi = np.sum(a[::2]*a[1::2]) + etha = np.sum(b[::2]*b[1::2]) + # print("xi = {}, etha = {}".format(xi, etha)) + ab = np.sum((a[::2]+b[1::2])*(a[1::2]+b[::2]))-xi-etha + else: + xi = np.sum(a[0:-1:2]*a[1::2]) + etha = np.sum(b[0:-1:2]*b[1::2]) + ab = np.sum((a[0:-1:2]+b[1::2])*(a[1::2]+b[0:-1:2]))-xi-etha+a[-1]*b[-1] + return ab + +def winograd(A, B): + m,n = np.shape(A) + n2,p = np.shape(B) + C = np.zeros((m,p)) + for i in range(np.shape(A)[0]): + for j in range(np.shape(B)[1]): + C[i,j] = winograd_inner(A[i,:], B[:,j]) + return C + +def winograd2(A, B): + m,n = np.shape(A) + n2,p = np.shape(B) + C = np.zeros((m,p)) + xi = np.zeros((m)) + eta = np.zeros((p)) + ab = 0 + for i in range(m): + for j in range(n//2): + xi[i] += A[i,2*j]*A[i,2*j+1] + + for i in range(p): + for j in range(n//2): + eta[i] += B[2*j,i]*B[2*j+1,i] + + if n%2==0: + for i in range(m): + for j in range(p): + ab = 0 + for k in range(n//2): + ab += (A[i,2*k]+B[2*k+1,j])*(A[i,2*k+1]+B[2*k,j]) + C[i,j] = ab-eta[j]-xi[i] + else: + for i in range(m): + for j in range(p): + ab = 0 + for k in range(n//2): + ab += (A[i,2*k]+B[2*k+1,j])*(A[i,2*k+1]+B[2*k,j]) + C[i,j] = ab-eta[j]-xi[i]+A[i,-1]*B[-1,j] + + return C + +def test_perfomance(n): + t_mm = [] + t_mm_dc = [] + t_mm_strassen = [] + t_wino = [] + t_np = [] + + for i in n: + A = np.random.randn(i, i) + B = np.random.randn(i, i) + # A = np.random.randint(-100, 100,(i, i)) + # B = np.random.randint(-100, 100,(i, i)) + + start = time.time() + C3 = strassen(A, B) + t_mm_strassen.append(time.time() - start) + + start = time.time() + C1 = MM(A, B) + t_mm.append(time.time() - start) + + start = time.time() + C2 = MM_dc(A, B) + t_mm_dc.append(time.time() - start) + + start = time.time() + C4 = winograd2(A, B) + t_wino.append(time.time() - start) + + start = time.time() + C = A@B + t_np.append(time.time() - start) + + plt.figure(figsize=(13,8)) + plt.rcParams['font.family'] = 'STIXGeneral' + plt.rc('axes', labelsize=23) + plt.rc('xtick', labelsize=23) + plt.rc('ytick', labelsize=23) + plt.plot(n, t_mm, label='Standard', lw=5) + plt.plot(n, t_mm_dc, label='Divide and conquer', lw=5) + plt.plot(n, t_mm_strassen, label='Strassen', lw=5) + plt.plot(n, t_wino, label='Winograd', lw=5) + plt.plot(n, t_np, label='NumPy A@B', lw=5) + plt.legend() + plt.xlabel("n") + plt.ylabel("time (s)") + plt.grid(True) + plt.tight_layout() + # plt.yscale('log') + plt.legend(fontsize=19) + plt.savefig('meas_' + str(max(n))+ '.pdf') + arr = np.array([n, t_mm, t_mm_dc, t_mm_strassen, t_wino, t_np]) + np.savetxt('meas_' + str(max(n))+ '.txt',arr) + return arr + + +def plot(num): + arr = np.loadtxt('meas_{}.txt'.format(num)) + n, t_mm, t_mm_dc, t_mm_strassen, t_wino, t_np = arr + plt.figure(figsize=(13,8)) + plt.rcParams['font.family'] = 'STIXGeneral' + plt.rc('axes', labelsize=23) + plt.rc('xtick', labelsize=23) + plt.rc('ytick', labelsize=23) + plt.plot(n, t_mm, label='3 For Loops', lw=5) + plt.plot(n, t_mm_dc, label='Divide and Conquer', lw=5) + plt.plot(n, t_mm_strassen, label='Strassen', lw=5) + # plt.plot(n, t_wino, label='Winograd', lw=5) + plt.plot(n, t_np, label='NumPy A@B', lw=5) + plt.legend() + plt.xlabel("n") + plt.ylabel("time (s)") + plt.grid(True) + plt.tight_layout() + # plt.yscale('log') + plt.legend(fontsize=19) + plt.savefig('meas_' + str(num)+ '.pdf') + return arr + +def plot_c_res(ave, num): + MM = np.loadtxt("meas/MM.txt", delimiter=',') + # winograd = np.loadtxt("meas/winograd.txt", delimiter=',') + blas = np.loadtxt("meas/blas.txt", delimiter=',') + MM_dc = np.loadtxt("meas/MM_dc.txt", delimiter=',') + strassen = np.loadtxt("meas/strassen.txt", delimiter=',') + + MM_t = MM[:,0] + MM_n = MM[:,1] + MM_t = np.mean(MM_t.reshape(-1,ave),axis=1) + MM_n = np.mean(MM_n.reshape(-1,ave),axis=1) + + MM_dc_t = MM_dc[:,0] + MM_dc_n = MM_dc[:,1] + MM_dc_t = np.mean(MM_dc_t.reshape(-1,ave),axis=1) + MM_dc_n = np.mean(MM_dc_n.reshape(-1,ave),axis=1) + + strassen_t = strassen[:,0] + strassen_n = strassen[:,1] + strassen_t = np.mean(strassen_t.reshape(-1,ave),axis=1) + strassen_n = np.mean(strassen_n.reshape(-1,ave),axis=1) + + # winograd_t = winograd[:,0] + # winograd_n = winograd[:,1] + # winograd_t = np.mean(winograd_t.reshape(-1,ave),axis=1) + # winograd_n = np.mean(winograd_n.reshape(-1,ave),axis=1) + + blas_t = blas[:,0] + blas_n = blas[:,1] + blas_t = np.mean(blas_t.reshape(-1,ave),axis=1) + blas_n = np.mean(blas_n.reshape(-1,ave),axis=1) + + def func(x, a,b): + return b*x**a + + # popt, pcov = curve_fit(func, blas_n, blas_t) + # popt1, pcov2 = curve_fit(func, blas_n, winograd_t) + # popt2, pcov2 = curve_fit(func, blas_n, MM_t) + + plt.figure(figsize=(13,8)) + plt.rcParams['font.family'] = 'STIXGeneral' + plt.rc('axes', labelsize=23) + plt.rc('xtick', labelsize=23) + plt.rc('ytick', labelsize=23) + plt.plot(MM_n, MM_t, label='3 For Loops', lw=5) + # plt.plot(winograd_n, winograd_t, label='Winograd MM', lw=5) + plt.plot(blas_n, blas_t, label='Blas', lw=5) + plt.plot(strassen_n, strassen_t, label='Strassen', lw=5) + plt.plot(MM_dc_n, MM_dc_t, label='Divide and Conquer', lw=5) + plt.xlabel("n") + plt.ylabel("time (s)") + plt.grid(True) + plt.tight_layout() + plt.legend(fontsize=19) + plt.savefig('c_meas_' + str(num)+ '.pdf') + + # plt.plot(blas_n, func(blas_n, *popt), 'r-', label='fit blas: a=%5.5f, b=%5.10f' % tuple(popt)) + # plt.plot(blas_n, func(blas_n, *popt1), 'r-', label='fit winograd: a=%5.5f, b=%5.10f' % tuple(popt1)) + # plt.plot(blas_n, func(blas_n, *popt2), 'r-', label='fit MM: a=%5.5f, b=%5.10f' % tuple(popt2)) + + plt.legend() + + +# test%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +if __name__ == '__main__': + plot_c_res(1, 4096) + + + # plot(8) + # n = np.logspace(1,10,10,base=2,dtype=(np.int)) + # n = np.arange(1,50,2) + A = np.random.randint(-10, 10, (5,3)) + B = np.random.randint(-10, 10, (3,5)) + + C = winograd2(A, B) + C_test = A@B + print(C) + print(C_test) + # print(np.equal(C, C_test)) + + # t_np = test_perfomance(n) + # C = strassen(A, B) + # C_test = A@B + + + # plot_c_res() + # def func(x, a): + # return x**a + + # popt, pcov = curve_fit(func, n, t_np, bounds=(2, 3)) + + + # plt.figure() + # plt.plot(n, t_np, 'b-', label='data') + # plt.plot(n, func(n, *popt), 'r-', label='fit: a=%5.3f' % tuple(popt)) + # plt.xlabel('x') + # plt.ylabel('y') + # plt.legend() +
\ No newline at end of file diff --git a/buch/papers/multiplikation/code/__pycache__/MM.cpython-38.pyc b/buch/papers/multiplikation/code/__pycache__/MM.cpython-38.pyc Binary files differnew file mode 100644 index 0000000..7768772 --- /dev/null +++ b/buch/papers/multiplikation/code/__pycache__/MM.cpython-38.pyc diff --git a/buch/papers/multiplikation/code/c_matrix.h b/buch/papers/multiplikation/code/c_matrix.h new file mode 100644 index 0000000..13df55d --- /dev/null +++ b/buch/papers/multiplikation/code/c_matrix.h @@ -0,0 +1,101 @@ +/* Seminar Matrizen, autogenerated File, Michael Schmid, 30/05/2021, 22:00:57 */ + +#include <stdint.h> +const int A0[][2] = + { + {-15,68}, + {49,86} + }; +const int B0[][2] = + { + {33,73}, + {38,-76} + }; +const double dB0[][2] = + { + {33,73}, + {38,-76} + }; +const double dA0[][2] = + { + {-15,68}, + {49,86} + }; +const int A1[][4] = + { + {75,-38,-32,-65}, + {37,74,-31,29}, + {15,-62,-20,-20}, + {-31,-35,-89,47} + }; +const int B1[][4] = + { + {71,90,78,-98}, + {4,63,12,-47}, + {11,-44,75,-69}, + {95,-15,64,23} + }; +const double dB1[][4] = + { + {71,90,78,-98}, + {4,63,12,-47}, + {11,-44,75,-69}, + {95,-15,64,23} + }; +const double dA1[][4] = + { + {75,-38,-32,-65}, + {37,74,-31,29}, + {15,-62,-20,-20}, + {-31,-35,-89,47} + }; +const int A2[][8] = + { + {80,42,3,-16,6,55,87,16}, + {-99,-14,21,-1,-94,-56,91,10}, + {-47,-55,-59,62,12,-53,87,-65}, + {-60,94,-67,23,-62,33,-63,-72}, + {12,-75,16,21,22,-37,1,16}, + {-100,-99,82,-66,2,64,-13,44}, + {59,-100,-90,8,36,-24,18,88}, + {73,-58,75,-100,-19,-29,85,-19} + }; +const int B2[][8] = + { + {-61,88,69,49,-53,47,73,45}, + {16,14,-88,-11,-67,-73,-20,43}, + {-60,-63,26,32,-29,18,-44,-69}, + {1,21,21,38,7,-100,-61,-76}, + {-90,95,-99,88,49,-80,27,-36}, + {24,-12,-47,-7,29,15,52,37}, + {-98,-76,29,76,-41,-75,97,79}, + {62,-90,-35,-14,-30,-42,-95,52} + }; +const double dB2[][8] = + { + {-61,88,69,49,-53,47,73,45}, + {16,14,-88,-11,-67,-73,-20,43}, + {-60,-63,26,32,-29,18,-44,-69}, + {1,21,21,38,7,-100,-61,-76}, + {-90,95,-99,88,49,-80,27,-36}, + {24,-12,-47,-7,29,15,52,37}, + {-98,-76,29,76,-41,-75,97,79}, + {62,-90,-35,-14,-30,-42,-95,52} + }; +const double dA2[][8] = + { + {80,42,3,-16,6,55,87,16}, + {-99,-14,21,-1,-94,-56,91,10}, + {-47,-55,-59,62,12,-53,87,-65}, + {-60,94,-67,23,-62,33,-63,-72}, + {12,-75,16,21,22,-37,1,16}, + {-100,-99,82,-66,2,64,-13,44}, + {59,-100,-90,8,36,-24,18,88}, + {73,-58,75,-100,-19,-29,85,-19} + }; +const int *Ap[3] = {(int*) A0,(int*) A1,(int*) A2}; +const int *Bp[3] = {(int*) B0,(int*) B1,(int*) B2}; +const double *dAp[3] = {(double*) dA0,(double*) dA1,(double*) dA2}; +const double *dBp[3] = {(double*) dB0,(double*) dB1,(double*) dB2}; +int n[3] = {2,4,8}; +int n_arrays = 3; diff --git a/buch/papers/multiplikation/code/c_meas_1024.pdf b/buch/papers/multiplikation/code/c_meas_1024.pdf Binary files differnew file mode 100644 index 0000000..95b68b5 --- /dev/null +++ b/buch/papers/multiplikation/code/c_meas_1024.pdf diff --git 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b/buch/papers/multiplikation/code/c_meas_8.pdf Binary files differnew file mode 100644 index 0000000..9682aca --- /dev/null +++ b/buch/papers/multiplikation/code/c_meas_8.pdf diff --git a/buch/papers/multiplikation/code/helper_class.py b/buch/papers/multiplikation/code/helper_class.py new file mode 100755 index 0000000..485fa76 --- /dev/null +++ b/buch/papers/multiplikation/code/helper_class.py @@ -0,0 +1,105 @@ +#!/usr/bin/env python3 +# -*- coding: utf-8 -*- +""" +Created on Fri Mar 12 09:02:48 2021 + +@author: nunigan +""" + +from datetime import datetime +import numpy as np + +class Helper(): + def __init__(self): + pass + + def write_c_matrix(self, n_array): + + with open('c_matrix.h', 'w') as file: + file.writelines('/* Seminar Matrizen, autogenerated File, Michael Schmid, {} */ \n \n'.format(datetime.now().strftime("%d/%m/%Y, %H:%M:%S"))) + + file.writelines('#include <stdint.h> \n') + + + + for k, n in enumerate(n_array): + A = np.random.randint(-100,100,(n,n)) + B = np.random.randint(-100,100,(n,n)) + file.writelines('const int A{}[][{}] = \n'.format(k, n)) + file.writelines(' {\n') + for i in range(n): + file.writelines(' {') + for j in range(n): + if j == n-1: + file.writelines('{}'.format(A[i,j])) + else: + file.writelines('{},'.format(A[i,j])) + if i == n-1: + file.writelines('}\n') + else: + file.writelines('},\n') + + file.writelines(' };\n') + + file.writelines('const int B{}[][{}] = \n'.format(k,n)) + file.writelines(' {\n') + for i in range(n): + file.writelines(' {') + for j in range(n): + if j == n-1: + file.writelines('{}'.format(B[i,j])) + else: + file.writelines('{},'.format(B[i,j])) + if i == n-1: + file.writelines('}\n') + else: + file.writelines('},\n') + + file.writelines(' };\n') + + file.writelines('const double dB{}[][{}] = \n'.format(k,n)) + file.writelines(' {\n') + for i in range(n): + file.writelines(' {') + for j in range(n): + if j == n-1: + file.writelines('{}'.format(B[i,j])) + else: + file.writelines('{},'.format(B[i,j])) + if i == n-1: + file.writelines('}\n') + else: + file.writelines('},\n') + + file.writelines(' };\n') + + file.writelines('const double dA{}[][{}] = \n'.format(k,n)) + file.writelines(' {\n') + for i in range(n): + file.writelines(' {') + for j in range(n): + if j == n-1: + file.writelines('{}'.format(A[i,j])) + else: + file.writelines('{},'.format(A[i,j])) + if i == n-1: + file.writelines('}\n') + else: + file.writelines('},\n') + + file.writelines(' };\n') + + file.writelines('const int *Ap[{}] = {{{}}}; \n'.format(len(n_array),",".join(['(int*) A'+str(element) for element in np.arange(len(n_array))]))) + file.writelines('const int *Bp[{}] = {{{}}}; \n'.format(len(n_array),",".join(['(int*) B'+str(element) for element in np.arange(len(n_array))]))) + file.writelines('const double *dAp[{}] = {{{}}}; \n'.format(len(n_array),",".join(['(double*) dA'+str(element) for element in np.arange(len(n_array))]))) + file.writelines('const double *dBp[{}] = {{{}}}; \n'.format(len(n_array),",".join(['(double*) dB'+str(element) for element in np.arange(len(n_array))]))) + file.writelines('int n[{}] = {{{}}}; \n'.format(len(n_array),",".join([str(element) for element in n_array]))) + file.writelines('int n_arrays = {};\n'.format(len(n_array))) + +# test%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +if __name__ == '__main__': + + helper = Helper() + # n = np.arange(2,10) + n = np.logspace(1,3,3,base=2,dtype=(np.int)) + C = helper.write_c_matrix(n) diff --git a/buch/papers/multiplikation/code/meas/MM.txt b/buch/papers/multiplikation/code/meas/MM.txt new file mode 100644 index 0000000..1a0cd5d --- /dev/null +++ b/buch/papers/multiplikation/code/meas/MM.txt @@ -0,0 +1,12 @@ +0.000000,2 +0.000000,4 +0.000002,8 +0.000011,16 +0.000080,32 +0.000653,64 +0.005397,128 +0.045147,256 +0.487710,512 +3.964180,1024 +128.863544,2048 +996.370209,4096 diff --git a/buch/papers/multiplikation/code/meas/MM_dc.txt b/buch/papers/multiplikation/code/meas/MM_dc.txt new file mode 100644 index 0000000..0d5580a --- /dev/null +++ b/buch/papers/multiplikation/code/meas/MM_dc.txt @@ -0,0 +1,12 @@ +0.000006,2 +0.000007,4 +0.000035,8 +0.000228,16 +0.001310,32 +0.007204,64 +0.034338,128 +0.267511,256 +2.131212,512 +17.177403,1024 +146.112874,2048 +1156.777565,4096 diff --git a/buch/papers/multiplikation/code/meas/blas.txt b/buch/papers/multiplikation/code/meas/blas.txt new file mode 100644 index 0000000..6b7cd0b --- /dev/null +++ b/buch/papers/multiplikation/code/meas/blas.txt @@ -0,0 +1,12 @@ +0.000001,2 +0.000000,4 +0.000001,8 +0.000003,16 +0.000021,32 +0.000164,64 +0.001240,128 +0.009657,256 +0.072523,512 +0.735149,1024 +6.895747,2048 +56.812183,4096 diff --git a/buch/papers/multiplikation/code/meas/strassen.txt b/buch/papers/multiplikation/code/meas/strassen.txt new file mode 100644 index 0000000..89cf41a --- /dev/null +++ b/buch/papers/multiplikation/code/meas/strassen.txt @@ -0,0 +1,12 @@ +0.000000,2 +0.000003,4 +0.000010,8 +0.000086,16 +0.000476,32 +0.003366,64 +0.025547,128 +0.184593,256 +1.248713,512 +9.007700,1024 +61.079879,2048 +424.493037,4096 diff --git a/buch/papers/multiplikation/code/meas/test/4096/MM.txt b/buch/papers/multiplikation/code/meas/test/4096/MM.txt new file mode 100644 index 0000000..25e40e1 --- /dev/null +++ b/buch/papers/multiplikation/code/meas/test/4096/MM.txt @@ -0,0 +1,12 @@ +0.000000,2 +0.000000,4 +0.000002,8 +0.000011,16 +0.000100,32 +0.000712,64 +0.005498,128 +0.046711,256 +0.489233,512 +4.006544,1024 +124.427496,2048 +993.405615,4096 diff --git a/buch/papers/multiplikation/code/meas/test/4096/strassen.txt b/buch/papers/multiplikation/code/meas/test/4096/strassen.txt new file mode 100644 index 0000000..eb2a496 --- /dev/null +++ b/buch/papers/multiplikation/code/meas/test/4096/strassen.txt @@ -0,0 +1,12 @@ +0.000007,2 +0.000007,4 +0.000029,8 +0.000199,16 +0.001414,32 +0.007583,64 +0.028096,128 +0.171662,256 +1.198323,512 +8.421896,1024 +58.803644,2048 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at={(0.03,0.97)}, + anchor=north west, + draw=white!80!black, + fill=white!89.8039215686275!black +}, +tick align=outside, +tick pos=left, +x grid style={white}, +xlabel={n}, +xmajorgrids, +xmin=-4.3, xmax=134.3, +xtick style={color=white!33.3333333333333!black}, +y grid style={white}, +ylabel={time (s)}, +ymajorgrids, +ymin=-0.0834965705871582, ymax=1.75356960296631, +ytick style={color=white!33.3333333333333!black} +] +\addplot [line width=2pt, color0] +table {% +2 1.57356262207031e-05 +4 5.96046447753906e-05 +8 0.000428915023803711 +16 0.00276041030883789 +32 0.0217020511627197 +64 0.160412073135376 +128 1.3419406414032 +}; +\addlegendentry{Standard MM} +\addplot [line width=2pt, color1] +table {% +2 6.43730163574219e-06 +4 6.69956207275391e-05 +8 0.00048065185546875 +16 0.00336766242980957 +32 0.0257236957550049 +64 0.231612205505371 +128 1.67006659507751 +}; +\addlegendentry{Divide and conquer MM} +\addplot [line width=2pt, color2] +table {% +2 2.90870666503906e-05 +4 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2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Einleitung \label{multiplikation:section:einleitung}} +\rhead{Einleitung} + +Die Multiplikation zweier Matrizen ist eine wichtige Operation die in verschiedensten Teilen der Mathematik Anwendung findet. +Die Beschreibung der Multiplikation aus der Definition 2.10 (\textcolor{blue} {Kein Hyperlink zu einer Definition?)}: + +Eine $m\times n$-Matrix $\mathbf{A}\in M_{m\times n}(\Bbbk)$ und eine +$n\times p$-Matrix $\mathbf{B}\in M_{n\times l}(\Bbbk)$ haben als Produkt +eine $n\times l$-Matrix $\mathbf{C}=\mathbf{AB}\in M_{n\times l}(\Bbbk)$ mit den +Koeffizienten +\begin{equation} +c_{ij} = \sum_{k=1}^n a_{ik} b_{kj}. +\label{multiplikation:eq:MM} +\end{equation} +Grafisch kann die Matrizenmultiplikation $AB=C$ wie in \ref{multiplikation:fig:mm_viz} visualisiert werden. +\begin{figure} + \center + \includegraphics[]{papers/multiplikation/images/mm_visualisation} + \caption{Matrizen Multiplikation} + \label{multiplikation:fig:mm_viz} +\end{figure} +Im Fall einer Matrizengr\"osse von $2\times 2$ +\begin{equation} + \begin{bmatrix} +A_{11} & A_{12}\\ +A_{21} & A_{22} +\end{bmatrix} +\begin{bmatrix} +B_{11} & B_{12}\\ +B_{21} & B_{22} +\end{bmatrix} += +\begin{bmatrix} +C_{11} & C_{12}\\ +C_{21} & C_{22} +\end{bmatrix} +\end{equation} +kann die Gleichung der einzelnen Terme +\begin{equation} \label{multiplikation:eq:MM_exp} +\begin{split} +C_{11} &= A_{11} \cdot B_{11} + A_{12} \cdot B_{21}\\ +C_{12} &= A_{11} \cdot B_{12} + A_{12} \cdot B_{22}\\ +C_{21} &= A_{21} \cdot B_{11} + A_{22} \cdot B_{21}\\ +C_{22} &= A_{21} \cdot B_{12} + A_{22} \cdot B_{22} +\end{split} +\end{equation} +explizit geschrieben werden. diff --git a/buch/papers/multiplikation/images/bigo.pdf b/buch/papers/multiplikation/images/bigo.pdf Binary files differnew file mode 100644 index 0000000..dfa2ba4 --- /dev/null +++ b/buch/papers/multiplikation/images/bigo.pdf diff --git a/buch/papers/multiplikation/images/bigo.tex b/buch/papers/multiplikation/images/bigo.tex new file mode 100644 index 0000000..e3293e4 --- /dev/null +++ b/buch/papers/multiplikation/images/bigo.tex @@ -0,0 +1,107 @@ +\documentclass[border=10pt,varwidth]{standalone} +\usepackage[left=25mm,right=25mm,top=25mm,bottom=25mm]{geometry} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{times} +\usepackage{geometry} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{mathrsfs} +\usepackage{amsfonts} +\usepackage{amsthm} +\usepackage{lipsum} +\usepackage{amscd} +\usepackage{graphicx} +\usepackage{fancyhdr} +\usepackage{textcomp} +\usepackage{pgfplots} +\usepackage{txfonts} +\usepackage[all]{xy} +\usepackage{paralist} +\usepackage[colorlinks=true]{hyperref} +\usepackage{array} +\usepackage{tikz} +\usepackage{slashed} +\usepackage{pdfpages} +\usepackage{cite} +\usepackage{url} +\usepackage{amsmath,amsfonts,amssymb} +\usepackage{tikz} +\usetikzlibrary{arrows,matrix,positioning} +\usetikzlibrary{overlay-beamer-styles} +\usetikzlibrary{matrix.skeleton} +\usetikzlibrary{automata,positioning} +\usetikzlibrary{decorations.text} +\usepackage{listings} +\usepackage{multirow} +\usepackage{color} + +\begin{document} + +\begin{tikzpicture} +\begin{axis}[ + axis lines = left, + xlabel = $n$ (Data Input), + ylabel = {$t$ (time)}, + legend pos=north east, + very thick, + ymax = 500, + yticklabels=\empty, + xticklabels=\empty, + scale only axis=true, + width=12cm, height=6cm, + ] +\addplot [ + domain= 1:20, + samples=100, + color=red, +] +{1}; +\addlegendentry{$\mathcal{O}(1)$} +\addplot [ + domain= 1:20, + samples=100, + color=green, +] +{x}; +\addlegendentry{$\mathcal{O}(n)$} +\addplot [ + domain= 1:20, + samples=100, + color=blue, +] +{x^2}; +\addlegendentry{$\mathcal{O}(n^2)$} +\addplot [ + domain= 1:10, + samples=100, + color=purple, +] +{x^3}; +\addlegendentry{$\mathcal{O}(n^3)$} +\addplot [ + domain= 1:10, + samples=100, + color=black, +] +{exp(x)}; +\addlegendentry{$\mathcal{O}(e^n)$} +\addplot [ + domain= 1:20, + samples=100, + color=orange, +] +{log2(x)}; +\addlegendentry{$\mathcal{O}(\log n)$} + +\addplot [ + domain= 1:20, + samples=100, + color=gray, +] +{x*log2(x)}; +\addlegendentry{$\mathcal{O}(n \log n)$} +\end{axis} +\end{tikzpicture} + +\end{document} diff --git a/buch/papers/multiplikation/images/mm_visualisation.pdf b/buch/papers/multiplikation/images/mm_visualisation.pdf Binary files differnew file mode 100644 index 0000000..9309df1 --- /dev/null +++ b/buch/papers/multiplikation/images/mm_visualisation.pdf diff --git a/buch/papers/multiplikation/images/mm_visualisation.tex b/buch/papers/multiplikation/images/mm_visualisation.tex new file mode 100644 index 0000000..6e8f789 --- /dev/null +++ b/buch/papers/multiplikation/images/mm_visualisation.tex @@ -0,0 +1,45 @@ + + \begin{tikzpicture}[ampersand replacement=\&] + + \matrix (A)[matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}] at (0,0) + { + A_{1,1} \& \cdots \& A_{1,k} \& \cdots \& A_{1,n} \\ + \vdots \& \& \vdots \& \& \vdots \\ + A_{i,1} \& \cdots \& A_{i,k} \& \cdots \& A_{i,n} \\ + \vdots \& \& \vdots \& \& \vdots \\ + A_{m,1} \& \cdots \& A_{m,k} \& \cdots \& A_{m,n} \\ + }; + + \node [right=0.1 of A] (mul) {$\cdot$}; + + + \matrix (B)[right=0.1 of mul, matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}] + { + B_{1,1} \& \cdots \& B_{1,j} \& \cdots \& B_{1,p} \\ + \vdots \& \& \vdots \& \& \vdots \\ + B_{k,1} \& \cdots \& B_{k,j} \& \cdots \& B_{k,p} \\ + \vdots \& \& \vdots \& \& \vdots \\ + B_{n,1} \& \cdots \& B_{n,j} \& \cdots \& B_{n,p} \\ + }; + + \node [right=0.1 of B] (eq) {$=$}; + + \matrix (C)[right=0.1 of eq, matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}] + { + C_{1,1} \& \cdots \& C_{1,j} \& \cdots \& C_{1,p} \\ + \vdots \& \& \vdots \& \& \vdots \\ + C_{i,1} \& \cdots \& C_{i,j} \& \cdots \& C_{i,p} \\ + \vdots \& \& \vdots \& \& \vdots \\ + C_{m,1} \& \cdots \& C_{m,j} \& \cdots \& C_{m,p} \\ + }; + + + \node[opacity=0.5, rounded corners=2pt, inner sep=-1pt, fill=green, fit=(A-3-1)(A-3-5)] {}; + \node[opacity=0.5, rounded corners=2pt, inner sep=-1pt, fill=blue, fit=(B-1-3)(B-5-3)] {}; + \node[opacity=0.5, rounded corners=2pt, inner sep=-1pt, fill=red, fit=(C-3-3)] {}; + + + \end{tikzpicture} + +\end{document} + diff --git a/buch/papers/multiplikation/images/strassen.pdf b/buch/papers/multiplikation/images/strassen.pdf Binary files differnew file mode 100644 index 0000000..9899dcb --- /dev/null +++ b/buch/papers/multiplikation/images/strassen.pdf diff --git a/buch/papers/multiplikation/images/strassen.tex b/buch/papers/multiplikation/images/strassen.tex new file mode 100644 index 0000000..797772b --- /dev/null +++ b/buch/papers/multiplikation/images/strassen.tex @@ -0,0 +1,140 @@ +\documentclass[border=10pt]{standalone} +\usepackage[left=25mm,right=25mm,top=25mm,bottom=25mm]{geometry} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{times} +\usepackage{geometry} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{mathrsfs} +\usepackage{amsfonts} +\usepackage{amsthm} +\usepackage{lipsum} +\usepackage{amscd} +\usepackage{graphicx} +\usepackage{fancyhdr} +\usepackage{textcomp} +\usepackage{pgfplots} +\usepackage{txfonts} +\usepackage[all]{xy} +\usepackage{paralist} +\usepackage[colorlinks=true]{hyperref} +\usepackage{array} +\usepackage{tikz} +\usepackage{slashed} +\usepackage{pdfpages} +\usepackage{cite} +\usepackage{url} +\usepackage{amsmath,amsfonts,amssymb} +\usepackage{tikz} +\usetikzlibrary{arrows,matrix,positioning} +\usetikzlibrary{overlay-beamer-styles} +\usetikzlibrary{matrix.skeleton} +\usetikzlibrary{automata,positioning} +\usetikzlibrary{decorations.text} +\usepackage{listings} +\usepackage{multirow} +\usepackage{color} + +\begin{document} + +\begin{tikzpicture}[ampersand replacement=\&] + +\foreach \i in {1,...,4} +{ + \small{ + \matrix (X\i)[matrix of math nodes,nodes in empty cells, + nodes = {draw, minimum size=10mm, + anchor=center, + inner sep=0pt, outer sep=0pt}, + column sep=-\pgflinewidth, + row sep=-\pgflinewidth, + ] at (0,-\i*5) + { + A_{11}B_{11} \& A_{12}B_{11} \& A_{21}B_{11} \& A_{22}B_{11} \\ + A_{11}B_{21} \& A_{12}B_{21} \& A_{21}B_{21} \& A_{22}B_{21} \\ + A_{11}B_{11} \& A_{12}B_{12} \& A_{21}B_{12} \& A_{22}B_{12} \\ + A_{11}B_{22} \& A_{12}B_{22} \& A_{21}B_{22} \& A_{22}B_{22} \\ + };} + + \foreach \j in {1,...,7} + { + \matrix(M\i\j)[matrix of math nodes,nodes in empty cells, + nodes = {draw, minimum size=10mm, + anchor=center, + inner sep=0pt, outer sep=0pt}, + column sep=-\pgflinewidth, + row sep=-\pgflinewidth, + ] at (\j*5,-\i*5) + { + \& \& \& \\ + \& \& \& \\ + \& \& \& \\ + \& \& \& \\ + }; + } +} + +\huge{ + \node at (-3,-20) {$C_{22}=$}; + \node at (-3,-15) {$C_{21}=$} ; + \node at (-3,-10) {$C_{12}=$} ; + \node at (-3,-5) {$C_{11}=$} ; + + \node at (5,-2) {I}; + \node at 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2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% + +\section{L\"osungsmethoden} +\rhead{L\"osungsmethoden} + +In diesem Abschnitt werden mehrere Algorithmen zur Berechnung der Matrizenmultiplikation vorgestellt, auch werden Libraries zur automatisierten Verwendung von vordefinierten Algorithmen gezeigt. + +\subsection{Standard Algorithmus} + +Der Standard Methode kann im Algorithmus \ref{multiplikation:alg:smm} entnommen werden. +Hierf\"ur wurde die Gleichung \eqref{multiplikation:eq:MM} direkt implementiert. +Die \texttt{For i} Schleife iteriert \"uber alle Zeilen der $\mathbf{A}$ Matrix, die \texttt{For j} Schleife iteriert \"uber alle Spalten der $\mathbf{B}$ Matrix und die \texttt{For k} Schleife iteriert \"uber alle Eintr\"age dieser Zeilen bzw. Spalten. + +\begin{algorithm}\caption{Matrix Multiplication} + \label{multiplikation:alg:smm} + \setlength{\lineskip}{7pt} + \begin{algorithmic}[1] + \Function{MM}{$\textbf{A}, \textbf{B}$} + \State $sum \gets 0$ + \State $n \gets columns(\textbf{A}) == rows(\textbf{B})$ + \State $m \gets rows(\textbf{A})$ + \State $p \gets columns(\textbf{B})$ + \State $\textbf{C} \gets zeros(m,p)$ + \For{$i = 0,1,2 \dots,m-1$} + \For{$j = 0,1,2 \dots,p-1$} + \State $sum \gets 0$ + \For{$k = 0,1,2 \dots,n-1$} + \State $sum \gets sum + \textbf{A}[i][k] \cdot \textbf{B}[k][j]$ + \EndFor + \State $\textbf{C}[i][j] \gets sum $ + \EndFor + \EndFor + \State \textbf{return} $\textbf{C}$ + \EndFunction + \end{algorithmic} +\end{algorithm} + +Die Laufzeit dieser Struktur mit drei \texttt{For} Schleifen ist $\mathcal{O}(n^3)$ + +\subsubsection{Divide and Conquer Methode} + +F\"ur gewisse Algorithmen f\"uhren \textit{Divide and Conquer} Ans\"atze zu markant besseren Laufzeiten. +Das bekannteste Beispiel ist wohl die \textit{Fast Fourier Transform} wobei die Laufzeit von $\mathcal{O}(n^2)$ zu $\mathcal{O}(n \log n)$ verbessert werden kann. + +Die Matrizenmultiplikation kann ebenfalls mit solch einem Ansatz berechnet werden. +Zur vereinfachten Veranschaulichung kann die Situation, mit $\mathbf{A}$ und $\mathbf{B}$ der gr\"osse $2^n \times 2^n$ verwendet werden. +Die Matrizen $\mathbf{A}$ und $\mathbf{B}$ werden in jeweils vier Blockmatrizen der gr\"osse $2^{n-1} \times 2^{n-1}$ +\begin{equation} +\mathbf{A}\mathbf{B}= +\begin{bmatrix} +\mathbf{A}_{11} & \mathbf{A}_{12}\\ +\mathbf{A}_{21} & \mathbf{A}_{22} +\end{bmatrix} +\begin{bmatrix} +\mathbf{B}_{11} & \mathbf{B}_{12}\\ +\mathbf{B}_{21} & \mathbf{B}_{22} +\end{bmatrix} += +\begin{bmatrix} +\mathbf{C}_{11} & \mathbf{C}_{12}\\ +\mathbf{C}_{21} & \mathbf{C}_{22} +\end{bmatrix} +\end{equation} +aufgeteilt. +Die Berechnung +\begin{equation} +\mathbf{C}_{ij} = \sum_{k=1}^n \mathbf{A}_{ik} \mathbf{B}_{kj} +\label{multiplikation:eq:MM_block} +\end{equation} +ist identisch zu der Gleichung \eqref{multiplikation:eq:MM}, wobei hier f\"ur die Multiplikation die Matrizenmultiplikation verwendet wird. + +Der Algorithmus \ref{multiplikation:alg:devide_mm} zeigt den \textit{Divide and Conquer} Ansatz, +Der Grundstruktur dieser Methode besteht aus dem rekursiven Aufruf der Funktion mit den erzeugten Blockmatrizen. +Der rekursive Aufruf wird bis zu der Gr\"osse der Matrizen von $N = 2 \times 2$ durchgef\"uhrt. +\begin{algorithm}\caption{Divide and Conquer Matrix Multiplication} + \setlength{\lineskip}{7pt} + \label{multiplikation:alg:devide_mm} + \begin{algorithmic} + \Function{MM}{$\textbf{A}, \textbf{B}, n$} + \If{$n = 2$} + \State $ \mathbf{C} \gets zeros(n, n)$ + \State $C[0, 0] \gets A[0][0]\cdot B[0][0]+A[0][1]\cdot B[1][0]$ + \State $C[0, 1] \gets A[0][0]\cdot B[0][1]+A[0][1]\cdot B[1][1]$ + \State $C[1, 0] \gets A[1][0]\cdot B[0][0]+A[1][1]\cdot B[1][0]$ + \State $C[1, 1] \gets A[1][0]\cdot B[0][1]+A[1][1]\cdot B[1][1]$ + \Else + \State $ m \gets n/2$ + \State $\mathbf{A11}, \mathbf{A12}, \mathbf{A21}, \mathbf{A22} \gets \mathbf{A}[:m][:m], \mathbf{A}[:m][m:], \mathbf{A}[m:][:m], \mathbf{A}[m:][m:]$ + \State $\mathbf{B11}, \mathbf{B12}, \mathbf{B21}, \mathbf{B22} \gets \mathbf{B}[:m][:m], \mathbf{B}[:m][m:], \mathbf{B}[m:][:m], \mathbf{B}[m:][m:]$ + + \State $\mathbf{C11} \gets \text{MM}(\mathbf{A11}, \mathbf{B11},n) + \text{MM}(\mathbf{A12}, \mathbf{B21},n)$ + \State $\mathbf{C12} \gets \text{MM}(\mathbf{A11},\mathbf{B12},n) + \text{MM}(\mathbf{A12}, \mathbf{B22},n)$ + \State $\mathbf{C21} \gets \text{MM}(\mathbf{A21}, \mathbf{B11},n) + \text{MM}(\mathbf{A22}, \mathbf{B21},n)$ + \State $\mathbf{C22} \gets \text{MM}(\mathbf{A21}, \mathbf{B12},n) + \text{MM}(\mathbf{A22}, \mathbf{B22},n)$ + \State $ C \gets vstack(hstack(C11, C12), hstack(C21, C22))$ + + \EndIf + \State \textbf{return} $\textbf{C}$ + + \EndFunction + \end{algorithmic} +\end{algorithm} + +Die Laufzeit dieser rekursiven Funktion kann mit dem \textit{Master Theorem} berechnet werden. +Ohne auf diesen vertieft einzugehen, bestimmt die Anzahl rekursiver Aufrufe der Funktion die Laufzeit. +In diesem Fall wird die Funktion pro Durchlauf acht mal rekursiv aufgerufen, dies f\"uhrt +\begin{equation} \label{multiplikation:eq:laufzeitdac} + \mathcal{T}(n) = + \begin{cases} + 1 & \text{if } n \leq 2\\ + 8 \cdot \mathcal{T}(\frac{n}{2}) + n^2 & \text{if } n > 2 + \end{cases} = \mathcal{O}(n^{\log_2 8}) = \mathcal{O}(n^{3}) +\end{equation} +zu einer kubischen Laufzeit. +Die Addition zweier Matrizen $\mathbf{A} + \mathbf{B} = \mathbf{C}$ hat eine Laufzeit von $\mathcal{O}(n^{2})$ und kann neben dem dominierendem Anteil von $\mathcal{O}(n^{3})$ ignoriert werden. +In diesem Fall hat der \textit{Divide and Conquer} Ansatz zu keiner Verbesserung gef\"uhrt. + + +\subsection{Strassen's Algorithmus} + +Strassen's Algorithmus \cite{multiplikation:strassen_1969} beschreibt die Matrizenmultiplikation mit einer Vielzahl von Additionen, Subtraktionen und Multiplikationen. +Die Grundlegenden Terme +\begin{equation} \label{multiplikation:eq:strassen} +\begin{split} +\text{\textbf{P}} &= (\mathbf{A}_{11} + \mathbf{A}_{22}) \cdot (\mathbf{B}_{11} + \mathbf{B}_{22}) \\ +\text{\textbf{Q}} &= (\mathbf{A}_{21} + \mathbf{A}_{22}) \cdot \mathbf{B}_{11} \\ +\text{\textbf{R}} &= \mathbf{A}_{11} \cdot (\mathbf{B}_{12}-\mathbf{B}_{22}) \\ +\text{\textbf{S}} &= \mathbf{A}_{22} \cdot (-\mathbf{B}_{11}+\mathbf{B}_{21}) \\ +\text{\textbf{T}} &= (\mathbf{A}_{11} + \mathbf{A}_{12}) \cdot \mathbf{B}_{22} \\ +\text{\textbf{U}} &= (-\mathbf{A}_{11} + \mathbf{A}_{21}) \cdot (\mathbf{B}_{11} + \mathbf{B}_{12}) \\ +\text{\textbf{V}} &= (\mathbf{A}_{12} - \mathbf{A}_{22}) \cdot (\mathbf{B}_{21} + \mathbf{B}_{22}) +\end{split} +\end{equation} +aus $\mathbf{A}$ und $\mathbf{B}$, werden f\"ur die Berechnung der Matrix $\mathbf{C}$ +\begin{equation} \label{multiplikation:eq:strassen2} +\begin{split} +\mathbf{C}_{11} &= \text{\textbf{P}} + \text{\textbf{S}} - \text{\textbf{T}} + \text{\textbf{V}} \\ +\mathbf{C}_{21} &= \text{\textbf{R}} + \text{\textbf{T}} \\ +\mathbf{C}_{12} &= \text{\textbf{Q}} + \text{\textbf{S}}\\ +\mathbf{C}_{22} &= \text{\textbf{P}} + \text{\textbf{R}} - \text{\textbf{Q}} + \text{\textbf{U}} +\end{split} +\end{equation} +gebraucht. +\begin{algorithm}\caption{Strassen Matrix Multiplication} + \label{multiplikation:alg:strassen} + \setlength{\lineskip}{7pt} + \begin{algorithmic} + \Function{strassen}{$\textbf{A}, \textbf{B}, n$} + \If{$n = 2$} + \State $ \mathbf{C} \gets zeros((n, n))$ + \State $P \gets (A[0][0]+A[1][1])\cdot( B[0][0]+B[1][1])$ + \State $Q \gets (A[1][0]+A[1][1])\cdot B[0][0]$ + \State $R \gets A[0][0]\cdot (B[0][1]-B[1][1])$ + \State $S \gets A[1][1]\cdot (B[1][0]-B[0][0])$ + \State $T \gets (A[0][0]+A[0][1])\cdot B[1][1]$ + \State $U \gets (A[1][0]-A[0][0])\cdot (B[0][0]+B[0][1])$ + \State $V \gets (A[0][1]-A[1][1])\cdot (B[1][0]+B[1][1])$ + \State $C[0][0] \gets P+S-T+V$ + \State $C[0][1] \gets R+T$ + \State $C[1][0] \gets Q+S$ + \State $C[1][1] \gets P+R-Q+U$ + \Else + \State $ m \gets n/2$ + \State $\mathbf{A11}, \mathbf{A12}, \mathbf{A21}, \mathbf{A22} \gets \mathbf{A}[:m][:m], \mathbf{A}[:m][m:], \mathbf{A}[m:][:m], \mathbf{A}[m:][m:]$ + \State $\mathbf{B11}, \mathbf{B12}, \mathbf{B21}, \mathbf{B22} \gets \mathbf{B}[:m][:m], \mathbf{B}[:m][m:], \mathbf{B}[m:][:m], \mathbf{B}[m:][m:]$ + + \State $ \mathbf{P} \gets \text{strassen}((\mathbf{A11}+ \mathbf{A22}),(\mathbf{B11}+\mathbf{B22}), m)$ + \State $ \mathbf{Q} \gets \text{strassen}((\mathbf{A21}+ \mathbf{A22}), \mathbf{B11},m)$ + \State $ \mathbf{R} \gets \text{strassen}( \mathbf{A11},(\mathbf{B12}- \mathbf{B22}),m)$ + \State $ \mathbf{S} \gets \text{strassen}( \mathbf{A22},(\mathbf{B21}- \mathbf{B11}),m)$ + \State $ \mathbf{T} \gets \text{strassen}((\mathbf{A11}+ \mathbf{A12}), \mathbf{B22},m)$ + \State $ \mathbf{U} \gets \text{strassen}((\mathbf{A21}- \mathbf{A11}),(\mathbf{B11}+\mathbf{B12}),m)$ + \State $ \mathbf{V} \gets \text{strassen}((\mathbf{A12}- \mathbf{A22}),(\mathbf{B21}+\mathbf{B22}),m)$ + + + + \State $\mathbf{C11} \gets \mathbf{P+S-T+V}$ + \State $\mathbf{C12} \gets \mathbf{R+T}$ + \State $\mathbf{C21} \gets \mathbf{Q+S}$ + \State $\mathbf{C22} \gets \mathbf{P+R-Q+U}$ + \State $ C \gets vstack(hstack(C11, C12), hstack(C21, C22))$ + + \EndIf + \State \textbf{return} $\textbf{C}$ + + \EndFunction + \end{algorithmic} +\end{algorithm} +Strassens's Methode wird in der Abbildung \ref{multiplikation:fig:strassen} grafisch dargestellt. +\begin{figure} + \center + \includegraphics[width=\linewidth]{papers/multiplikation/images/strassen.pdf} + \caption{Strassen's Algorithmus} + \label{multiplikation:fig:strassen} +\end{figure} + +Die Funktion wird sieben mal rekursiv aufgerufen. +Dies f\"uhrt zu einer Laufzeit von +\begin{equation} \label{multiplikation:eq:laufzeitstrassen} +\mathcal{T}(n) = +\begin{cases} +1 & \text{if } n \leq 2\\ +7 \cdot \mathcal{T}(\frac{n}{2}) + n^2 & \text{if } n > 2 +\end{cases} = \mathcal{O}(n^{\log_2 7}) = \mathcal{O}(n^{2.8074}) +\end{equation} +und ist somit schneller als die Standard Methode. + +\subsection{Winograd's Algorithmus} + +Ein weiterer Ansatz lieferte Shmuel Winograd im Jahre 1968 \cite{multiplikation:winograd_1968}. +Er zeigte einen neuen Algorithmus f\"ur das +\begin{equation} + \langle x,y \rangle = \sum_{i=1}^{n}x_i y_i +\end{equation} +Skalarprodukt. +F\"ur jeden Vektor berechne +\begin{equation} + \xi = \sum_{j=1}^{ \lfloor n/2 \rfloor} x_{2j-1} \cdot x_{2j} +\end{equation} +und +\begin{equation} + \eta = \sum_{j=1}^{ \lfloor n/2 \rfloor} y_{2j-1} \cdot y_{2j}. +\end{equation} +Das Skalarprodukt ist nun geben mit +\begin{equation} + \langle x,y \rangle = + \begin{cases} + \displaystyle \quad \sum_{j=1}^{ \lfloor n/2 \rfloor} (x_{2j-1} + y_{2j})(x_{2j}+y_{2j-1})-\xi - \eta & \text{if $n$ is even}\\ + \displaystyle \quad \sum_{j=1}^{ \lfloor n/2 \rfloor} (x_{2j-1} + y_{2j})(x_{2j}+y_{2j-1})-\xi - \eta + x_n y_n & \text{if $n$ is odd}. + \end{cases} +\end{equation} + +Angenommen man hat $N$ Vektoren mit welchen man $T$ Skalarprodukte berechnen m\"ochte. +Daf\"ur werden $N\lfloor n/2 \rfloor + T\lfloor (n+1)/2 \rfloor $ Multiplikationen ben\"otigt. +Eine Matrizenmultiplikation mit $\mathbf{A}$ einer $m \times n$ und $\mathbf{B}$ einer $n \times p$ Matrix, entspricht $N=m+p$ Vektoren mit welchen man $T=mp$ Skalarprodukte berechnet. +Dies f\"uhrt zu +\begin{equation} + (m+p) \left \lfloor \frac{n}{2} \right \rfloor + mp \left \lfloor \frac{n+1}{2} \right \rfloor = \frac{mn}{2} + \frac{pn}{2} + \frac{mpn}{2} + \frac{mp}{2} +\end{equation} +Multiplikationen. +Wenn $m,p,n$ gross werden, dominiert der Term $\frac{mpn}{2}$ und es werden $\frac{mpn}{2}$ Multiplikationen ben\"otigt. +Was im Vergleich zu den $mpn$ Multiplikation der Standard Methode nur die H\"alfte ist. +Die Implementation kann im Algorithmus \ref{multiplikation:alg:winograd} entnommen werden. + +\begin{algorithm}\caption{Winograd Matrix Multiplication} + \setlength{\lineskip}{7pt} + \label{multiplikation:alg:winograd} + \begin{algorithmic} + \Function{Winograd}{$\textbf{A}, \textbf{B}, n$} + \State $ m \gets rows(\mathbf{A})$ + \State $ n \gets columns(\mathbf{A}) == rows(\mathbf{B})$ + \State $ p \gets columns(\mathbf{B})$ + \State $ \mathbf{\xi} \gets zeros(m)$ + \State $ \mathbf{\eta} \gets zeros(p)$ + + + \For{$i = 0,1,2 \dots,m-1$} + \For{$j = 0,1,2 \dots,\lfloor n/2 \rfloor-1$} + \State $\xi[i] \gets \xi[i]+A[i,2 j]A[i,2 j+1]$ + \EndFor + \EndFor + + \For{$i = 0,1,2 \dots,p-1$} + \For{$j = 0,1,2 \dots,\lfloor n/2 \rfloor-1$} + \State $\eta[i] \gets \eta[i]+B[2 j,i]B[2 j+1,i]$ + \EndFor + \EndFor + + \If{$n \% 2 == 0$} + \For{$i = 0,1,2 \dots,m-1$} + \For{$j = 0,1,2 \dots,p-1$} + \State $ab \gets 0$ + \For{$k = 0,1,2 \dots,\lfloor n/2 \rfloor-1$} + \State $ab \gets ab + (A[i,2k]+B[2k+1,j])(A[i,2k+1]+B[2k,j])$ + \EndFor + \State $C[i,j] \gets ab-\eta[j]-\xi[i]$ + \EndFor + \EndFor + \Else + \For{$i = 0,1,2 \dots,n-1$} + \For{$j = 0,1,2 \dots,n-1$} + \State $ab \gets 0$ + \For{$k = 0,1,2 \dots,\lfloor n/2 \rfloor-1$} + \State $ab \gets ab + (A[i,2k]+B[2k+1,j])(A[i,2k+1]+B[2k,j])$ + \EndFor + \State $C[i,j] \gets ab-\eta[j]-\xi[i]+A[i,-1]B[-1,j]$ + \EndFor + \EndFor + \EndIf + \State \textbf{return} $\textbf{C}$ + + \EndFunction + \end{algorithmic} +\end{algorithm} + +\subsection{Weitere Algorithmen} + +\textcolor{red}{TODO: BLAS} + +\section{Implementation} +\rhead{Implementation} +\textcolor{red}{TODO: messresultate} + +\section{Fazit} +\rhead{Fazit} diff --git a/buch/papers/multiplikation/main.tex b/buch/papers/multiplikation/main.tex index 42f2768..8d0a8df 100644..100755 --- a/buch/papers/multiplikation/main.tex +++ b/buch/papers/multiplikation/main.tex @@ -1,36 +1,18 @@ +% !TEX root = ../../buch.tex % % main.tex -- Paper zum Thema <multiplikation> % -% (c) 2020 Hochschule Rapperswil +% (c) 2021 Hochschule Rapperswil % -\chapter{Thema\label{chapter:multiplikation}} -\lhead{Thema} +\chapter{Schnelle Matrizen Multiplikation\label{chapter:multiplikation}} +\lhead{FMM} \begin{refsection} -\chapterauthor{Hans Muster} +\chapterauthor{Michael Schmid} -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} -\input{papers/multiplikation/teil0.tex} -\input{papers/multiplikation/teil1.tex} -\input{papers/multiplikation/teil2.tex} -\input{papers/multiplikation/teil3.tex} +\input{papers/multiplikation/einlteung.tex} +\input{papers/multiplikation/problemstellung.tex} +\input{papers/multiplikation/loesungsmethoden.tex} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/multiplikation/packages.tex b/buch/papers/multiplikation/packages.tex index e4173c0..e4173c0 100644..100755 --- a/buch/papers/multiplikation/packages.tex +++ b/buch/papers/multiplikation/packages.tex diff --git a/buch/papers/multiplikation/papers/Strassen_GPU.pdf b/buch/papers/multiplikation/papers/Strassen_GPU.pdf Binary files differnew file mode 100755 index 0000000..4ce7625 --- /dev/null +++ b/buch/papers/multiplikation/papers/Strassen_GPU.pdf diff --git a/buch/papers/multiplikation/papers/Strassen_original_1969.pdf b/buch/papers/multiplikation/papers/Strassen_original_1969.pdf Binary files differnew file mode 100755 index 0000000..b647fc0 --- /dev/null +++ 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file mode 100644 index 0000000..200d244 --- /dev/null +++ b/buch/papers/multiplikation/presentation/common.tex @@ -0,0 +1,79 @@ +% +% common.tex -- gemeinsame Definitionen +% +% (c) 2021 Michael Schmid, OST Campus Rapperswil +% +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{epic} +\usepackage{color} +\usepackage{array} +\usepackage{algorithm} +\usepackage{ifthen} +\usepackage{adjustbox} +\usepackage[noend]{algpseudocode} +\usepackage{neuralnetwork} +\usepackage{amsmath} +\usepackage{lmodern} +\usepackage{tikz} +\usetikzlibrary{decorations.text} +\usetikzlibrary{arrows,matrix,positioning} +\usetikzlibrary{overlay-beamer-styles} +\usetikzlibrary{matrix.skeleton} +\usepackage{pgfplots} +\usepackage{listings} +\usepackage{svg} + +\definecolor{codegreen}{rgb}{0,0.6,0} +\definecolor{codegray}{rgb}{0.5,0.5,0.5} +\definecolor{codepurple}{rgb}{0.58,0,0.82} +\definecolor{backcolour}{rgb}{0.95,0.95,0.92} +\definecolor{ost}{rgb}{164,0,136} + +\lstdefinestyle{mystyle}{ + backgroundcolor=\color{backcolour}, + commentstyle=\color{codegreen}, + keywordstyle=\color{magenta}, + numberstyle=\tiny\color{codegray}, + stringstyle=\color{codepurple}, + basicstyle=\footnotesize, + breakatwhitespace=false, + breaklines=true, + captionpos=b, + keepspaces=true, + numbers=left, + numbersep=2pt, + showspaces=false, + showstringspaces=false, + showtabs=false, + tabsize=2 +} + +\usetikzlibrary{fit} +\tikzset{% + highlight/.style={rectangle,rounded corners,fill=red!15,draw,fill opacity=0.5,inner sep=0pt} +} +\newcommand{\tikzmark}[2]{\tikz[overlay,remember picture,baseline=(#1.base)] \node (#1) {#2};} +% +\newcommand{\Highlight}[1][submatrix]{% + \tikz[overlay,remember picture]{ + \node[highlight,fit=(left.north west) (right.south east)] (#1) {};} +} + + +\lstset{style=mystyle} +\lstdefinestyle{mystyle}{ + morekeywords={cwt,contourf,datetick} +} + + +\usetikzlibrary{shapes.geometric} +\mode<beamer>{% +\usetheme[]{Frankfurt}} +\beamertemplatenavigationsymbolsempty +\title[]{Fast Matrix Multiplication} +\author[]{Michael Schmid} +\usecolortheme[named=ost]{structure} + +\date[]{31.05.2021} +\newboolean{presentation} diff --git a/buch/papers/multiplikation/presentation/presentation.nav b/buch/papers/multiplikation/presentation/presentation.nav new file mode 100644 index 0000000..2a01568 --- /dev/null +++ b/buch/papers/multiplikation/presentation/presentation.nav @@ -0,0 +1,59 @@ +\headcommand {\slideentry {0}{0}{1}{1/1}{}{0}} +\headcommand {\beamer@framepages {1}{1}} +\headcommand {\beamer@sectionpages {1}{1}} +\headcommand {\beamer@subsectionpages {1}{1}} +\headcommand {\sectionentry {1}{Big $\mathcal {O}$}{2}{Big $\mathcal {O}$}{0}} +\headcommand {\slideentry {1}{0}{1}{2/4}{}{0}} +\headcommand {\beamer@framepages {2}{4}} +\headcommand {\slideentry {1}{0}{2}{5/6}{}{0}} +\headcommand {\beamer@framepages {5}{6}} +\headcommand {\slideentry {1}{0}{3}{7/8}{}{0}} +\headcommand {\beamer@framepages {7}{8}} +\headcommand {\slideentry {1}{0}{4}{9/10}{}{0}} 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0000000..2a4af45 --- /dev/null +++ b/buch/papers/multiplikation/presentation/presentation.tex @@ -0,0 +1,12 @@ +% +% MathSem-yyy-xxx.tex -- Präsentation +% +% (c) 2021 Michael Schmid, OST campus Rapperswil +% + +\documentclass[aspectratio=169]{beamer} +\input{common.tex} +%\setboolean{presentation}{true} +\begin{document} +\input{slides/slides.tex} +\end{document} diff --git a/buch/papers/multiplikation/presentation/slides/algo.tex b/buch/papers/multiplikation/presentation/slides/algo.tex new file mode 100644 index 0000000..0c3d130 --- /dev/null +++ b/buch/papers/multiplikation/presentation/slides/algo.tex @@ -0,0 +1,111 @@ +\begin{frame} + \frametitle{Algorithm} + \begin{columns} + \begin{column}{0.6\textwidth} + \begin{algorithm}[H]\caption{Square Matrix Multiplication} + \setlength{\lineskip}{7pt} + \begin{algorithmic}[1] + \Function{MM}{$\textbf{A}, \textbf{B}, \textbf{C}$} + \State $sum \gets 0$ + \State $n \gets columns(\textbf{A}) == rows(\textbf{B})$ + \State $m \gets rows(\textbf{A})$ + \State $p \gets columns(\textbf{B})$ + + \For{$i = 0,1,2 \dots,m-1$} + \For{$j = 0,1,2 \dots,p-1$} + \State $sum \gets 0$ + \For{$k = 0,1,2 \dots,n-1$} + \State $sum \gets sum + \textbf{A}[i][k] \cdot \textbf{B}[k][j]$ + \EndFor + \State $\textbf{C}[i][j] \gets sum $ + \EndFor + \EndFor + \State \textbf{return} $\textbf{C}$ + \EndFunction + \end{algorithmic} + \end{algorithm} +\end{column} +\begin{column}{0.4\textwidth} + \scalebox{0.6}{\parbox{\linewidth}{ + + \begin{tikzpicture}[ampersand replacement=\&,remember picture,overlay] + + \matrix (A)[matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}] at (2,-2.8) + { + A_{1,1} \& \cdots \& A_{1,k} \& \cdots \& A_{1,n} \\ + \vdots \& \& \vdots \& \& \vdots \\ + A_{i,1} \& \cdots \& A_{i,k} \& \cdots \& A_{i,n} \\ + \vdots \& \& \vdots \& \& \vdots \\ + A_{m,1} \& \cdots \& A_{m,k} \& \cdots \& A_{m,n} \\ + }; + + \matrix (B)[matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}] at (7.5,1.2) + { + B_{1,1} \& \cdots \& B_{1,j} \& \cdots \& B_{1,p} \\ + \vdots \& \& \vdots \& \& \vdots \\ + B_{k,1} \& \cdots \& B_{k,j} \& \cdots \& B_{k,p} \\ + \vdots \& \& \vdots \& \& \vdots \\ + B_{n,1} \& \cdots \& B_{n,j} \& \cdots \& B_{n,p} \\ + }; + + \matrix (C)[matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}] at (7.5,-2.8) + { + C_{1,1} \& \cdots \& C_{1,j} \& \cdots \& C_{1,p} \\ + \vdots \& \& \vdots \& \& \vdots \\ + C_{i,1} \& \cdots \& C_{i,j} \& \cdots \& C_{i,p} \\ + \vdots \& \& \vdots \& \& \vdots \\ + C_{m,1} \& \cdots \& C_{m,j} \& \cdots \& C_{m,p} \\ + }; + + + \begin{scope}[on background layer] + \node[opacity=0.5, rounded corners=2pt, inner sep=-1pt, fill=green, fit=(A-3-1)(A-3-5)] {}; + \node[opacity=0.5, rounded corners=2pt, inner sep=-1pt, fill=blue, fit=(B-1-3)(B-5-3)] {}; + \node[opacity=0.5, rounded corners=2pt, inner sep=-1pt, fill=red, fit=(C-3-3)] {}; + + \end{scope} + + + + + \end{tikzpicture} + }} + \end{column} +\end{columns} +\end{frame} + + +\begin{frame} + \frametitle{Algorithm} + +\begin{columns} + \begin{column}{0.6\textwidth} +\begin{algorithm}[H]\caption{Square Matrix Multiplication} + \setlength{\lineskip}{7pt} + \begin{algorithmic}[1] + \Function{MM}{$\textbf{A}, \textbf{B}, \textbf{C}$} + \State $sum \gets 0$ + \State $n \gets columns(\textbf{A}) == rows(\textbf{B})$ + \State $m \gets rows(\textbf{A})$ + \State $p \gets columns(\textbf{B})$ + + \For{$i = 0,1,2 \dots,m-1$} + \For{$j = 0,1,2 \dots,p-1$} + \State $sum \gets 0$ + \For{$k = 0,1,2 \dots,n-1$} + \State $sum \gets sum + \textbf{A}[i][k] \cdot \textbf{B}[k][j]$ + \EndFor + \State $\textbf{C}[i][j] \gets sum $ + \EndFor + \EndFor + \State \textbf{return} $\textbf{C}$ + \EndFunction + \end{algorithmic} +\end{algorithm} +\end{column} +\begin{column}{0.4\textwidth} +\Huge$\mathcal{O}(n^3)$ +\end{column} +\end{columns} + +\end{frame} diff --git a/buch/papers/multiplikation/presentation/slides/bigO.tex b/buch/papers/multiplikation/presentation/slides/bigO.tex new file mode 100644 index 0000000..d425da8 --- /dev/null +++ b/buch/papers/multiplikation/presentation/slides/bigO.tex @@ -0,0 +1,251 @@ + +\begin{frame} + \frametitle{Big $\mathcal{O}$ notation} +\begin{itemize} + \item <1-> Time complexity of an algorithm + \item <2-> How many multiplications in a function + \item <3-> Drop Constants +\end{itemize} +\end{frame} + + +\begin{frame} + \frametitle{Big $\mathcal{O}$ notation} + \onslide<1->{ + + \begin{algorithm}[H]\caption{Foo 1} + \setlength{\lineskip}{7pt} + \begin{algorithmic}[1] + \Function{foo}{$a, b$} + \State \textbf{return} $a+b$ + \EndFunction + \end{algorithmic} + \end{algorithm} +} +\onslide<2->{ +$\mathcal{O}(1)$ + } +\end{frame} + +\begin{frame} + \frametitle{Big $\mathcal{O}$ notation} + \onslide<1->{ + + \begin{algorithm}[H]\caption{Foo 2} + \setlength{\lineskip}{7pt} + \begin{algorithmic}[1] + \Function{foo}{$a, b$} + \State $ x \gets a+b $ + \State $ y \gets a \cdot b $ + \State \textbf{return} $x+y$ + \EndFunction + \end{algorithmic} + \end{algorithm} +} +\onslide<2->{ +$\mathcal{O}(1) + \mathcal{O}(1) = 2\mathcal{O}(1) = \mathcal{O}(1) $ + } +\end{frame} + +\begin{frame} + \frametitle{Big $\mathcal{O}$ notation} + \onslide<1->{ + + \begin{algorithm}[H]\caption{Foo 3} + \setlength{\lineskip}{7pt} + \begin{algorithmic}[1] + \Function{foo}{$\mathbf{A}, \mathbf{B}$,n} + \State $ sum \gets 0$ + \For{$i = 0,1,2 \dots,n$} + \State $ sum \gets sum + A[i] \cdot B[i] $ + \EndFor + + \State \textbf{return} $sum$ + + \EndFunction + \end{algorithmic} + \end{algorithm} +} +\onslide<2->{ +$\mathcal{O}(n)$ + } +\end{frame} + +\begin{frame} + \frametitle{Big $\mathcal{O}$ notation} + \onslide<1->{ + + \begin{algorithm}[H]\caption{Foo 4} + \setlength{\lineskip}{7pt} + \begin{algorithmic}[1] + \Function{foo}{$\mathbf{A}, \mathbf{B}$,n} + \State $ sum \gets 0$ + \For{$i = 0,1,2 \dots,n$} + \For{$j = 0,1,2 \dots,n$} + \State $ sum \gets sum + A[i] \cdot B[j] $ + \EndFor + \EndFor + \State \textbf{return} $sum$ + \EndFunction + \end{algorithmic} + \end{algorithm} +} +\onslide<2->{ +$\mathcal{O}(n^2)$ + } +\end{frame} + +% \begin{frame} +% \frametitle{Big $\mathcal{O}$ notation} +% \onslide<1->{ +% +% \begin{algorithm}[H]\caption{Fibonacci} +% \setlength{\lineskip}{7pt} +% \begin{algorithmic}[1] +% \Function{fib}{$n$} +% \If{$n <= 1$} +% \State \textbf{return} $1$ +% \Else +% \State \textbf{return} fib($n-1$) + fib($n-2$) +% \EndIf +% +% \EndFunction +% \end{algorithmic} +% \end{algorithm} +% } +% \onslide<2->{ +% \[ +% \langle x,y \rangle = +% \begin{cases} +% \displaystyle $\mathcal{O}(1)$ & \text{if $n \leq 2$}\\ +% \displaystyle $ 2 \mathcal{T}(\frac{n}{2})$ & \text{if $n > 2$} +% \end{cases} +% \] } +% \end{frame} + + +\begin{frame} + \frametitle{Big $\mathcal{O}$ notation} +\begin{tikzpicture} +\begin{axis}[ + axis lines = left, + xlabel = $n$ (Data Input), + ylabel = {$t$ (time)}, + legend pos=north east, + very thick, + ymax = 20, + yticklabels=\empty, + xticklabels=\empty, + scale only axis=true, + width=12cm, height=6cm, + ] +%Below the red parabola is defined +\addplot [ + domain= 1:6, + samples=100, + color=red, +] +{1}; +\addlegendentry{$\mathcal{O}(1)$} +%Here the blue parabloa is defined +\addplot [ + domain= 1:6, + samples=100, + color=green, +] +{x}; +\addlegendentry{$\mathcal{O}(n)$} +\addplot [ + domain= 1:6, + samples=100, + color=blue, +] +{x^2}; +\addlegendentry{$\mathcal{O}(n^2)$} +\addplot [ + domain= 1:6, + samples=100, + color=purple, +] +{x^3}; +\addlegendentry{$\mathcal{O}(n^3)$} +\addplot [ + domain= 1:3, + samples=100, + color=black, +] +{exp(x)}; +\addlegendentry{$\mathcal{O}(e^n)$} +\addplot [ + domain= 1:6, + samples=100, + color=orange, +] +{log2(x)}; +\addlegendentry{$\mathcal{O}(\log n)$} +\end{axis} +\end{tikzpicture} + +\end{frame} + +\begin{frame} + \frametitle{Big $\mathcal{O}$ notation} +\begin{tikzpicture} +\begin{axis}[ + axis lines = left, + xlabel = $n$ (Data Input), + ylabel = {$t$ (time)}, + legend pos=north east, + very thick, + ymax = 500, + yticklabels=\empty, + xticklabels=\empty, + scale only axis=true, + width=12cm, height=6cm, + ] +\addplot [ + domain= 1:20, + samples=100, + color=red, +] +{1}; +\addlegendentry{$\mathcal{O}(1)$} +\addplot [ + domain= 1:20, + samples=100, + color=green, +] +{x}; +\addlegendentry{$\mathcal{O}(n)$} +\addplot [ + domain= 1:20, + samples=100, + color=blue, +] +{x^2}; +\addlegendentry{$\mathcal{O}(n^2)$} +\addplot [ + domain= 1:10, + samples=100, + color=purple, +] +{x^3}; +\addlegendentry{$\mathcal{O}(n^3)$} +\addplot [ + domain= 1:10, + samples=100, + color=black, +] +{exp(x)}; +\addlegendentry{$\mathcal{O}(e^n)$} +\addplot [ + domain= 1:20, + samples=100, + color=orange, +] +{log2(x)}; +\addlegendentry{$\mathcal{O}(\log n)$} +\end{axis} +\end{tikzpicture} + +\end{frame} diff --git a/buch/papers/multiplikation/presentation/slides/blas.tex b/buch/papers/multiplikation/presentation/slides/blas.tex new file mode 100644 index 0000000..ed498a3 --- /dev/null +++ b/buch/papers/multiplikation/presentation/slides/blas.tex @@ -0,0 +1,18 @@ +\begin{frame} +\frametitle{BLAS, LAPACK} +\begin{itemize} + \item Basic Linear Algebra Subprograms + \begin{itemize} + \item $\mathbf{y} = \alpha \mathbf{x}+\mathbf{y}$ + \item $\mathbf{y} = \alpha \mathbf{A}\mathbf{x}+ \beta \mathbf{y}$ + \item $\mathbf{C} = \alpha \mathbf{A}\mathbf{B}+ \beta \mathbf{C}$ + + \end{itemize} + \item Linear Algebra Package + \begin{itemize} + \item QR decomposition + \item Singular value decomposition + \item Eigenvalues + \end{itemize} +\end{itemize} +\end{frame} diff --git a/buch/papers/multiplikation/presentation/slides/conclusuion.tex b/buch/papers/multiplikation/presentation/slides/conclusuion.tex new file mode 100644 index 0000000..e69de29 --- /dev/null +++ b/buch/papers/multiplikation/presentation/slides/conclusuion.tex diff --git a/buch/papers/multiplikation/presentation/slides/logo.pdf b/buch/papers/multiplikation/presentation/slides/logo.pdf Binary files differnew file mode 100644 index 0000000..d78ca88 --- /dev/null +++ b/buch/papers/multiplikation/presentation/slides/logo.pdf diff --git a/buch/papers/multiplikation/presentation/slides/meas.tex b/buch/papers/multiplikation/presentation/slides/meas.tex new file mode 100644 index 0000000..489c010 --- /dev/null +++ b/buch/papers/multiplikation/presentation/slides/meas.tex @@ -0,0 +1,42 @@ +\begin{frame} + \frametitle{Measurements Python} + \only<1>{ + \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/meas_8.pdf}} + \only<2>{ + \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/meas_16.pdf}} + \only<3>{ + \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/meas_32.pdf}} + \only<4>{ + \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/meas_64.pdf}} + \only<5>{ + \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/meas_128.pdf}} + \only<6>{ + \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/meas_256.pdf}} + \only<7>{ + \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/meas_512.pdf}} + \only<8>{ + \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/meas_1024.pdf}} +\end{frame} + + +\begin{frame} + \frametitle{Measurements C} + \only<1>{ + \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/c_meas_8.pdf}} + \only<2>{ + \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/c_meas_16.pdf}} + \only<3>{ + \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/c_meas_32.pdf}} + \only<4>{ + \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/c_meas_64.pdf}} + \only<5>{ + \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/c_meas_128.pdf}} + \only<6>{ + \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/c_meas_256.pdf}} + \only<7>{ + \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/c_meas_512.pdf}} + \only<8>{ + \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/c_meas_1024.pdf}} + \only<9>{ + \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/c_meas_2048.pdf}} +\end{frame} diff --git a/buch/papers/multiplikation/presentation/slides/nn.tex b/buch/papers/multiplikation/presentation/slides/nn.tex new file mode 100644 index 0000000..e74e970 --- /dev/null +++ b/buch/papers/multiplikation/presentation/slides/nn.tex @@ -0,0 +1,97 @@ + +\begin{frame} + \frametitle{Neural Network} + \centering +\newcommand{\inputnum}{4} + +% Hidden layer neurons'number +\newcommand{\hiddennumA}{5} +\newcommand{\hiddennumB}{6} + +% Output layer neurons'number +\newcommand{\outputnum}{4} + +\begin{tikzpicture} + + +% Input Layer +\foreach \i in {1,...,\inputnum} +{ + \node[circle, + minimum size = 6mm, + fill=blue!30] (Input-\i) at (0,-\i) {}; +} + +% Hidden Layer1 +\foreach \i in {1,...,\hiddennumA} +{ + \node[circle, + minimum size = 6mm, + fill=red!50, + yshift=(\hiddennumA-\inputnum)*5 mm + ] (Hidden1-\i) at (2.5,-\i) {}; +} + +% Hidden Layer2 +\foreach \i in {1,...,\hiddennumB} +{ + \node[circle, + minimum size = 6mm, + fill=red!50, + yshift=(\hiddennumB-\inputnum)*5 mm + ] (Hidden2-\i) at (5,-\i) {}; +} + +% Output Layer +\foreach \i in {1,...,\outputnum} +{ + \node[circle, + minimum size = 6mm, + fill=green!50, + yshift=(\outputnum-\inputnum)*5 mm + ] (Output-\i) at (7.5,-\i) {}; +} + +% Connect neurons In-Hidden +\foreach \i in {1,...,\inputnum} +{ + \foreach \j in {1,...,\hiddennumA} + { + \draw[->, shorten >=1pt] (Input-\i) -- (Hidden1-\j); + } +} + +% Connect neurons In-Hidden +\foreach \i in {1,...,\hiddennumA} +{ + \foreach \j in {1,...,\hiddennumB} + { + \draw[->, shorten >=1pt] (Hidden1-\i) -- (Hidden2-\j); + } +} + +% Connect neurons Hidden-Out +\foreach \i in {1,...,\hiddennumB} +{ + \foreach \j in {1,...,\outputnum} + { + \draw[->, shorten >=1pt] (Hidden2-\i) -- (Output-\j); + } +} + +% Inputs +\foreach \i in {1,...,\inputnum} +{ + \draw[<-, shorten <=1pt] (Input-\i) -- ++(-1,0) + node[left]{\LARGE{$x_{\i}$}}; +} + +% Outputs +\foreach \i in {1,...,\outputnum} +{ + \draw[->, shorten <=1pt] (Output-\i) -- ++(1,0) + node[right]{\LARGE{$y_{\i}$}}; +} + +\end{tikzpicture} +\end{frame} diff --git a/buch/papers/multiplikation/presentation/slides/parcomp.tex b/buch/papers/multiplikation/presentation/slides/parcomp.tex new file mode 100644 index 0000000..1ba39ee --- /dev/null +++ b/buch/papers/multiplikation/presentation/slides/parcomp.tex @@ -0,0 +1,66 @@ +% !TEX root = presentation.tex + +\begin{frame} + \frametitle{Vector-Matrix Multiplication} +\center{ + \begin{tikzpicture}[ampersand replacement=\&] + + \matrix (A)[matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}] + { + A_{1,1} \& A_{1,2} \& A_{1,3} \& A_{1,4} \\ + }; + + \matrix (B)[matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}] at (5,-0.95) + { + B_{1,1} \& B_{1,2} \& B_{1,3} \& B_{1,4} \& B_{1,5} \\ + B_{2,1} \& B_{2,2} \& B_{2,3} \& B_{2,4} \& B_{2,5} \\ + B_{3,1} \& B_{3,2} \& B_{3,3} \& B_{3,4} \& B_{3,5} \\ + B_{4,1} \& B_{4,2} \& B_{4,3} \& B_{4,4} \& B_{4,5} \\ + }; + + \matrix (C)[matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}] at (5,-3) + { + C_{1,1} \& C_{1,2} \& C_{1,3} \& C_{1,4} \& C_{1,5}\\ + }; + + \foreach \i in {1,...,4} + { + \pgfmathtruncatemacro{\ii}{\i+1} + \onslide<\ii>{ + + \foreach \j in {1,...,5} + { + \draw[thick] (A-1-\i.south) to [out=-90,in=135]node[visible on=<\i->, anchor=north]{} (B-\i-\j.center); + + } + } + } + + + \end{tikzpicture} +} +\end{frame} + + +\begin{frame} + \frametitle{DSP Architecture} +\scalebox{2}{ + \begin{tikzpicture} + \node (mul) at (0,0) [circle,draw=black,inner sep=0pt,minimum size=0.5cm] {X}; + \node (mac) at (2,0) [circle,draw=black,inner sep=0pt,minimum size=0.5cm] {\textbf{+}}; + + \node at (-2,0.3) {$A[n]$}; + \node at (0.4,2) {$B[n]$}; + \node at (4,0.3) {$C[n]$}; + + \draw[thick, ->] (-2,0) --++ (mul); + \draw[thick, ->] (0,2) --++ (mul); + \draw[thick, ->] (mul) -- (mac); + \draw[thick] (mac) --++ (1,0) node (i) {}; + \draw[thick, ->] (i.center) --++ (0,1) --++ (-1,0) -- (mac); + \draw[thick, ->] (i.center) --++ (1,0); + + + \end{tikzpicture} + } +\end{frame} diff --git a/buch/papers/multiplikation/presentation/slides/slides.tex b/buch/papers/multiplikation/presentation/slides/slides.tex new file mode 100644 index 0000000..64edb86 --- /dev/null +++ b/buch/papers/multiplikation/presentation/slides/slides.tex @@ -0,0 +1,15 @@ +% !TEX root = presentation.tex +\begin{frame} +\titlepage +\end{frame} +% +\section{Big $\mathcal{O}$} +\input{slides/BigO.tex} +\section{Strassen's Algorithm} +\input{slides/strassen.tex} +% \input{slides/nn.tex} +\section{Measurements} +\input{slides/meas.tex} +% \input{slides/parcomp.tex} +\section{How To Matrix Multiply} +\input{slides/blas.tex} diff --git a/buch/papers/multiplikation/presentation/slides/strassen.tex b/buch/papers/multiplikation/presentation/slides/strassen.tex new file mode 100644 index 0000000..c3398d5 --- /dev/null +++ b/buch/papers/multiplikation/presentation/slides/strassen.tex @@ -0,0 +1,429 @@ +\begin{frame} + \frametitle{Strassen's Algorithm} + \includegraphics[page=1,width=\textwidth,height=0.8\textheight,keepaspectratio]{../papers/Strassen_original_1969.pdf} + \includegraphics[page=2,width=\textwidth,height=0.8\textheight,keepaspectratio]{../papers/Strassen_original_1969.pdf} \includegraphics[page=3,width=\textwidth,height=0.8\textheight,keepaspectratio]{../papers/Strassen_original_1969.pdf} + \end{frame} + +\begin{frame} + \frametitle{Strassen's Algorithm} + \centering + \large +\onslide<1->{ + $ + \mathbf{A B = C} + $ +} + +\onslide<2->{ + + +\medskip + $ + \begin{bmatrix} + A_{11} & A_{12}\\ + A_{21} & A_{22} + \end{bmatrix} + \begin{bmatrix} + B_{11} & B_{12}\\ + B_{21} & B_{22} + \end{bmatrix} + = + \begin{bmatrix} + C_{11} & C_{12}\\ + C_{21} & C_{22} + \end{bmatrix} + $ + } + + + \onslide<3->{ + +\medskip +$ +C_{11} = A_{11} \cdot B_{11} + A_{12} \cdot B_{21} +$ + +$ +C_{12} = A_{11} \cdot B_{12} + A_{12} \cdot B_{22} +$ + +$ +C_{21} = A_{21} \cdot B_{11} + A_{22} \cdot B_{21} +$ + +$ +C_{22} = A_{21} \cdot B_{12} + A_{22} \cdot B_{22} +$ +} +\end{frame} + +\input{slides/algo.tex} + + + +\begin{frame} + \frametitle{Strassen's Algorithm} + \begin{columns} + \begin{column}{0.5\textwidth} + \onslide<1->{ + \large + \begin{math} + \begin{aligned} + \text{I} &= (A_{11} + A_{22}) \cdot (B_{11} + B_{22}) \\ + \text{II} &= (A_{21} + A_{22}) \cdot B_{11} \\ + \text{III} &= A_{11} \cdot (B_{12}-B_{22}) \\ + \text{IV} &= A_{22} \cdot (-B_{11}+B_{21}) \\ + \text{V} &= (A_{11} + A_{12}) \cdot B_{22} \\ + \text{VI} &= (-A_{11} + A_{21}) \cdot (B_{11} + B_{12}) \\ + \text{VII} &= (A_{12} - A_{22}) \cdot (B_{21} + B_{22}) \\ + \end{aligned} + \end{math} + } + \end{column} + + \begin{column}{0.5\textwidth} + \onslide<2->{ + \large + \begin{math} + \begin{aligned} + C_{11} &= \text{I} + \text{IV} - \text{V} + \text{VII} \\ + C_{21} &= \text{II} + \text{IV} \\ + C_{12} &= \text{III} + \text{V}\\ + C_{22} &= \text{I} + \text{III} - \text{II} + \text{VI} \\ + \end{aligned} + \end{math} + } + \end{column} +\end{columns} + +\onslide<3->{ + +\bigskip +\centering +\tiny +\begin{math} +\begin{aligned} + C_{11} &= (A_{11} + A_{22}) \cdot (B_{11} + B_{22}) + A_{22} \cdot (-B_{11}+B_{21}) - (A_{11} + A_{12}) \cdot B_{22} + (A_{12} - A_{22}) \cdot (B_{21} + B_{22}) \\ + C_{11} &= A_{11}B_{11} + A_{11}B_{22} + A_{22}B_{11} + A_{22}B_{22} -A_{22}B_{11}+A_{22}B_{21} - A_{11}B_{22} - A_{12}B_{22}+ A_{12}B_{21} + A_{12}B_{22} - A_{22}B_{21} - A_{22}B_{22} \\ + C_{11} &= A_{11}B_{11} + A_{12}B_{21} +\end{aligned} +\end{math} +} + +\end{frame} + + +\begin{frame} +\begin{adjustbox}{width=\textwidth} +\begin{tikzpicture}[ampersand replacement=\&] + + \foreach \i in {1,...,4} + { + \small{ + \matrix (X\i)[matrix of math nodes,nodes in empty cells, + nodes = {draw, minimum size=10mm, + anchor=center, + inner sep=0pt, outer sep=0pt}, + column sep=-\pgflinewidth, + row sep=-\pgflinewidth, + ] at (0,-\i*5) + { + A_{11}B_{11} \& A_{12}B_{11} \& A_{21}B_{11} \& A_{22}B_{11} \\ + A_{11}B_{21} \& A_{12}B_{21} \& A_{21}B_{21} \& A_{22}B_{21} \\ + A_{11}B_{11} \& A_{12}B_{12} \& A_{21}B_{12} \& A_{22}B_{12} \\ + A_{11}B_{22} \& A_{12}B_{22} \& A_{21}B_{22} \& A_{22}B_{22} \\ + };} + + \foreach \j in {1,...,7} + { + \matrix(M\i\j)[matrix of math nodes,nodes in empty cells, + nodes = {draw, minimum size=10mm, + anchor=center, + inner sep=0pt, outer sep=0pt}, + column sep=-\pgflinewidth, + row sep=-\pgflinewidth, + ] at (\j*5,-\i*5) + { + \& \& \& \\ + \& \& \& \\ + \& \& \& \\ + \& \& \& \\ + }; + } + } + +\huge{ + \node at (-3,-20) {$C_{22}=$}; + \node at (-3,-15) {$C_{21}=$} ; + \node at (-3,-10) {$C_{12}=$} ; + \node at (-3,-5) {$C_{11}=$} ; + + \node at (5,-2) {I}; + \node at (10,-2) {II}; + \node at (15,-2) {III}; + \node at (20,-2) {IV}; + \node at (25,-2) {V}; + \node at (30,-2) {VI}; + \node at (35,-2) {VII}; + } + + + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X1-1-1)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X1-2-2)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X2-3-1)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X2-4-2)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X3-1-3)] {}; + \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X3-2-4)] {}; + \node[opacity=0.5, rounded 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+\end{tikzpicture} +\end{adjustbox} +\end{frame} + + +\begin{frame} + \frametitle{Strassen's Algorithm} + \begin{columns} + \begin{column}{0.5\textwidth} + \large + \begin{math} + \begin{aligned} + \text{I} &= (A_{11} + A_{22}) \cdot (B_{11} + B_{22}) \\ + \text{II} &= (A_{21} + A_{22}) \cdot B_{11} \\ + \text{III} &= A_{11} \cdot (B_{12}-B_{22}) \\ + \text{IV} &= A_{22} \cdot (-B_{11}+B_{21}) \\ + \text{V} &= (A_{11} + A_{12}) \cdot B_{22} \\ + \text{VI} &= (-A_{11} + A_{21}) \cdot (B_{11} + B_{12}) \\ + \text{VII} &= (A_{12} - A_{22}) \cdot (B_{21} + B_{22}) \\ + \end{aligned} + \end{math} + + \end{column} + + \begin{column}{0.5\textwidth} + \large + \begin{math} + \begin{aligned} + C_{11} &= \text{I} + \text{IV} - \text{V} + \text{VII} \\ + C_{21} &= \text{II} + \text{IV} \\ + C_{12} &= \text{III} + \text{V}\\ + C_{22} &= \text{I} + \text{III} - \text{II} + \text{VI} \\ + \end{aligned} + \end{math} + + \end{column} +\end{columns} +\end{frame} + + + +\begin{frame} + \frametitle{Strassen's Algorithm} + +\begin{columns} + \begin{column}{0.5\textwidth} +\large +\begin{math} +\begin{aligned} +\text{\textbf{I}} &= (\mathbf{A_{11}} + \mathbf{A_{22}}) \cdot (\mathbf{B_{11}} + \mathbf{B_{22}}) \\ +\text{\textbf{II}} &= (\mathbf{A_{21}} + \mathbf{A_{22}}) \cdot \mathbf{B_{11}} \\ +\text{\textbf{III}} &= \mathbf{A_{11}} \cdot (\mathbf{B_{12}}-\mathbf{B_{22}}) \\ +\text{\textbf{IV}} &= \mathbf{A_{22}} \cdot (-\mathbf{B_{11}}+\mathbf{B_{21}}) \\ +\text{\textbf{V}} &= (\mathbf{A_{11}} + \mathbf{A_{12}}) \cdot \mathbf{B_{22}} \\ +\text{\textbf{VI}} &= (-\mathbf{A_{11}} + \mathbf{A_{21}}) \cdot (\mathbf{B_{11}} + \mathbf{B_{12}}) \\ +\text{\textbf{VII}} &= (\mathbf{A_{12}} - \mathbf{A_{22}}) \cdot (\mathbf{B_{21}} + \mathbf{B_{22}}) \\ +\end{aligned} +\end{math} + +\end{column} + +\begin{column}{0.5\textwidth} + \large + \begin{math} + \begin{aligned} + \mathbf{C_{11}} &= \text{\textbf{I}} + \text{\textbf{IV}} - \text{\textbf{V}} + \text{\textbf{VII}} \\ + \mathbf{C_{21}} &= \text{\textbf{II}} + \text{\textbf{IV}} \\ + \mathbf{C_{12}} &= \text{\textbf{III}} + \text{\textbf{V}}\\ + \mathbf{C_{22}} &= \text{\textbf{I}} + \text{\textbf{III}} - \text{\textbf{II}} + \text{\textbf{VI}} \\ + \end{aligned} + \end{math} + +\end{column} +\end{columns} + +\end{frame} + +\begin{frame} + \frametitle{Algorithm} + \onslide<1->{ + + \scalebox{0.45}{\parbox{\linewidth}{ + \begin{algorithm}[H]\caption{Strassen Matrix Multiplication} + \setlength{\lineskip}{7pt} + \begin{algorithmic}[1] + \Function{strassen}{$\textbf{A}, \textbf{B}, n$} + \If{$n = 2$} + \State $ \mathbf{C} \gets zeros((n, n))$ + \State $P \gets (A[0][0]+A[1][1])\cdot( B[0][0]+B[1][1])$ + \State $Q \gets (A[1][0]+A[1][1])\cdot B[0][0]$ + \State $R \gets A[0][0]\cdot (B[0][1]-B[1][1])$ + \State $S \gets A[1][1]\cdot (B[1][0]-B[0][0])$ + \State $T \gets (A[0][0]+A[0][1])\cdot B[1][1]$ + \State $U \gets (A[1][0]-A[0][0])\cdot (B[0][0]+B[0][1])$ + \State $V \gets (A[0][1]-A[1][1])\cdot (B[1][0]+B[1][1])$ + \State $C[0][0] \gets P+S-T+V$ + \State $C[0][1] \gets R+T$ + \State $C[1][0] \gets Q+S$ + \State $C[1][1] \gets P+R-Q+U$ + \Else + \State $ m \gets n/2$ + \State $\mathbf{A11}, \mathbf{A12}, \mathbf{A21}, \mathbf{A22} \gets \mathbf{A}[:m][:m], \mathbf{A}[:m][m:], \mathbf{A}[m:][:m], \mathbf{A}[m:][m:]$ + \State $\mathbf{B11}, \mathbf{B12}, \mathbf{B21}, \mathbf{B22} \gets \mathbf{B}[:m][:m], \mathbf{B}[:m][m:], \mathbf{B}[m:][:m], \mathbf{B}[m:][m:]$ + + \State $ \mathbf{P} \gets \text{strassen}((\mathbf{A11}+ \mathbf{A22}),(\mathbf{B11}+\mathbf{B22}), m)$ + \State $ \mathbf{Q} \gets \text{strassen}((\mathbf{A21}+ \mathbf{A22}), \mathbf{B11},m)$ + \State $ \mathbf{R} \gets \text{strassen}( \mathbf{A11},(\mathbf{B12}- \mathbf{B22}),m)$ + \State $ \mathbf{S} \gets \text{strassen}( \mathbf{A22},(\mathbf{B21}- \mathbf{B11}),m)$ + \State $ \mathbf{T} \gets \text{strassen}((\mathbf{A11}+ \mathbf{A12}), \mathbf{B22},m)$ + \State $ \mathbf{U} \gets \text{strassen}((\mathbf{A21}- \mathbf{A11}),(\mathbf{B11}+\mathbf{B12}),m)$ + \State $ \mathbf{V} \gets \text{strassen}((\mathbf{A12}- \mathbf{A22}),(\mathbf{B21}+\mathbf{B22}),m)$ + + + + \State $\mathbf{C11} \gets \mathbf{P+S-T+V}$ + \State $\mathbf{C12} \gets \mathbf{R+T}$ + \State $\mathbf{C21} \gets \mathbf{Q+S}$ + \State $\mathbf{C22} \gets \mathbf{P+R-Q+U}$ + \State $ C \gets vstack((hstack((C11, C12)), hstack((C21, C22))))$ + + \EndIf + \State \textbf{return} $\textbf{C}$ + + \EndFunction + \end{algorithmic} + \end{algorithm} + }}} +% \[ +% \mathcal{T}(n) = \left\{\begin{array}{lr} +% 1, & \text{if} n \leq 2\\ +% 7 \mathcal{T}(\frac{n}{2}) + n^2, & \text{if} n > 2\\ +% \end{array}\right\} +% \] +\only<2>{ + $ + \mathcal{T}(n) = + \begin{cases} + 1 & \text{if } n \leq 2\\ + 7 \cdot \mathcal{T}(\frac{n}{2}) + n^2 & \text{if } n > 2 + \end{cases} = \mathcal{O}(n^{\log_2 7})$ + +} +\only<3>{ + $ + \mathcal{T}(n) = + \begin{cases} + 1 & \text{if } n \leq 2\\ + 7 \cdot \mathcal{T}(\frac{n}{2}) + n^2 & \text{if } n > 2 + \end{cases} = \mathcal{O}(n^{2.81})$ + +} + +\end{frame} + +\begin{frame} + \frametitle{Algorithm} + \onslide<1->{ + + \scalebox{0.45}{\parbox{\linewidth}{ + \begin{algorithm}[H]\caption{Strassen Matrix Multiplication} + \setlength{\lineskip}{7pt} + \begin{algorithmic}[1] + \Function{MM}{$\textbf{A}, \textbf{B}, n$} + \If{$n = 2$} + \State $ \mathbf{C} \gets zeros((n, n))$ + \State $C[0, 0] \gets A[0][0]*B[0][0]+A[0][1]*B[1][0]$ + \State $C[0, 1] \gets A[0][0]*B[0][1]+A[0][1]*B[1][1]$ + \State $C[1, 0] \gets A[1][0]*B[0][0]+A[1][1]*B[1][0]$ + \State $C[1, 1] \gets A[1][0]*B[0][1]+A[1][1]*B[1][1]$ + \Else + \State $ m \gets n/2$ + \State $\mathbf{A11}, \mathbf{A12}, \mathbf{A21}, \mathbf{A22} \gets \mathbf{A}[:m][:m], \mathbf{A}[:m][m:], \mathbf{A}[m:][:m], \mathbf{A}[m:][m:]$ + \State $\mathbf{B11}, \mathbf{B12}, \mathbf{B21}, \mathbf{B22} \gets \mathbf{B}[:m][:m], \mathbf{B}[:m][m:], \mathbf{B}[m:][:m], \mathbf{B}[m:][m:]$ + + \State $\mathbf{C11} \gets \text{MM}(\mathbf{A11}, \mathbf{B11}) + \text{MM}(\mathbf{A12}, \mathbf{B21})$ + \State $\mathbf{C12} \gets \text{MM}(\mathbf{A11},\mathbf{B12}) + \text{MM}(\mathbf{A12},\mathbf{B22})$ + \State $\mathbf{C21} \gets \text{MM}(\mathbf{A21}, \mathbf{B11}) + \text{MM}(\mathbf{A22}, \mathbf{B21})$ + \State $\mathbf{C22} \gets \text{MM}(\mathbf{A21}, \mathbf{B12}) + \text{MM}(\mathbf{A22}, \mathbf{B22})$ + \State $ C \gets vstack((hstack((C11, C12)), hstack((C21, C22))))$ + + \EndIf + \State \textbf{return} $\textbf{C}$ + + \EndFunction + \end{algorithmic} + \end{algorithm} + \bigskip + \bigskip + \bigskip + \bigskip + \bigskip + }}} + +\only<2>{ + + + $ + \mathcal{T}(n) = + \begin{cases} + 1 & \text{if } n \leq 2\\ + 8 \cdot \mathcal{T}(\frac{n}{2}) + n^2 & \text{if } n > 2 + \end{cases} = \mathcal{O}(n^{\log_2 8})$ + +} +\only<3>{ + $ + \mathcal{T}(n) = + \begin{cases} + 1 & \text{if } n \leq 2\\ + 8 \cdot \mathcal{T}(\frac{n}{2}) + n^2 & \text{if } n > 2 + \end{cases} = \mathcal{O}(n^{3})$ + +} + +\end{frame} diff --git a/buch/papers/multiplikation/presentation/tikz/algo.pdf b/buch/papers/multiplikation/presentation/tikz/algo.pdf Binary files differnew file mode 100644 index 0000000..752f42e --- /dev/null +++ b/buch/papers/multiplikation/presentation/tikz/algo.pdf diff --git a/buch/papers/multiplikation/presentation/tikz/algo.tex b/buch/papers/multiplikation/presentation/tikz/algo.tex new file mode 100644 index 0000000..0b2c567 --- /dev/null +++ b/buch/papers/multiplikation/presentation/tikz/algo.tex @@ -0,0 +1,52 @@ +\documentclass[border=10pt]{article} +\usepackage[left=25mm,right=25mm,top=25mm,bottom=25mm]{geometry} +\usepackage[utf8]{inputenc} +\usepackage[T1]{fontenc} +\usepackage{times} +\usepackage{geometry} +\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{mathrsfs} +\usepackage{amsfonts} +\usepackage{amsthm} +\usepackage{lipsum} +\usepackage{amscd} +\usepackage{graphicx} +\usepackage{fancyhdr} +\usepackage{textcomp} +\usepackage{txfonts} +\usepackage[all]{xy} +\usepackage{paralist} +\usepackage[colorlinks=true]{hyperref} +\usepackage{array} +\usepackage{tikz} +\usepackage{slashed} +\usepackage{pdfpages} +\usepackage{cite} +\usepackage{url} +\usepackage{algorithm} +\usepackage[noend]{algpseudocode} +\usepackage{listings} +\usepackage{multirow} +\usepackage{color} + +\begin{document} + +\begin{algorithm}[H]\caption{Square Matrix Multiplication} + \setlength{\lineskip}{7pt} + \begin{algorithmic}[1] + \Function{MM}{$\textbf{A}, \textbf{B}, \textbf{C}, n$} + \State $sum \gets 0$ + \For{$i = 0,1,2 \dots,n-1$} + \For{$j = 0,1,2 \dots,n-1$} + \State $sum \gets 0$ + \For{$k = 0,1,2 \dots,n-1$} + \State $sum \gets sum + \textbf{A}[i][k] \cdot \textbf{B}[k][j]$ + \EndFor + \State $\textbf{C}[i][j] \gets sum $ + \EndFor + \EndFor + \EndFunction + \end{algorithmic} +\end{algorithm} +\end{document} diff --git a/buch/papers/multiplikation/problemstellung.tex b/buch/papers/multiplikation/problemstellung.tex new file mode 100755 index 0000000..b20a791 --- /dev/null +++ b/buch/papers/multiplikation/problemstellung.tex @@ -0,0 +1,104 @@ +% +% teil1.tex -- Beispiel-File für das Paper +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{Problemstellung} +\rhead{Problemstellung} +Dank der breiten Anwendung der Matrizenmultiplikation ist eine effiziente L\"osung dieser Operation von grosser Bedeutung. +Das Ziel dieses Papers ist verschiedenen Algorithmen der Matrizenmultiplikation vorzustellen. +Wobei gezielt auf Algorithmen, welche das Problem schneller als der Standard Algorithmus L\"osen eingegangen wird. + +\subsection{Big $\mathcal{O}$ Notation} +Die Big $\mathcal{O}$ Notation beschreibt die Laufzeitkomplexit\"at eines Algorithmus \cite{multiplikation:bigo}. +$f(x) \in \mathcal{O}(g(x))$ besagt das die Funktion $f$ nicht wesentlich schneller w\"achst als $g$ wenn $x \rightarrow \infty$. +Vereinfacht werden f\"ur Algorithmen die folgende Notation verwendet: +\begin{itemize} + \item $f \in \mathcal{O}(1) \rightarrow f$ ist beschr\"ankt + \item $f \in \mathcal{O}(n) \rightarrow f$ w\"achst linear + \item $f \in \mathcal{O}(n^2) \rightarrow f$ w\"achst quadratisch + \item $f \in \mathcal{O}(\log n) \rightarrow f$ w\"achst logarithmisch + \item $f \in \mathcal{O}(n \log n) \rightarrow f$ hat super-lineares Wachstum + \item $f \in \mathcal{O}(e^n) \rightarrow f$ w\"achst exponentiell + \item usw. +\end{itemize} + +In der Abbildung \ref{multiplikation:fig:bigo} k\"onnen die Verschiedenen Laufzeiten miteinander verglichen werden. + +\begin{figure} + \center + \includegraphics[]{papers/multiplikation/images/bigo} + \caption{Verschiedene Laufzeiten} + \label{multiplikation:fig:bigo} +\end{figure} + +\subsubsection{Beispiel Algorithmen} +\paragraph{Beschr\"ankter Algorithmus} + +Ein Beispiel eines Beschr\"ankter Verhalten $\mathcal{O}(1)$, kann im Algorithmus \ref{multiplikation:alg:b1} entnommen werden. + +\begin{algorithm}\caption{} + \label{multiplikation:alg:b1} + \setlength{\lineskip}{7pt} + \begin{algorithmic} + \Function{B1}{$a, b$} + \State \textbf{return} $a+b$ + \EndFunction + \end{algorithmic} +\end{algorithm} + +Wobei Konstanten nicht beachtet werden, der Algorithmus \ref{multiplikation:alg:b2} f\"uhrt ebenso zu $\mathcal{O}(1)$ und nicht zu $\mathcal{O}(2)$. + +\begin{algorithm}\caption{} + \label{multiplikation:alg:b2} + \setlength{\lineskip}{7pt} + \begin{algorithmic} + \Function{B2}{$a, b$} + \State $ x \gets a+b $ + \State $ y \gets a \cdot b $ + \State \textbf{return} $x+y$ + \EndFunction + \end{algorithmic} +\end{algorithm} + +\paragraph{Linearer Algorithmus} + +Folgender Algorithmus \ref{multiplikation:alg:l1} hat ein lineares $\mathcal{O}(n)$ Verhalten. + +\begin{algorithm}\caption{} + \setlength{\lineskip}{7pt} + \begin{algorithmic} + \label{multiplikation:alg:l1} + \Function{L}{$\mathbf{A}, \mathbf{B}$,n} + \State $ sum \gets 0$ + \For{$i = 0,1,2 \dots,n$} + \State $ sum \gets sum + A[i] \cdot B[i] $ + \EndFor + + \State \textbf{return} $sum$ + + \EndFunction + \end{algorithmic} +\end{algorithm} + +\paragraph{Quadratischer Algorithmus} + +Folgender Algorithmus \ref{multiplikation:alg:q1} hat ein quadratisches $\mathcal{O}(n^2)$ Verhalten. + +\begin{algorithm}[H]\caption{} + \label{multiplikation:alg:q1} + \setlength{\lineskip}{7pt} + \begin{algorithmic} + \Function{Q}{$\mathbf{A}, \mathbf{B}$,n} + \State $ sum \gets 0$ + \For{$i = 0,1,2 \dots,n$} + \For{$j = 0,1,2 \dots,n$} + \State $ sum \gets sum + A[i] \cdot B[j] $ + \EndFor + \EndFor + \State \textbf{return} $sum$ + \EndFunction + \end{algorithmic} +\end{algorithm} + + diff --git a/buch/papers/multiplikation/references.bib b/buch/papers/multiplikation/references.bib index 7149fb1..9d76e8e 100644..100755 --- a/buch/papers/multiplikation/references.bib +++ b/buch/papers/multiplikation/references.bib @@ -33,3 +33,33 @@ url = {https://doi.org/10.1016/j.acha.2017.11.004} } +@article{multiplikation:winograd_1968, + title={A New Algorithm for Inner Product}, + volume={C-17}, + DOI={10.1109/tc.1968.227420}, + number={7}, + journal={IEEE Transactions on Computers}, + author={Winograd, S.}, + year={1968}, + pages={693–694} +} + +@article{multiplikation:strassen_1969, + title={Gaussian elimination is not optimal}, + volume={13}, + DOI={10.1007/bf02165411}, + number={4}, + journal={Numerische Mathematik}, + author={Strassen, Volker}, + year={1969}, + pages={354–356} +} + +@online{multiplikation:bigo, + title = {Big O notation}, + url = {https://en.wikipedia.org/wiki/Big_O_notation}, + date = {2021-07-27}, + year = {2021}, + month = {7}, + day = {27} +} diff --git a/buch/papers/multiplikation/teil0.tex b/buch/papers/multiplikation/teil0.tex deleted file mode 100644 index 082b7f5..0000000 --- a/buch/papers/multiplikation/teil0.tex +++ /dev/null @@ -1,22 +0,0 @@ -% -% einleitung.tex -- Beispiel-File für die Einleitung -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 0\label{multiplikation:section:teil0}} -\rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{multiplikation:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. - -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. - - diff --git a/buch/papers/multiplikation/teil1.tex b/buch/papers/multiplikation/teil1.tex deleted file mode 100644 index 0a6903a..0000000 --- a/buch/papers/multiplikation/teil1.tex +++ /dev/null @@ -1,55 +0,0 @@ -% -% teil1.tex -- Beispiel-File für das Paper -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 1 -\label{multiplikation:section:teil1}} -\rhead{Problemstellung} -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 -\begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{multiplikation:equation1} -\end{equation} -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? - -\subsection{De finibus bonorum et malorum -\label{multiplikation:subsection:finibus}} -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 \eqref{000tempmlate:equation1}. - -Et harum quidem rerum facilis est et expedita distinctio -\ref{multiplikation:section:loesung}. -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 -\ref{multiplikation:section:folgerung}. -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. - - diff --git a/buch/papers/multiplikation/teil2.tex b/buch/papers/multiplikation/teil2.tex deleted file mode 100644 index efbf31a..0000000 --- a/buch/papers/multiplikation/teil2.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil2.tex -- Beispiel-File für teil2 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 2 -\label{multiplikation: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? - -\subsection{De finibus bonorum et malorum -\label{multiplikation: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. - - diff --git a/buch/papers/multiplikation/teil3.tex b/buch/papers/multiplikation/teil3.tex deleted file mode 100644 index f58508b..0000000 --- a/buch/papers/multiplikation/teil3.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil3.tex -- Beispiel-File für Teil 3 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 3 -\label{multiplikation:section:teil3}} -\rhead{Teil 3} -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? - -\subsection{De finibus bonorum et malorum -\label{multiplikation:subsection:malorum}} -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. 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+\usepackage{amsmath} +\usepackage{amssymb} +\usepackage{mathrsfs} +\usepackage{amsfonts} +\usepackage{amsthm} +\usepackage{lipsum} +\usepackage{amscd} +\usepackage{graphicx} +\usepackage{fancyhdr} +\usepackage{textcomp} +\usepackage{txfonts} +\usepackage[all]{xy} +\usepackage{paralist} +\usepackage[colorlinks=true]{hyperref} +\usepackage{array} +\usepackage{tikz} +\usepackage{slashed} +\usepackage{pdfpages} +\usepackage{cite} +\usepackage{url} +\usepackage{amsmath,amsfonts,amssymb} +\usepackage{tikz} +\usetikzlibrary{arrows,matrix,positioning} +\usetikzlibrary{overlay-beamer-styles} +\usetikzlibrary{matrix.skeleton} +\usetikzlibrary{automata,positioning} +\usepackage{listings} +\usepackage{multirow} +\usepackage{color} + +\begin{document} + +$ +A= +\begin{bmatrix} +A_{11} & A_{12}\\ +A_{21} & A_{22} +\end{bmatrix}, +B= +\begin{bmatrix} +B_{11} & B_{12}\\ +B_{21} & B_{22} +\end{bmatrix}, +C= +\begin{bmatrix} +C_{11} & C_{12}\\ +C_{21} & C_{22} +\end{bmatrix} +$ + +\medskip +$ +A \cdot B = C +$ + +\medskip +$ +C_{11} = A_{11} \cdot B_{11} + A_{12} \cdot B_{21}\\ +C_{12} = A_{11} \cdot B_{12} + A_{12} \cdot B_{22}\\ +C_{21} = A_{21} \cdot B_{11} + A_{22} \cdot B_{21}\\ +C_{22} = A_{21} \cdot B_{12} + A_{22} \cdot B_{22} +$ + +\medskip +\begin{math} +\begin{aligned} +\text{I} &= (A_{11} + A_{22}) \cdot (B_{11} + B_{22}) \\ +\text{II} &= (A_{21} + A_{22}) \cdot B_{11} \\ +\text{III} &= A_{11} \cdot (B_{12}-B_{22}) \\ +\text{IV} &= A_{22} \cdot (-B_{11}+B_{21}) \\ +\text{V} &= (A_{11} + A_{12}) \cdot B_{22} \\ +\text{VI} &= (-A_{11} + A_{21}) \cdot (B_{11} + B_{12})) \\ +\text{VII} &= (A_{12} - A_{22}) \cdot (B_{21} + B_{22}) \\ +\end{aligned} +\end{math} + + +\medskip +\begin{math} +\begin{aligned} +C_{11} &= \text{I} + \text{IV} - \text{V} + \text{VII} \\ +C_{21} &= \text{II} + \text{IV} \\ +C_{12} &= \text{III} + \text{V}\\ +C_{22} &= \text{I} + \text{III} - \text{II} + \text{VI} \\ +\end{aligned} +\end{math} + + +\medskip +\begin{math} +\begin{aligned} +C_{11} &= \text{II} + \text{IV} \\ +C_{11} &= (A_{11} + A_{22}) \cdot (B_{11} + B_{22}) + A_{22} \cdot (-B_{11}+B_{21}) - (A_{11} + A_{12}) \cdot B_{22} + (A_{12} - A_{22}) \cdot (B_{21} + B_{22})C_{21} \\ +C_{11} &= A_{11}B_{11} + A_{11}B_{22} + A_{22}B_{11} + A_{22}B_{22} -A_{22}B_{11}+A_{22}B_{21} - A_{11}B_{22} - A_{12}B_{22}+ A_{12}B_{21} + A_{12}B_{22} - A_{22}B_{21} - A_{22}B_{22} \\ +C_{11} &= A_{11}B_{11} + A_{12}B_{21} +\end{aligned} +\end{math} + +\section{Winograd} + +$ +x_1 y_1 + x_2 y_2 = (x_1 +y_2)(y_1 + x_2)-x_1 x_2 - y_1 y_2 +$ + +$ +x = (x_1, \cdots, x_n), y=(y_1, \cdots, y_n) +$ + +\[ +\xi = \sum_{j=1}^{ \lfloor n/2 \rfloor} x_{2j-1} \cdot x_{2j} +\] + +\[ +\eta = \sum_{j=1}^{ \lfloor n/2 \rfloor} y_{2j-1} \cdot y_{2j} +\] + +\[ +\langle x,y \rangle = +\begin{cases} + \displaystyle \sum_{j=1}^{ \lfloor n/2 \rfloor} (x_{2j-1} + y_{2j})(x_{2j}+y_{2j-1})-\xi - \eta & \text{if $n$ is even}\\ +\displaystyle \sum_{j=1}^{ \lfloor n/2 \rfloor} (x_{2j-1} + y_{2j})(x_{2j}+y_{2j-1})-\xi - \eta + x_n y_n & \text{if $n$ is odd} +\end{cases} +\] + +\end{document} diff --git a/buch/papers/multiplikation/tikz_formulas/algo_graph.fdb_latexmk b/buch/papers/multiplikation/tikz_formulas/algo_graph.fdb_latexmk new file mode 100644 index 0000000..ddfa880 --- /dev/null +++ b/buch/papers/multiplikation/tikz_formulas/algo_graph.fdb_latexmk @@ -0,0 +1,245 @@ +# Fdb version 3 +["pdflatex"] 1621585121 "algo_graph.tex" "algo_graph.pdf" "algo_graph" 1621585184 + "/dev/null" 1621583990 0 d41d8cd98f00b204e9800998ecf8427e "" + "/etc/texmf/web2c/texmf.cnf" 1619433543 475 c0e671620eb5563b2130f56340a5fde8 "" + "/usr/share/texlive/texmf-dist/fonts/enc/dvips/base/8r.enc" 1165713224 4850 80dc9bab7f31fb78a000ccfed0e27cab "" + "/usr/share/texlive/texmf-dist/fonts/map/fontname/texfonts.map" 1577235249 3524 cb3e574dea2d1052e39280babc910dc8 "" + "/usr/share/texlive/texmf-dist/fonts/tfm/jknappen/ec/ecrm1000.tfm" 1136768653 3584 adb004a0c8e7c46ee66cad73671f37b4 "" + 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0000000..87c164c --- /dev/null +++ b/buch/papers/munkres/figures/bipartiter_graph.png diff --git a/buch/papers/munkres/main.tex b/buch/papers/munkres/main.tex index 4dd20fa..e5282dc 100644 --- a/buch/papers/munkres/main.tex +++ b/buch/papers/munkres/main.tex @@ -3,34 +3,18 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:munkres}} -\lhead{Thema} +\chapter{Das Zuordnungsproblem und der Munkres-Algorithmus\label{chapter:munkres}} +\lhead{Das Zuordnungsproblem und der Munkres-Algorithmus} \begin{refsection} -\chapterauthor{Hans Muster} +\chapterauthor{Marc Kühne} -Ein paar Hinweise für die korrekte Formatierung des Textes -\begin{itemize} -\item -Absätze werden gebildet, indem man eine Leerzeile einfügt. -Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet. -\item -Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende -Optionen werden gelöscht. -Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen. -\item -Beginnen Sie jeden Satz auf einer neuen Zeile. -Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen -in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt -anzuwenden. -\item -Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren -Übersicht wegen, aber auch um GIT die Arbeit zu erleichtern. -\end{itemize} \input{papers/munkres/teil0.tex} \input{papers/munkres/teil1.tex} \input{papers/munkres/teil2.tex} \input{papers/munkres/teil3.tex} +\input{papers/munkres/teil4.tex} +\input{papers/munkres/teil5.tex} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/munkres/teil0.tex b/buch/papers/munkres/teil0.tex index de522c7..0578429 100644 --- a/buch/papers/munkres/teil0.tex +++ b/buch/papers/munkres/teil0.tex @@ -3,20 +3,8 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 0\label{munkres:section:teil0}} -\rhead{Teil 0} -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua \cite{munkres:bibtex}. -At vero eos et accusam et justo duo dolores et ea rebum. -Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum -dolor sit amet. - -Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam -nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam -erat, sed diam voluptua. -At vero eos et accusam et justo duo dolores et ea rebum. Stet clita -kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit -amet. - +\section{Einleitung\label{munkres:section:teil0}} +\rhead{Einleitung} +Im Bereich der Unternehmensplanung (Operations Research) gibt es verschiedene Fragestellungen. Eine davon ist das sogenannte Transportproblem. Zum Transport einheitlicher Objekte von mehreren Angebots- zu mehreren Nachfrageorten ist ein optimaler, d. h. kostenminimaler Plan zu finden, wobei die vorhandenen und zu liefernden Mengen an den einzelnen Standorten gegeben sowie die jeweiligen Transportkosten pro Einheit zwischen allen Standorten bekannt sind. +Nun gibt es im Bereich des klassischen Transportproblems Sonderfälle. Ein Sonderfall ist z.B. das Zuordnungsproblem. diff --git a/buch/papers/munkres/teil1.tex b/buch/papers/munkres/teil1.tex index f4f5e39..c13732c 100644 --- a/buch/papers/munkres/teil1.tex +++ b/buch/papers/munkres/teil1.tex @@ -3,53 +3,56 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 1 +\section{Beschrieb des Zuordnungsproblems \label{munkres:section:teil1}} \rhead{Problemstellung} -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 -\begin{equation} -\int_a^b x^2\, dx -= -\left[ \frac13 x^3 \right]_a^b -= -\frac{b^3-a^3}3. -\label{munkres:equation1} -\end{equation} -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? +Das spezielle an einem Zuordnungsproblem ist, dass es an jedem Ort nur eine Einheit angeboten bzw. nachgefragt wird. Es werden hier nicht Mengen möglichst kostenminimal von einem zum anderen +Ort transportiert, sondern es geht um die kostenminimale Zuordnung von z.B. Personen, oder Bau-Materialien auf bestimmte Orte, Stellen oder Aufgaben. +Um dieses Problem in einer einfachen, händischen Art und Weise zu lösen wurde der Munkres-Algorithmus, auch die Ungarische Methode genannt, entwickelt. Diese Methode ist ein weiteres Hauptthema dieses Kapitels. + +\subsection{Zuordnungsproblem an einem konkreten Beispiel +\label{munkres:subsection:bonorum}} -\subsection{De finibus bonorum et malorum -\label{munkres:subsection:finibus}} -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 \eqref{000tempmlate:equation1}. +\subsection{Zuordnungsproblem abstrakt +\label{munkres:subsection:bonorum}} + +Es sind alle Angebots- und Bedarfsmengen gleich 1 +\begin{equation} +a_{i}=b_{j}=1 +\end{equation} -Et harum quidem rerum facilis est et expedita distinctio -\ref{munkres:section:loesung}. -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 -\ref{munkres:section:folgerung}. -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. +\subsection{alternative Darstellungen des Zuordnungsproblems +\label{munkres:subsection:bonorum}} +\begin{equation} +Netzwerk +\end{equation} +\begin{equation} +Matrix +\end{equation} +\begin{equation} +Bitpartiter Graph +\end{equation} +Ein bipartiter Graph ist ein mathematisches Modell für Beziehungen +zwischen den Elementen zweier Mengen. +Es eignet sich sehr gut zur Untersuchung von Zuordnungsproblemen» +\begin{figure} +\centering +\includegraphics[width=5cm]{papers/munkres/figures/Netzwerkdarstellung} +\caption{Typische Netzwerkdarstellung eines Zuordnungsproblems.} +\label{munkres:Vr2} +\end{figure} +\begin{figure} +\centering +\includegraphics[width=5cm]{papers/munkres/figures/Matrixdarstellung} +\caption{Typische 4x4 Matrixdarstellung eines Zuordnungsproblems.} +\label{munkres:Vr2} +\end{figure} +\begin{figure} +\centering +\includegraphics[width=5cm]{papers/munkres/figures/bipartiter_graph} +\caption{$K_{3,3}$ vollständig bipartiter Graph mit 3 Knoten pro Teilmenge.} +\label{munkres:Vr2} +\end{figure} diff --git a/buch/papers/munkres/teil2.tex b/buch/papers/munkres/teil2.tex index 23536b9..9a44cd4 100644 --- a/buch/papers/munkres/teil2.tex +++ b/buch/papers/munkres/teil2.tex @@ -3,38 +3,11 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 2 +\section{Schwierigkeit der Lösung (Permutationen) \label{munkres: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{Schwierigkeit der Lösung (Permutationen)} -\subsection{De finibus bonorum et malorum -\label{munkres: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. +Eine Permutation ist eine Anordnung von Objekten in einer bestimmten Reihenfolge oder eine Umordnung von Objekten aus einer vorgegebenen Reihung. Ist eine maximale Zuordnung (maximales Matching) gefunden, so steht in jeder Zeile und jeder Spalte der Matrix genau ein Element, das zur optimalen Lösung gehört, eine solche Gruppe von Positionen wird auch als Transversale der Matrix bezeichnet. +Die Problemstellung kann auch so formuliert werden, dass man die Zeilen- oder die Spaltenvektoren so umordnet soll, dass die Summe der Elemente in der Hauptdiagonale maximal wird. Hieraus wird sofort ersichtlich, dass es in einer n×n-Matrix genau so viele Möglichkeiten gibt, die Zeilen- bzw. Spaltenvektoren zu ordnen, wie es Permutationen von n Elementen gibt, also n!. Außer bei kleinen Matrizen ist es nahezu aussichtslos, die optimale Lösung durch Berechnung aller Möglichkeiten zu finden. Schon bei einer 10×10-Matrix gibt es nahezu 3,63 Millionen (3.628.800) zu berücksichtigender Permutationen. diff --git a/buch/papers/munkres/teil3.tex b/buch/papers/munkres/teil3.tex index b67ad74..cd47c92 100644 --- a/buch/papers/munkres/teil3.tex +++ b/buch/papers/munkres/teil3.tex @@ -3,38 +3,44 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Teil 3 +\section{Der Munkres-Algorithmus (Ungarische Methode) \label{munkres:section:teil3}} -\rhead{Teil 3} -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{Der Munkres-Algorithmus (Ungarische Methode)} -\subsection{De finibus bonorum et malorum +Mit der ungarischen Methode können also lineare Optimierungsprobleme gelöst +werden, die bei gewichteten Zuordnungen in bipartiten Graphen entstehen. +Mit ihr kann die eindeutige Zuordnung von Objekten aus zwei Gruppen so +optimiert werden, dass die Gesamtkosten minimiert werden bzw.~der +Gesamtgewinn maximiert werden kann. + +\subsection{Geschichte +\label{munkres:subsection:malorum}} +Die Ungarische Methode wurde 1955 von Harold Kuhn entwickelt und veröffentlicht. +Der Name ``Ungarische Methode'' ergab sich, weil der Algorithmus +weitestgehend auf den früheren Arbeiten zweier ungarischer Mathematiker +basierte: Dénes Kőnig und Jenő Egerváry. +James Munkres überprüfte den Algorithmus im Jahr 1957 und stellte fest, +dass der Algorithmus (stark) polynomiell ist. +Seitdem ist der Algorithmus auch als Kuhn-Munkres oder +Munkres-Zuordnungsalgorithmus bekannt. +Die Zeitkomplexität des ursprünglichen Algorithmus war $O(n^4)$, +später wurde zudem festgestellt, dass er modifiziert werden kann, +um eine $O(n^3)$-Laufzeit zu erreichen. + +\subsection{Besondere Leistung der Ungarischen Methode \label{munkres:subsection:malorum}} -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. +Es ist ein kombinatorischer Optimierungsalgorithmus, der das Zuordnungsproblem +in polynomieller Zeit löst. +Der Begriff polynomielle Laufzeit bedeutet, dass die Laufzeit des Programms +wie $n^2$, $n^3$, $n^4$, etc.~wächst und vernünftig skaliert. +\subsection{Beispiel eines händischen Verfahrens +\label{munkres:subsection:malorum}} + +\begin{figure} +\centering +\includegraphics[width=14cm]{papers/munkres/figures/beispiel_munkres} +\caption{Händisches Beispiel des Munkres Algorithmus.} +\label{munkres:Vr2} +\end{figure} diff --git a/buch/papers/munkres/teil4.tex b/buch/papers/munkres/teil4.tex new file mode 100644 index 0000000..9a27227 --- /dev/null +++ b/buch/papers/munkres/teil4.tex @@ -0,0 +1,9 @@ +% +% teil4.tex -- Beispiel-File für Teil 4 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{- +\label{munkres:section:teil4}} +\rhead{-} + diff --git a/buch/papers/munkres/teil5.tex b/buch/papers/munkres/teil5.tex new file mode 100644 index 0000000..b938c50 --- /dev/null +++ b/buch/papers/munkres/teil5.tex @@ -0,0 +1,8 @@ +% +% teil5.tex -- Beispiel-File für Teil 5 +% +% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\section{- +\label{munkres:section:teil5}} +\rhead{-} diff --git a/buch/papers/punktgruppen/Makefile b/buch/papers/punktgruppen/Makefile index f92dc95..03ad15a 100644 --- a/buch/papers/punktgruppen/Makefile +++ b/buch/papers/punktgruppen/Makefile @@ -11,11 +11,16 @@ SOURCES := \ symmetry.tex TIKZFIGURES := \ + tikz/atoms-grid-still.tex \ + tikz/atoms-grid-force.tex \ + tikz/atoms-piezo-still.tex \ + tikz/atoms-piezo-force-vertical.tex \ + tikz/atoms-piezo-force-horizontal.tex \ tikz/combine-symmetries.tex \ tikz/lattice.tex \ - tikz/piezo-atoms.tex \ tikz/piezo.tex \ tikz/projections.tex \ + tikz/stereographic-projections.tex \ tikz/symmetric-shapes.tex FIGURES := $(patsubst tikz/%.tex, figures/%.pdf, $(TIKZFIGURES)) @@ -28,7 +33,7 @@ figures/%.pdf: tikz/%.tex pdflatex --output-directory=figures $< .PHONY: standalone -standalone: standalone.tex $(SOURCES) +standalone: standalone.tex $(SOURCES) $(FIGURES) mkdir -p standalone cd ../..; \ pdflatex \ diff --git a/buch/papers/punktgruppen/Makefile.inc b/buch/papers/punktgruppen/Makefile.inc index 8cde9d7..fbb073e 100644 --- a/buch/papers/punktgruppen/Makefile.inc +++ b/buch/papers/punktgruppen/Makefile.inc @@ -11,8 +11,15 @@ dependencies-punktgruppen = \ papers/punktgruppen/crystals.tex \ papers/punktgruppen/piezo.tex \ papers/punktgruppen/references.bib \ - papers/punktgruppen/tikz/combine-symmetries.tex \ + papers/punktgruppen/tikz/atoms-grid-force.tex \ + papers/punktgruppen/tikz/atoms-grid-still.tex \ + papers/punktgruppen/tikz/atoms-piezo-force-horizontal.tex \ + papers/punktgruppen/tikz/atoms-piezo-force-vertical.tex \ + papers/punktgruppen/tikz/atoms-piezo-still.tex \ + papers/punktgruppen/tikz/combine-symmetries.tex \ papers/punktgruppen/tikz/lattice.tex \ papers/punktgruppen/tikz/piezo-atoms.tex \ papers/punktgruppen/tikz/piezo.tex \ - papers/punktgruppen/tikz/projections.tex + papers/punktgruppen/tikz/projections.tex \ + papers/punktgruppen/tikz/stereographic-projections.tex \ + papers/punktgruppen/tikz/symmetric-shapes.tex diff --git a/buch/papers/punktgruppen/crystals.tex b/buch/papers/punktgruppen/crystals.tex index 1aec16f..21e29c9 100644 --- a/buch/papers/punktgruppen/crystals.tex +++ b/buch/papers/punktgruppen/crystals.tex @@ -1,6 +1,6 @@ \section{Kristalle} %einleitung sollte noch an das ende von der Symmetrie angepasst werden -Unter dem Begriff Kristall sollte sich jeder ein Bild machen können. +Unter dem Begriff Kristall sollte sich jeder ein Bild machen können. Wir werden uns aber nicht auf sein Äusseres fokussieren, sondern was ihn im Inneren ausmacht. Die Innereien eines Kristalles sind glücklicherweise relativ einfach definiert. \begin{definition}[Kristall] @@ -12,117 +12,151 @@ Die Innereien eines Kristalles sind glücklicherweise relativ einfach definiert. \includegraphics[]{papers/punktgruppen/figures/lattice} \caption{ Zweidimensionales Kristallgitter. - \texttt{TODO: make wider and shorter} \label{fig:punktgruppen:lattice} } \end{figure} \subsection{Kristallgitter} Ein zweidimensionales Beispiel eines solchen Muster ist Abbildung \ref{fig:punktgruppen:lattice}. -Für die Überschaubarkeit haben wir ein simples Motiv eines einzelnen grauen Punktes gewählt und betrachten dies nur in Zwei Dimensionen. -Die eingezeichneten Vektoren $\vec{a}$ und $\vec{b}$ sind die kleinstmöglichen Schritte im Raum bis sich das Kristallgitter wiederholt. -Wird ein beliebiger grauer Gitterpunkt in \ref{fig:punktgruppen:lattice} gewählt -und um eine ganzzahlige Linearkombination von $\vec{a}$ und $\vec{b}$ verschoben, -endet er zwangsweise auf einem Gitterpunkt, wenn nicht wieder am selben Ort. -Im Dreidimensionalen-Raum können alle Gitterpunkte mit derselben Idee und einem zusätzlichen Vektor $\vec{c}$ also +Für die Überschaubarkeit haben wir ein simples Motiv eines einzelnen grauen Punktes dargestellt und betrachten dies nur in zwei Dimensionen. +Die eingezeichneten Vektoren \(\vec{a}_1\) und \(\vec{a}_2\) sind die kleinstmöglichen Schritte im Raum bis sich das Kristallgitter wiederholt. +Wird ein beliebiger grauer Gitterpunkt in \ref{fig:punktgruppen:lattice} gewählt und um eine ganzzahlige Linearkombination von \(\vec{a}_1\) und \(\vec{a}_2\) verschoben, endet er zwangsweise auf einem Gitterpunkt, wenn nicht wieder am selben Ort. +Im dreidimensionalen Raum können alle Gitterpunkte mit derselben Idee und einem zusätzlichen Vektor \(\vec{c}\) also \[ - \vec{r} = n_1 \vec{a} + n_2 \vec{b} + n_3 \vec{c} + \vec{r} = n_1 \vec{a}_1 + n_2 \vec{a}_2 + n_3 \vec{a}_3 = \sum_i n_i \vec{a}_i \] -erreicht werden sofern $\{n_1,n_2,n_3\} \in \mathbb{Z}$ sind. -Sind die Vektoren $\vec{a}$ , $\vec{b}$ , $\vec{c}$ gegeben , -ist ein Kristallgitter eindeutig beschrieben, weswegen sie auch als Grundvektoren bekannt sind. +erreicht werden sofern \(n_1,n_2,n_3 \in \mathbb{Z}\) sind. +Sind die Vektoren \(\vec{a}_1\), \(\vec{a}_2\), \(\vec{a}_3\) gegeben, ist ein Kristallgitter eindeutig beschrieben, weswegen sie auch als Grundvektoren bekannt sind. -\subsection{Translationssymmetrie} +\subsection{Translationssymmetrie} Da sich das ganze Kristallgitter wiederholt, wiederholen sich auch dessen Eigenschaften periodisch mit den Grundvektoren. -Sollte man sich auf einem Gitterpunkt in einem Kristall aufhalten, ist es unmöglich zu wissen, auf welchem Gitterpunkt man sich befindet, -da die Umgebungen aller Punkte Identisch sind. -Mit anderen worten: Jedes Kristallgitter $ G $ ist \emph{Translationssymmetrisch} in der Translation +Sollte man sich auf einem Gitterpunkt in einem Kristall aufhalten, ist es unmöglich zu wissen, auf welchem Gitterpunkt man sich befindet, da die Umgebungen aller Punkte identisch sind. +Mit anderen Worten: Jedes Kristallgitter $ G $ ist \emph{Translationssymmetrisch} in der Translation \[ - Q_i(G) = G + \vec{a_i} -\] wobei der Vektor $a_i$ ein Grundvektor sein muss. + \vec{Q}_i(G) = G + \vec{a}_i +\] wobei der Vektor $\vec{a}_i$ ein Grundvektor sein muss. Da die Translationssymmetrie beliebig oft mit allen Grundvektoren angewendet werden kann, können wir auch sagen, dass alle Verschiebungen um eine Linearkombination -der Vektoren $\vec{a}$ , $\vec{b}$ und $\vec{c}$ erlaubt sind oder kurz, um $\vec{r}$. +der Vektoren $\vec{a}_1$ , $\vec{a}_2$ und $\vec{a}_3$ erlaubt sind oder kurz, um $\vec{r}$. Verschiebungen um $\vec{r}$ bewirken demnach keine Veränderungen, solange wir ein unendlich grosses Kristallgitter verschieben. -\subsection{Limitierte Kristallsymmetrien} +\subsection{Limitierte Kristallsymmetrien} \label{txt:punktgruppen:Translationssymmetrie} Die Translationssymmetrie ist wohl keine grosse Überraschung, wenn man die Abbildung \ref{fig:punktgruppen:lattice} betrachtet. - Was nicht direkt ersichtlich ist, ist das auch wenn die Grundvektoren frei gewählt werden können, - können nur Rotationssymmetrische Kristalle bestimmter Rotationswinkel erzeugt werden. - + Was nicht direkt ersichtlich ist, dass bei beliebigen Grundvektoren nicht beliebige Symmetrien erstellt werden können. + Die geforderte Translationssymmetrie eines Kristalles schränkt weitere Symmetrien deutlich ein. + \begin{figure} \centering \includegraphics[]{papers/punktgruppen/figures/combine-symmetries} \caption{ Translations und Rotationssymmetrisches Kristallgitter - \texttt{TODO: make wider and change color (yellow)} } \label{fig:punktgruppen:rot-geometry} \end{figure} - \subsubsection{Translationssymmetrie $Q$ in Kombination mit Rotationssymmetrie $C_\alpha$} % Müssen uns auf eine schreibweise für Symmetrie Operationen einigen oder sicher am Ende überprüfen - In Abbildung \ref{fig:punktgruppen:rot-geometry} Sehen wir Gitterpunkte und deren Zusammenhänge. +\begin{satz} + Die Rotationssymmetrien eines Kristalls sind auf 2-fach, 3-fach, 4-fach und 6-fach beschränkt. + Mit anderen Worten: Es sind nur Drehwinkel von + 0\(^{\circ}\), + 60\(^{\circ}\), + 90\(^{\circ}\), + 120\(^{\circ}\) und + 180\(^{\circ}\) + erlaubt. +\end{satz} + +\begin{proof} + In Abbildung \ref{fig:punktgruppen:rot-geometry} sehen wir Gitterpunkte und deren Zusammenhänge. \begin{itemize} - \item $A$ ist unser erster Gitterpunkt. - - \item $A'$ ist gegeben, weil wir $A$ mit der Translation $Q$ um einen Grundvektor verschieben und wir wissen, - dass nach einer Translation wieder ein Gitterpunkt an der Verschobenen Stelle sein muss. - \item $B$ entsteht, weil wir die Rotationssymmetrie $C_\alpha$ auf den Punkt $A$ anwenden. - Dadurch dreht sich das ganze Gitter um den Winkel $\alpha$. - Für uns bedeutet dies lediglich, dass unser zweiter Punkt $A'$ abgedreht wird. - An der neuen Position von $A'$ muss also auch ein Punkt sein, um die Rotationssymmetrie zu erfüllen. - \item $B$ ist unser Name für diesen neuen Punkt. - Da auch die Eigenschaften des Kristallgittes periodisch mit dem Gitter sein müssen, dürfen wir $C_\alpha$ auch auf $A'$ anwenden. - Also wenden wir $C_\alpha$ invertiert - \footnote{Eine Rotationssymmetrie muss auch in die inverse Richtung funktionieren. - Genauere Überlegungen hierzu werden dem Leser überlassen, da sich die Autoren nicht explizit mit dieser Frage Auseinander gesetzt haben.} - auch auf $A'$ an. - Dies dreht $A$ auf einen neuen Punkt. - \item $B'$ ist kein zufälliger Name für diesen neuen Punkt, denn wir wissen, dass zwischen allen Punkten eine Translationssymmetrie bestehen muss. - Die Translationssymmetrie zwischen $B$ und $B'$ ist hier als $Q'$ bezeichnet. + \item \(A\) ist unser erster Gitterpunkt. + + \item \(A'\) ist gegeben, weil wir \(A\) mit der Translation \(\vec{Q}\) um einen Grundvektor verschieben und wir wissen, + dass nach einer Translation wieder ein Gitterpunkt an der verschobenen Stelle sein muss. + \item \(B\) entsteht, weil wir die Rotationssymmetrie \(C_n\) auf den Punkt \(A\) anwenden. + Dadurch dreht sich das ganze Gitter um den Winkel \(360^\circ/n\). + Für uns bedeutet dies lediglich, dass unser zweiter Punkt \(A'\) abgedreht wird. + An der neuen Position \(B\) von \(A'\) muss also auch ein Punkt des Gitters sein, um die Rotationssymmetrie zu erfüllen. + \item \(B\) ist unser Name für diesen neuen Punkt. + Da auch die Eigenschaften des Kristallgittes periodisch mit dem Gitter sein müssen, dürfen wir \(C_n\) auch auf \(A'\) anwenden. + Also wenden wir \(C_n\) invertiert\footnote{Eine Rotationssymmetrie muss auch in die inverse Richtung funktionieren. + Genauere Überlegungen hierzu werden dem Leser überlassen, da sich die Autoren nicht explizit mit dieser Frage Auseinander gesetzt haben.} + auch auf \(A'\) an. + Dies dreht \(A\) auf einen neuen Punkt. + \item \(B'\) ist kein zufälliger Name für diesen neuen Punkt, denn wir wissen, dass zwischen allen Punkten eine Translationssymmetrie bestehen muss. + Die Translationssymmetrie zwischen \(B\) und \(B'\) ist hier als \(\vec{Q}'\) bezeichnet. \end{itemize} Mit den gegebenen Punkten lassen sich geometrische Folgerungen ziehen. - Wir beginnen, indem wir die Länge der Translation $Q$ mit jener von $Q'$ vergleichen. - Aus Abbildung \ref{fig:punktgruppen:rot-geometry} ist ersichtlich, dass $|Q| = |Q'|+ 2x$. - Ist $Q$ ein Grundvektor so muss $|Q'|$ ein ganzes vielfaches von $|Q|$ sein. Also + Wir beginnen, indem wir die Länge der Verschiebung \(|\vec{Q}| = Q\) setzen und \(|\vec{Q}'| = Q'\). + Aus Abbildung \ref{fig:punktgruppen:rot-geometry} ist ersichtlich, dass \(Q' = Q + 2x\). + Da \(\vec{Q}\) eine Translation um ein Grundvektor ist , muss \(\vec{Q}'\) ein ganzes vielfaches von \(\vec{Q}\) sein. + Demnach auch die Längen \[ - |Q'| = n|Q| = |Q| + 2x + Q' = nQ = Q + 2x \] - Die Strecke $x$ lässt sich auch mit hilfe der Trigonometrie und dem angenommenen Rotationswinkel $\alpha$ ausdrücken: + Die Strecke \(x\) lässt sich auch mit hilfe der Trigonometrie und dem angenommenen Rotationswinkel \(\alpha\) ausdrücken: \[ - n|Q| = |Q| + 2|Q|\sin(\alpha - \pi/2) + nQ = Q + 2Q\sin(\alpha - \pi/2) \] - Wir können mit $|Q|$ dividieren um unabhängig von der Läge des Grundvektors zu werden, - was auch Sinn macht, da eine Skalierung eines Kristalles seine Symmetrieeigenschaften nicht tangieren soll. + Wir können durch \(Q\) dividieren um unabhängig von der Läge des Grundvektors zu werden, was auch Sinn macht, + da eine Skalierung eines Kristalles seine Symmetrieeigenschaften nicht tangiert. Zusätzlich können wir den Sinusterm vereinfachen. \[ - n = 1 - 2\cos\alpha + n = 1 - 2\cos\alpha \quad\iff\quad \alpha = \cos^{-1}\left(\frac{1-n}{2}\right) \] Dies schränkt die möglichen Rotationssymmetrien auf - \[ + \( \alpha \in \left\{ 0^\circ, 60^\circ, 90^\circ, 120^\circ, 180^\circ\right\} - \] + \) ein. +\end{proof} \begin{figure} \centering - \includegraphics[]{papers/punktgruppen/figures/projections} - \caption{Kristallklassen mit zugehöriger Schönfliesnotation} - \label{fig:punktgruppen:Kristallkassen} + \includegraphics[height=6cm]{papers/punktgruppen/figures/stereographic-projections} + \caption{ + Stereografische Projektion einer \(C_{i}\) Symmetrie. Es wird eine Linie vom magentafarbenen Punkt auf der oberen Hälfte der Kugel zum Südpol gezogen. + Wo die Linie die Ebene schneidet (\(z = 0\)), ist die Projektion des Punktes. + Die Koordinaten der Projektionen sind einfach zu berechnen: ein Punkt auf eine Kugel mit Radius \(r\) mit den Koordinaten \(x, y, z,\) wird auf \(xr/(r + z), yr/(r + z)\) projiziert. + Für den orangefarbenen Punkt unterhalb des Äquators wird die Linie zum Nordpol gezogen und die Projektionsformel hat stattdessen einen Nenner von \(r - z\). + } + \label{fig:punktgruppen:stereographic-projections} \end{figure} \subsection{Kristallklassen} + Vorgehend wurde gezeigt, dass in einem zweidimensionalen Kristallgitter nicht alle Symmetrien möglich sind. -Mit weiteren ähnlichen überlegungen gezeigt werden kann, dass Kristalle im dreidimensionalen Raum -\footnote{Alle $17$ möglichen zweidimensionalen Symmetrien sind als Wandmustergruppen bekannt} -nur auf genau $32$ Arten punktsymmetrisch sein können. -Diese $32$ möglichen Punktsymmetrien scheinen durchaus relevant zu sein, denn sie werden unter anderem als Kristallklassen bezeichnet. -Eine mögliche Art, die Klassen zu benennen ist nacht dem Mathematiker Arthur Moritz Schönflies, -welcher sich mit der Klasifizierung dieser Symmetrien auseinandergesetzt hat. -Auf der Abbildung \ref{fig:punktgruppen:Kristallkassen} sind die möglichen Punktsymmetrien mit deren Schönfliesnotation aufgelistet. -Als Darstellungsmethode wurde die stereographische Projektion gewählt, wobei $5$ Klassen aus Gründen der Überschaubarkeit nicht gezeichnet wurden. + Mit weiteren ähnlichen Überlegungen kann gezeigt werden, dass Kristalle im dreidimensionalen Raum nur auf genau 32 Arten rein punktsymmetrische Symmetriegruppen bilden können. + Diese 32 möglichen Symmetriegruppen scheinen durchaus relevant zu sein, denn sie werden unter anderem als Kristallklassen bezeichnet. + Die 32 möglichen Kristallklassen sind auf Abbildung \ref{fig:punktgruppen:Kristallkassen} zu sehen. + Die Darstellung von dreidimensionalen Punktsymmetrien wurde mit der stereographischen Projektion ermöglicht (siehe Abbildung \ref{fig:punktgruppen:stereographic-projections}), wobei die gestrichelten Klassen aus Gründen der Überschaubarkeit nicht im Detail gezeichnet wurden. + + +\begin{figure} + \centering + \includegraphics[]{papers/punktgruppen/figures/projections} + \caption{Kristallklassen mit zugehörigem Schönflies-Symbol} + \label{fig:punktgruppen:Kristallkassen} +\end{figure} + +\subsubsection{Schönflies-Symbilok} + +Jede der 32 Kristallklassen auf der Abbildung \ref{fig:punktgruppen:Kristallkassen} ist mit ihrem zugehörigen Schöönflies-Symbol bezeichnet. + Die Schönflies-Symbolik stammt von dem Mathematiker Arthur Moritz Schönflies, welcher sich unter anderem mit der Klasifizierung der Punktgruppen auseinandergesetzt hat. + Er hat Untergruppen gebildet, welche als Grossbuchstaben in Abbildung \ref{fig:punktgruppen:Kristallkassen} zu sehen sind. + Da nicht alle Symmetriegruppen in Kristallen möglich sind, werden nicht alle Untergruppen von Schönflies verwendet. + Es ist nur die Drehgruppe \(C\), Diedergruppe \(D\), Drehspiegelgruppe \(S\), Tetraedergruppe \(T\) und die Oktaedergruppe \(O\). + Für die eindeutige zuweisung in eine Kristallklasse werden noch identifizierende Merkmale als Subskript notiert. + Bei der Untergruppe \(C\) werden beispielsweise die möglichen Rotationssymmetrien gezeigt. + Dank Abschintt \ref{txt:punktgruppen:Translationssymmetrie} wissen wir, wieso auf \(C\) nur ganz bestimmte Subskripte folgen, weil das Subskript \(n\) von \(C_n\) zeigt, dass es sich um eine \(n\)-fache Rotationssymmetrie handelt. + Daher darf \(C_5\) auf der Abbildung \ref{fig:punktgruppen:Kristallkassen} nicht vorkommen darf, da \(360^\circ/5 = 72^\circ\) was nach Abschnitt \ref{txt:punktgruppen:Translationssymmetrie} in einem Kristall keine mögliche Rotationssymmetrie ist. + Sind im Subskript Buchstaben, definieren diese weitere Symmetrieeigenschaften der Klasse. + Wie zum Beispiel ein Inversionszentrum\footnote{Ein Objekt mit Inversionszentrum ist Punktsymmetrisch im Inversionszentrum.} \(i\) oder eine horizontale\footnote{Als Orientierungspunkt wird die Symmetrieachse höchster Ordnung (\(n\)) als vertikal definiert} Spiegelachse \(h\). + Zu beachten ist jedoch, dass manche Symmetriegruppen mit mehreren Schönflies-Symbolen beschieben werden können. + \(C_{3i}\) beschreibt genau das selbe wie \(S_6\), da eine dreifache Rotationssymmetrie mit einem Inversionszentrum einer sechsfachen Drehspiegelsymmetrie entspricht. + +%% vim:spell spelllang=de showbreak=.. breakindent linebreak: diff --git a/buch/papers/punktgruppen/figures/atoms-grid-force.pdf b/buch/papers/punktgruppen/figures/atoms-grid-force.pdf Binary files differnew file mode 100644 index 0000000..b3e6215 --- /dev/null +++ b/buch/papers/punktgruppen/figures/atoms-grid-force.pdf diff --git a/buch/papers/punktgruppen/figures/atoms-grid-still.pdf 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a/buch/papers/punktgruppen/intro.tex b/buch/papers/punktgruppen/intro.tex index 24212e7..b6a72b5 100644 --- a/buch/papers/punktgruppen/intro.tex +++ b/buch/papers/punktgruppen/intro.tex @@ -1,14 +1,26 @@ \section{Einleitung} Es gibt viele Möglichkeiten sich in Kristallen zu verlieren. -Auch wen man nur die mathematischen Betrachtunngsweisen berüksichtigt, hat man noch viel zu viele Optionen sich mit Kristallen zu beschäftigen. -In diesem Kapitel ist daher der Fokus ``nur'' auf die Symmetrie gelegt. -Zu beginn werden wir zeigen was eine Symmetrie ausmacht und dass sie noch weit mehr in sich verbirgt als nur schön auszusehen. -Die vorgestellten Symmetrien sind äusserst gut geeignet um die Grundeigenschaften eines Kristalles zu Beschreiben. -Mit etwas kiffligen geometrischen Überlegungen kann man zeigen wass in der Welt der Kristallographie alles möglich ist oder nicht. -Die Einschränkungen sind durchaus wilkommen, dank ihnen halten sich die möglichen Kristallgitter in Grenzen und Lassen sich Kategorisieren. -Kategorien sind nicht nur für einen besseren Überblich nützlich, sondern kann man aus ihnen auch auf Physikalische Eigenschaften schliessen, als spannendes Beispiel: Die Piezoelektrizität. -Die Piezoelektrizität ist vielleicht noch nicht jedem bekannt, sie versteckt sich aber in diversen Altagsgegenständen zum Beispiel sorgen sie in den meisten Feuerzeugen für die Zündung. -Ein Funken Interesse ist hoffentlich geweckt um sich mit dem scheinbar trivialen thema der Symmetrie auseinander zu setzten. +Auch wen man nur die mathematischen Betrachtungsweisen berücksichtigt, +hat man noch viel zu viele Optionen sich mit Kristallen zu beschäftigen. +In diesem Kapitel wird daher der Fokus ``nur'' auf die Symmetrie gelegt. +Zu Beginn werden wir zeigen was eine Symmetrie ausmacht und +dass sie noch weit mehr in sich verbirgt als nur schön auszusehen. +Die vorgestellten Symmetrien sind äusserst gut geeignet, +um die Grundeigenschaften eines Kristalles zu beschreiben. +Mit etwas kniffligen geometrischen Überlegungen kann man zeigen, +was in der Welt der Kristallographie alles möglich ist oder nicht. +Einschränkungen in Kristallsymmetrien sind durchaus willkommen, +da dank ihnen sich die möglichen Kristallgitter in Grenzen halten +und sich kategorisieren lassen. +Kategorien sind nicht nur für einen besseren Überblick nützlich, +sondern kann man aus ihnen auch auf Physikalische Eigenschaften schliessen. +Als spannendes Beispiel: Die Piezoelektrizität. +Die Piezoelektrizität ist vielleicht noch nicht jedem bekannt, +sie versteckt sich aber in diversen Altagsgegenständen +zum Beispiel sorgen sie in den meisten Feuerzeugen für die Zündung. +Hiermit ist hoffentlich ein Funken Interesse geweckt +um sich mit dem scheinbar trivialen Thema der Symmetrie auseinander zu setzten. +%% vim:linebreak breakindent showbreak=.. spell spelllang=de: diff --git a/buch/papers/punktgruppen/main.tex b/buch/papers/punktgruppen/main.tex index a6e246c..ea19421 100644 --- a/buch/papers/punktgruppen/main.tex +++ b/buch/papers/punktgruppen/main.tex @@ -18,6 +18,7 @@ \nocite{punktgruppen:pinter-algebra} \nocite{punktgruppen:sands-crystal} \nocite{punktgruppen:lang-elt2} +\nocite{punktgruppen:ouchem} \printbibliography[heading=subbibliography] \end{refsection} diff --git a/buch/papers/punktgruppen/piezo.tex b/buch/papers/punktgruppen/piezo.tex index e6b595a..6defcdc 100644 --- a/buch/papers/punktgruppen/piezo.tex +++ b/buch/papers/punktgruppen/piezo.tex @@ -19,10 +19,17 @@ Der Aufbau und somit auch die Symmetrie des Kristalles sind daher relevant für \begin{figure} \centering - \includegraphics[]{papers/punktgruppen/figures/piezo-atoms} + \begin{tabular}{c |c} + \subfigure[][\label{fig:punktgruppen:atoms-piezo}]{\includegraphics{papers/punktgruppen/figures/atoms-piezo-still}} & + \subfigure[][\label{fig:punktgruppen:atoms-grid}]{\includegraphics{papers/punktgruppen/figures/atoms-grid-still}} \\ + \subfigure[][\label{fig:punktgruppen:atoms-piezo-fv}]{\includegraphics{papers/punktgruppen/figures/atoms-piezo-force-vertical}} + \hspace{2mm} + \subfigure[][\label{fig:punktgruppen:atoms-piezo-fh}]{\includegraphics{papers/punktgruppen/figures/atoms-piezo-force-horizontal}} + \hspace{3mm} & \hspace{3mm} + \subfigure[][\label{fig:punktgruppen:atoms-grid-f}]{\includegraphics{papers/punktgruppen/figures/atoms-grid-force}} \\ + \end{tabular} \caption{ Kristallstrukturen mit und ohne piezoelektrischer Eigenschaft. - \texttt{TODO: adapt figure for paper with subfigure markers.} } \label{fig:punktgruppen:atomPiezo} \end{figure} @@ -32,26 +39,29 @@ Die Polarisation resultiert über eine gesamte Oberfläche eines Kristalles, ent Wir wollen dazu die verschiedenen Kristallstrukturen auf Abbildung \ref{fig:punktgruppen:atomPiezo} diskutieren. In Abbildung \ref{fig:punktgruppen:atomPiezo} gilt für alle Strukturen, dass rote Kreise Positive Ionen und blaue negative Ionen repräsentieren. %liste oder anderes format?.. -Struktur$(a)$ zeigt ein piezoelektrisches Material in Ruhe. Struktur $(b)$ ist dasselbe Kristallgitter, jedoch wird es senkrecht belastet. +Struktur \subref{fig:punktgruppen:atoms-piezo} zeigt ein piezoelektrisches Material in Ruhe. +Struktur \subref{fig:punktgruppen:atoms-piezo-fv} ist dasselbe Kristallgitter, jedoch wird es senkrecht belastet. Eingezeichnet ist auch das elektrische Feld, welches entsteht, weil mitlleren Ladungsträger weiter auseinander gerdrückt werden. -Als hilfe zur Vorstellung kann man $(b)$ zwischen zwei leitende Platten setzen, -so wird ersichtlich, dass mit wachsendem Druck eine negative Ladung an die rechte Platte gedrückt wird, -während sich die positiven Ionen weiter entfernen. -$(d)$ ist nicht piezoelektrisch. -Dies wird ersichtlich, wenn man $(d)$ unterdruck setzt und sich die Struktur zu $(e)$ verformt. -Setzt man $(e)$ gedanklich auch zwischen zwei leitende Platten scheint es als würden rechts mehr Positive Ionen in die Platte gedrückt werden -und links umgekehrt. -Dies ist aber nicht mehr der Fall, wenn der Kristall nach oben und periodisch wiederholt. -Struktur $(c)$ zeigt $(a)$ in unter horizontaler Belastung. -Was in zwischen $(b)$ und $(c)$ zu beobachten ist, ist dass das entstandene Ladungsdifferenz orthogonal zu der angelegten Kraft entsteht, -im Gegensatz zu $(b)$. -Daraus kann man schlissen, dass $(a)$ keine Rotationssymmetrie von $90^\circ$ besitzen kann, weil die Eigenschaften ändern bei einer $90^\circ$ Drehung. -Das Fehlen dieser Rotationssymmetrie kann mit betrachten von $(a)$ bestätigt werden. +Als hilfe zur Vorstellung kann man \subref{fig:punktgruppen:atoms-piezo-fv} zwischen zwei leitende Platten setzen, so wird ersichtlich, +dass mit wachsendem Druck eine negative Ladung an die rechte Platte gedrückt wird, während sich die positiven Ionen weiter entfernen. +\subref{fig:punktgruppen:atoms-grid} ist nicht piezoelektrisch. +Dies wird ersichtlich, wenn man \subref{fig:punktgruppen:atoms-grid} unterdruck setzt und sich die Struktur zu \subref{fig:punktgruppen:atoms-grid-f} verformt. +Setzt man \subref{fig:punktgruppen:atoms-grid-f} gedanklich auch zwischen zwei leitende Platten, +scheint es als würden rechts mehr Positive Ionen in die Platte gedrückt werden und links umgekehrt. +Dies ist aber nicht mehr der Fall, wenn die Struktur sich nach oben und unten periodisch wiederholt. +Struktur \subref{fig:punktgruppen:atoms-piezo-fh} zeigt \subref{fig:punktgruppen:atoms-piezo} in unter horizontaler Belastung. +Was zwischen \subref{fig:punktgruppen:atoms-piezo-fv} und \subref{fig:punktgruppen:atoms-piezo-fh} zu beobachten ist, +ist dass das entstandene Ladungsdifferenz orthogonal zu der angelegten Kraft entsteht, +im Gegensatz zu \subref{fig:punktgruppen:atoms-piezo-fh}. +Daraus kann man schlissen, dass \subref{fig:punktgruppen:atoms-piezo} keine Rotationssymmetrie von $90^\circ$ besitzen kann, +weil die Eigenschaften ändern bei einer $90^\circ$ Drehung. +Das Fehlen dieser Rotationssymmetrie kann mit betrachten von \subref{fig:punktgruppen:atoms-piezo} bestätigt werden. -\subsection{Punktsymmetrie}\footnote{In der Literatur wird ein Punktsymmetrisches Kristallgitter oft als Kristallgitter mit Inversionszentrum bezeichnet.} +\subsection{Punktsymmetrie} Piezoelektrische Kristalle können nicht Punktsymmetrisch sein. Kristallgitter, bei welchen eine Punktspiegelung eine symmetrische Operation ist, können keine piezoelektrische Kristalle bilden. -Auf Abbildung \ref{fig:punktgruppen:atomPiezo} ist bewusst $(a)$ ein nicht Punktsymmetrischer Kristall mit einem Punktsymmetrischen $(d)$ verglichen worden. +Auf Abbildung \ref{fig:punktgruppen:atomPiezo} ist bewusst \subref{fig:punktgruppen:atoms-piezo} ein nicht Punktsymmetrischer Kristall +mit einem Punktsymmetrischen \subref{fig:punktgruppen:atoms-grid}verglichen worden. Als vereinfachte Erklärung kann mann sich wieder das Bild vor augen führen, eines Kristalles, welcher unter Druck auf der einen Seite negative und der anderen Seite positive Ionen an seine Oberfläche verdrängt. Spiegelt man nun den Kristall um den Gitterpunkt in der mitte des Kristalles, so würden die negativen Ionen auf den Positiven auf der anderen seite landen, @@ -70,5 +80,4 @@ Sollten Sie also eines Tages in die Situation geraten, in welcher Sie zwei versc und ein piezoelektrisches Feuerzeug bauen müssen, wobei Sie aber wissen, dass einer eine Punktsymmetrie aufweist, versuche sie es mit dem anderen. -Ich muss aber anmerken, dass aus den $21$ möglichen Kristallsymmetrien ohne Punktsymmetrie einer nicht piezoelektrisch ist. -ein wenig glück brauchen Sie also immer noch. + diff --git a/buch/papers/punktgruppen/references.bib b/buch/papers/punktgruppen/references.bib index 9edb8bd..a29640c 100644 --- a/buch/papers/punktgruppen/references.bib +++ b/buch/papers/punktgruppen/references.bib @@ -33,3 +33,12 @@ inseries = {Vorlesungsskript zum Modul ELT}, } +@online{punktgruppen:ouchem, + title = {Symmetry in Crystallography}, + author = {Dept. of Chemistry \& Biochemistry, Chemical Crystallography Laboratory, University of Oklahoma}, + year = {2019}, + month = {11}, + day = {17}, + url = {http://archive.today/2021.07.22-083802/http://xrayweb.chem.ou.edu/notes/symmetry.html}, + urldate = {2021-07-22}, +} diff --git a/buch/papers/punktgruppen/symmetry.tex b/buch/papers/punktgruppen/symmetry.tex index 1dc6f98..0bb4aec 100644 --- a/buch/papers/punktgruppen/symmetry.tex +++ b/buch/papers/punktgruppen/symmetry.tex @@ -1,175 +1,134 @@ \section{Symmetrie} Das Wort Symmetrie ist sehr alt und hat sich seltsamerweise von seinem -ursprünglichen griechischen Wort -\(\mathrm{\Sigma\nu\mu\mu\varepsilon\tau\rho\iota\alpha}\) -\footnote{\emph{Symmetr\'ia}: ein gemeinsames Mass habend, gleichmässig, -verhältnismässig} fast nicht verändert. In der Alltagssprache mag es ein -locker definierter Begriff sein, aber in der Mathematik hat Symmetrie eine sehr -präzise Bedeutung. +ursprünglichen griechischen Wort \(\mathrm{\Sigma\upsilon\mu\mu\varepsilon\tau\rho\iota\alpha}\)\footnote{\emph{Symmetr\'ia}: ein gemeinsames Mass habend, gleichmässig,verhältnismässig} fast nicht verändert. +In der Alltagssprache mag es ein locker definierter Begriff sein, in der Mathematik hat Symmetrie jedoch eine sehr präzise Bedeutung. \begin{definition}[Symmetrie] - Ein mathematisches Objekt wird als symmetrisch bezeichnet, wenn es unter einer - bestimmten Operation invariant ist. + Ein mathematisches Objekt wird als symmetrisch bezeichnet, wenn es unter einer bestimmten Operation invariant ist. \end{definition} -Die intuitivsten Beispiele kommen aus der Geometrie, daher werden wir mit -einigen geometrischen Beispielen beginnen. Wie wir jedoch später sehen werden, -ist das Konzept der Symmetrie eigentlich viel allgemeiner. +Die intuitivsten Beispiele kommen aus der Geometrie, daher werden wir mit einigen geometrischen Beispielen beginnen. +Wie wir jedoch später sehen werden, ist das Konzept der Symmetrie eigentlich viel allgemeiner. \begin{figure} - \centering - \includegraphics{papers/punktgruppen/figures/symmetric-shapes} - \caption{ - Beispiele für geometrisch symmetrische Formen. - \label{fig:punktgruppen:geometry-example} - } + \centering + \includegraphics{papers/punktgruppen/figures/symmetric-shapes} + \caption{ + Beispiele für geometrisch symmetrische Formen. + \label{fig:punktgruppen:geometry-example} + } \end{figure} \subsection{Geometrische Symmetrien} -In Abbildung \ref{fig:punktgruppen:geometry-example} haben wir einige Formen, -die offensichtlich symmetrisch sind. Zum Beispiel hat das Quadrat eine Gerade, an -deren es gespiegelt werden kann, ohne sein Aussehen zu verändern. Regelmässige -Polygone mit \(n\) Seiten sind auch gute Beispiele, um eine diskrete -Rotationssymmetrie zu veranschaulichen, was bedeutet, dass eine Drehung um -einen Punkt um einen bestimmten Winkel \(360^\circ/n\) die Figur unverändert -lässt. Das letzte Beispiel auf der rechten Seite ist eine unendliche -Rotationssymmetrie. Sie wird so genannt, weil es unendlich viele Werte für -\(\alpha \in \mathbb{R}\) gibt, die die Form unverändert lassen. Dies ist -hoffentlich ausreichend, um die Bedeutung hinter der Notation zu verstehen, die -nun eingeführt wird. - -% Vieleicht eine kurze Einführung in für die Definition, ich habe das gefühl, dass in der Definition die Symmetrie-Operation und die Gruppe auf einmal erklährt wird -\subsubsection{Symetriegruppe} -\texttt{TODO: review this paragraph, explain what is \(\mathds{1}\).} +In Abbildung \ref{fig:punktgruppen:geometry-example} haben wir einige Formen, die offensichtlich symmetrisch sind. +Zum Beispiel hat das Quadrat eine Gerade, an deren es gespiegelt werden kann, ohne sein Aussehen zu verändern. +Regelmässige Polygone mit \(n\) Seiten sind auch gute Beispiele, um eine diskrete Rotationssymmetrie zu veranschaulichen, was bedeutet, dass eine Drehung um einen Punkt um einen bestimmten Winkel \(360^\circ/n\) die Figur unverändert lässt. +Das letzte Beispiel auf der rechten Seite ist eine unendliche Rotationssymmetrie. Sie wird so genannt, weil es unendlich viele Werte für \(\alpha \in \mathbb{R}\) gibt, die die Form unverändert lassen. Ein Objekt kann mehr als nur eine Symmetrie aufweisen. -Als Beispiel, kann das Quadrat in Abbildung \ref{fig:punktgruppen:geometry-example} -nicht nur um $\sigma$ sondern auch Diagonal gespiegelt werden oder um $90^\circ$ gedreht werden. -Fässt man die möglichen Symmetrien zusammen, entsteht eine Symmetriegruppe. +Als Beispiel, kann das Quadrat in Abbildung \ref{fig:punktgruppen:geometry-example} nicht nur um \(\sigma\) sondern auch Diagonal gespiegelt werden oder um \(90^\circ\) gedreht werden. +Fasst man die möglichen Symmetrien zusammen, entsteht eine Symmetriegruppe. \begin{definition}[Symmetriegruppe] - Sei \(g\) eine Operation, die ein mathematisches Objekt unverändert lässt. - Bei einer anderen Operation \(h\) definieren wir die Komposition \(h\circ g\) - als die Anwendung der Operationen nacheinander. Alle Operationen bilden unter - Komposition eine Gruppe, die Symmetriegruppe genannt wird. -\end{definition} % ich lese diese Definition ein wenig holprig, vieleicht können wir sie zusammen anschauen + \(g\) und \(h\) sein umkehrbare Operationen, die ein mathematisches Objekt unverändert lassen. + Die Komposition \(h\circ g\) definieren wir als die Anwendung der Operationen nacheinander. + Alle möglichen Operationen bilden unter Komposition eine Gruppe, die Symmetriegruppe genannt wird. +\end{definition} + +Eine Gruppe benötigt ausserdem auch zwingend ein neutrales Element, welches wir mit \(\mathds{1}\) bezeichnen. +Die Anwendung der neutralen Operation ist gleichbedeutend damit, alles unverändert zu lassen. +\(\mathds{1}\) ist auch äquivalent dazu, eine Operation anzuwenden und sie dann rückgängig zu machen (ihre Inverse anzuwenden). + Die Definition der Symmetriegruppe ist mit der Kompositionsoperation gegeben, es wird aber auch oft als Multiplikation geschrieben. +Das liegt daran, dass in manchen Fällen die Zusammensetzung algebraisch durch eine Multiplikation berechnet wird. +Die Verwendung einer multiplikativen Schreibweise ermöglicht es, einige Ausdrücke kompakter zu schreiben, z.B. +durch Verwendung von Potenzen \(r^n = r\circ r \circ \cdots r\circ r\) für eine wiederholte Komposition. -% Nach meinem Geschmack könne es hier auch eine einleitung wie mein Beispiel geben dammit man den Text flüssiger lesen kann \begin{definition}[Zyklische Untergruppe, Erzeuger] - Sei \(g\) ein Element einer Symmetriegruppe \(G\). Alle möglichen - Kompositionen von \(g\) und \(g^{-1}\) bilden eine sogenannte zyklische - Untergruppe von \(G\), und \(g\) wird ihr Erzeuger genannt. Die erzeugte - Untergruppe \(\langle g \rangle\) wird mit spitzen Klammern um den Erzeuger - bezeichnet. + \(g\) sei ein Element einer Symmetriegruppe \(G\). + Alle möglichen Kompositionen von \(g\) und \(g^{-1}\) bilden eine sogenannte zyklische Untergruppe von \(G\), wobei \(g\) Erzeuger der Untergruppe genannt wird. + Die von \(g\) erzeugte Untergruppe \(\langle g \rangle = \left\{ g^k : k \in \mathbb{Z} \right\}\) wird mit spitzen Klammern bezeichnet. \end{definition} +\begin{beispiel} + Um die Syntax zu verstehen, betrachten wir eine durch \(a\) erzeugte Gruppe \(G = \langle a \rangle\). + Das bedeutet, dass \(G\) die Elemente \(a, aa, aaa, \ldots\) sowie \(a^{-1}, a^{-1}a^{-1}, \ldots\) und ein neutrales Element \(\mathds{1} = aa^{-1}\) enthält. +\end{beispiel} +\begin{beispiel} + Als anschaulicheres Beispiel, können wir eine Zyklische Untergruppe des \(n\)-Gon formalisieren. + Wir bezeichnen mit \(r\) eine Drehung im Gegenuhrzeigersinn von \(360^\circ/n\) um einen Punkt. + Diese Definition reicht aus, um die gesamte Symmetriegruppe + \[ + C_n = \langle r \rangle + = \left\{\mathds{1}, r, r^2, \ldots, r^{n-1}\right\} + \] + der Drehungen eines \(n\)-Gons zu erzeugen. + Das liegt daran, dass wir durch die mehrfache Verwendung von \(r\) jeden Winkel erzeugen k\"onnen, der die Rotationssymmetrie bewahrt. + In ähnlicher Weise, aber weniger interessant enthält die Reflexionssymmetriegruppe \(\langle\sigma\rangle\) nur \(\left\{\mathds{1}, \sigma\right\}\), weil \(\sigma^2 = \mathds{1}\). +\end{beispiel} -Mit dem oben Gesagten können wir das \(n\)-Gon Beispiel formalisieren. -Bezeichnen wir mit \(r\) eine Drehung im Gegenuhrzeigersinn von \(360^\circ/n\) -um einen Punkt. Diese Definition reicht aus, um die gesamte Symmetriegruppe -\[ - C_n = \langle r \rangle - = \left\{\mathds{1}, r, r^2, \ldots, r^{n-1}\right\} -\] -der Drehungen eines \(n\)-Gons zu definieren. Das liegt daran, -dass wir durch die mehrfache Verwendung von \(r\) jeden Winkel erzeugen, der -die Rotationssymmetrie bewahrt. Hier die Potenzen von \(r\) sind als -wiederholte Komposition gemeint, dass heisst \(r^n = r\circ r \circ \cdots -r\circ r\). Wenn wir diese Idee nun erweitern, können wir mit einem -Erzeugendensystemen komplexere Strukturen aufbauen. +Wenn wir diese Idee nun erweitern, können wir mit einem Erzeugendensystemen +komplexere Strukturen aufbauen. \begin{definition}[Erzeugendensysteme] - % please fix this unreadable mess - Jede Gruppe kann durch eines oder mehrere ihrer Elemente generiert werden. - Wir lassen \(g_1, g_2, \ldots, g_n\) erzeugenden Elemente einer - Symmetriegruppe sein. Da es mehrere Erzeuger gibt, müssen auch die - sogenannte Definitionsgleichungen gegeben werden, die die - Multiplikationstabelle vollständig definieren. Die Gleichungen sind ebenfalls - in den Klammern angegeben. Die erzeugende Elementen zusammen mit der - Definitionsgleichungen bauen ein Erzeugendensysteme. + Jede disktrete Gruppe kann durch eines oder mehrere ihrer Elemente generiert werden. + Wir lassen \(g_1, g_2, \ldots, g_n\) erzeugenden Elemente einer Symmetriegruppe sein. + Da es mehrere Erzeuger gibt, müssen auch die sogenannte Definitionsgleichungen gegeben werden, die die Multiplikationstabelle vollständig definieren. + Die Gleichungen sind ebenfalls in den Klammern angegeben. + Die erzeugende Elementen zusammen mit der Definitionsgleichungen bauen ein Erzeugendensysteme. \end{definition} +\begin{beispiel} + Wir werden nun alle Symmetrien eines \(n\)-Gons beschreiben, was bedeutet, dass wir die Operationen \(r\) und \(\sigma\) kombinieren. + Die Definitionsgleichungen sind \(r^n = \mathds{1}\), \(\sigma^2 = \mathds{1}\) und \((\sigma r)^2 = \mathds{1}\). + Die ersten beiden sind ziemlich offensichtlich. + Die letzte wird oft auch als Inversion bezeichnet, weil die Anwendung von \(\sigma r\) dasselbe ist wie das Ziehen einer Linie von einem Punkt, die durch den Ursprung geht, und das Verschieben des Punktes auf die andere Seite des Nullpunkts. + Wenn man dies zweimal macht, geht man zurück zum Anfangspunkt. + Daraus ergibt sich die so genannte Diedergruppe + \begin{align*} + D_n &= \langle r, \sigma : r^n = \sigma^2 = (\sigma r)^2 = \mathds{1} \rangle \\ + &= \left\{ + \mathds{1}, r, \ldots, r^{n-1}, \sigma, \sigma r, \ldots, \sigma r^{n-1} + \right\}. + \end{align*} +\end{beispiel} -\texttt{TODO: should put examples for generators?} \\ - -Die Reflexionssymmetriegruppe ist nicht so interessant, da sie nur -\(\left\{\mathds{1}, \sigma\right\}\) enthält. Kombiniert man sie jedoch mit -der Rotation, erhält man die so genannte Diedergruppe -\[ - D_n = \langle r, \sigma : r^{n-1} = \sigma^2 = (\sigma r)^2 = \mathds{1} \rangle - = \left\{ - \mathds{1}, r, \ldots, r^{n-1}, \sigma, \sigma r, \ldots, \sigma r^{n-1} - \right\}. -\] - -Die Symmetrieoperationen, die wir bis jetzt besprochen haben, haben immer -mindestens einen Punkt gehabt, der wieder auf sich selbst abgebildet wird. Im -Fall der Rotation war es der Drehpunkt, bei der Spiegelung die Punkte der -Spiegelachse. Dies ist jedoch keine Voraussetzung für eine Symmetrie, da es -Symmetrien gibt, die jeden Punkt zu einem anderen Punkt verschieben können. -Diesen Spezialfall, bei dem mindestens ein Punkt unverändert bleibt, nennt man -Punktsymmetrie. +Die Symmetrieoperationen, die wir bis jetzt besprochen haben, haben immer mindestens einen Punkt gehabt, der wieder auf sich selbst abgebildet wird. +Im Fall der Rotation war es der Drehpunkt, bei der Spiegelung die Punkte der Spiegelachse. +Dies ist jedoch keine Voraussetzung für eine Symmetrie, da es Symmetrien gibt, die jeden Punkt zu einem anderen Punkt verschieben können. + Diesen Spezialfall, bei dem immer mindestens ein Punkt unverändert bleibt, nennt man Punktsymmetrie. \begin{definition}[Punktgruppe] - Wenn jede Operation in einer Symmetriegruppe die Eigenschaft hat, mindestens - einen Punkt unverändert zu lassen, sagt man, dass die Symmetriegruppe eine - Punktgruppe ist. + Wenn es einen Punkt gibt, der von jeder Gruppenoperation unverändert gelassen wird, ist die Symmetriegruppe eine Punktgruppe. \end{definition} \subsection{Algebraische Symmetrien} -Wir haben nun unseren Operationen Symbole gegeben, mit denen es tatsächlich -möglich ist, Gleichungen zu schreiben. Die naheliegende Frage ist dann, könnte -es sein, dass wir bereits etwas haben, das dasselbe tut? Natürlich, ja. +Wir haben nun unseren Operationen Symbole gegeben, mit denen es tatsächlich möglich ist, Gleichungen zu schreiben. +Die anschliesende Frage ist dann, ob wir bereits mathematische Objekte haben, mit denen wir Gleichungen schreiben, die sich auf die gleiche Weise verhalten. +Die Antwort lautet natürlich ja. Um es formaler zu beschreiben, werden wir einige Begriffe einführen. \begin{definition}[Gruppenhomomorphismus] - Seien \(G\) und \(H\) Gruppe mit unterschiedlicher Operation \(\diamond\) - bzw. \(\star\). Ein Homomorphismus\footnote{ Für eine ausführlichere - Diskussion siehe \S\ref{buch:grundlagen:subsection:gruppen} im Buch.} ist - eine Funktion \(f: G \to H\), so dass für jedes \(a, b \in G\) gilt - \(f(a\diamond b) = f(a) \star f(b)\). Man sagt, dass der Homomorphismus - \(f\) \(G\) in \(H\) transformiert. + \(G\) und \(H\) seien Gruppen mit unterschiedlichen Operationen \(\diamond\) bzw. + \(\star\). + Ein Homomorphismus\footnote{ Für eine ausführlichere Diskussion siehe \S\ref{buch:grundlagen:subsection:gruppen} im Buch.} ist eine Funktion \(f: G \to H\), so dass für jedes \(a, b \in G\) gilt \(f(a\diamond b) = f(a) \star f(b)\). + Man sagt, dass der Homomorphismus \(f\) \(G\) in \(H\) transformiert. \end{definition} \begin{beispiel} - Die Rotationssymmetrie des Kreises \(C_\infty\), mit einem unendlichen - Kontinuum von Werten \(\alpha \in \mathbb{R}\), entspricht perfekt dem - komplexen Einheitskreis. Der Homomorphismus \(\phi: C_\infty \to \mathbb{C}\) - ist durch die Eulersche Formel \(\phi(r) = e^{i\alpha}\) gegeben. + Die Rotationssymmetrie des Kreises \(C_\infty\), mit einem unendlichen Kontinuum von Werten \(\alpha \in \mathbb{R}\), entspricht perfekt dem komplexen Einheitskreis. + Der Homomorphismus \(\phi: C_\infty \to \mathbb{C}\) ist durch die Eulersche Formel \(\phi(r) = e^{i\alpha}\) gegeben. \end{beispiel} \begin{definition}[Darstellung einer Gruppe] - Die Darstellung einer Gruppe ist ein Homomorphismus, der eine Symmetriegruppe - auf eine Menge von Matrizen abbildet. - \[ - \Phi: G \to \operatorname{GL}_n(\mathbb{R}). - \] - Äquivalent kann man sagen, dass ein Element aus der Symmetriegruppe auf einen - Vektorraum \(V\) wirkt, indem man definiert \(\Phi : G \times V \to V\). + Die Darstellung einer Gruppe ist ein Homomorphismus, der eine Symmetriegruppe auf eine Menge von Matrizen abbildet. + \[ + \Phi: G \to \operatorname{GL}_n(\mathbb{R}). + \] + Äquivalent kann man sagen, dass ein Element aus der Symmetriegruppe auf einen Vektorraum \(V\) wirkt, indem man definiert \(\Phi : G \times V \to V\). \end{definition} \begin{beispiel} - Die Elemente \(r^k \in C_n\), wobei \(0 < k < n\), stellen abstrakt eine - Drehung von \(2\pi k/n\) um den Ursprung dar. Die mit der Matrix - \[ - \Phi(r^k) = \begin{pmatrix} - \cos(2\pi k/n) & -\sin(2\pi k/n) \\ - \sin(2\pi k/n) & \cos(2\pi k/n) - \end{pmatrix} - \] - definierte Funktion von \(C_n\) nach \(O(2)\) ist eine Darstellung von - \(C_n\). In diesem Fall ist die erste Gruppenoperation die Komposition und - die zweite die Matrixmultiplikation. Man kann überprüfen, dass \(\Phi(r^2 - \circ r) = \Phi(r^2)\Phi(r)\). + Die Elemente \(r^k \in C_n\), wobei \(0 < k < n\), stellen abstrakt eine Drehung von \(2\pi k/n\) um den Ursprung dar. + Die mit der Matrix + \[ + \Phi(r^k) = \begin{pmatrix} + \cos(2\pi k/n) & -\sin(2\pi k/n) \\ + \sin(2\pi k/n) & \cos(2\pi k/n) + \end{pmatrix} + \] + definierte Funktion von \(C_n\) nach \(O(2)\) ist eine Darstellung von \(C_n\). + In diesem Fall ist die erste Gruppenoperation die Komposition und die zweite die Matrixmultiplikation. + Man kann überprüfen, dass \(\Phi(r^2 \circ r) = \Phi(r^2)\Phi(r)\). \end{beispiel} - -\texttt{TODO: rewrite section on translational symmetry.} -%% TODO: title / fix continuity -% Um das Konzept zu illustrieren, werden wir den umgekehrten Fall diskutieren: -% eine Symmetrie, die keine Punktsymmetrie ist, die aber in der Physik sehr -% nützlich ist, nämlich die Translationssymmetrie. Von einem mathematischen -% Objekt \(U\) wird gesagt, dass es eine Translationssymmetrie \(Q(x) = x + a\) -% hat, wenn es die Gleichung -% \[ -% U(x) = U(Q(x)) = U(x + a), -% \] -% für ein gewisses \(a\), erfüllt. Zum Beispiel besagt das erste Newtonsche -% Gesetz, dass ein Objekt, auf das keine Kraft einwirkt, eine -% zeitranslationsinvariante Geschwindigkeit hat, d.h. wenn \(\vec{F} = \vec{0}\) -% dann \(\vec{v}(t) = \vec{v}(t + \tau)\). - -% \subsection{Sch\"onflies notation} - -% vim:ts=2 sw=2 spell spelllang=de: diff --git a/buch/papers/punktgruppen/tikz/atoms-grid-force.tex b/buch/papers/punktgruppen/tikz/atoms-grid-force.tex new file mode 100644 index 0000000..05742cf --- /dev/null +++ b/buch/papers/punktgruppen/tikz/atoms-grid-force.tex @@ -0,0 +1,42 @@ +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} + +\usetikzlibrary{arrows} +\usetikzlibrary{intersections} +\usetikzlibrary{math} +\usetikzlibrary{positioning} +\usetikzlibrary{arrows.meta} +\usetikzlibrary{shapes.misc} +\usetikzlibrary{calc} + +\begin{document} + \begin{tikzpicture}[ + >=latex, + node distance = 2mm, + charge/.style = { + circle, draw = black, thick, + minimum size = 5mm + }, + positive/.style = { fill = red!50 }, + negative/.style = { fill = blue!50 }, + ] + + \matrix[nodes = { charge }, row sep = 5mm, column sep = 1cm] { + \node[positive] (NW) {}; & \node[negative] (N) {}; & \node [positive] (NE) {}; \\ + \node[negative] (W) {}; & \node[positive] {}; & \node [negative] (E) {}; \\ + \node[positive] (SW) {}; & \node[negative] (S) {}; & \node [positive] (SE) {}; \\ + }; + + \foreach \d in {NW, N, NE} { + \draw[orange, very thick, <-] (\d) to ++(0,.7); + } + + \foreach \d in {SW, S, SE} { + \draw[orange, very thick, <-] (\d) to ++(0,-.7); + } + + \draw[gray, dashed] (W) to (N) to (E) to (S) to (W); + \end{tikzpicture} +\end{document} diff --git a/buch/papers/punktgruppen/tikz/atoms-grid-still.tex b/buch/papers/punktgruppen/tikz/atoms-grid-still.tex new file mode 100644 index 0000000..4e43856 --- /dev/null +++ b/buch/papers/punktgruppen/tikz/atoms-grid-still.tex @@ -0,0 +1,33 @@ +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} + +\usetikzlibrary{arrows} +\usetikzlibrary{intersections} +\usetikzlibrary{math} +\usetikzlibrary{positioning} +\usetikzlibrary{arrows.meta} +\usetikzlibrary{shapes.misc} +\usetikzlibrary{calc} + +\begin{document} + \begin{tikzpicture}[ + >=latex, + node distance = 2mm, + charge/.style = { + circle, draw = black, thick, + minimum size = 5mm + }, + positive/.style = { fill = red!50 }, + negative/.style = { fill = blue!50 }, + ] + + \matrix[nodes = { charge }, row sep = 8mm, column sep = 8mm] { + \node[positive] {}; & \node[negative] (N) {}; & \node [positive] {}; \\ + \node[negative] (W) {}; & \node[positive] {}; & \node [negative] (E) {}; \\ + \node[positive] {}; & \node[negative] (S) {}; & \node [positive] {}; \\ + }; + \draw[gray, dashed] (W) to (N) to (E) to (S) to (W); + \end{tikzpicture} +\end{document} diff --git a/buch/papers/punktgruppen/tikz/atoms-piezo-force-horizontal.tex b/buch/papers/punktgruppen/tikz/atoms-piezo-force-horizontal.tex new file mode 100644 index 0000000..e4c3f93 --- /dev/null +++ b/buch/papers/punktgruppen/tikz/atoms-piezo-force-horizontal.tex @@ -0,0 +1,47 @@ +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} + +\usetikzlibrary{arrows} +\usetikzlibrary{intersections} +\usetikzlibrary{math} +\usetikzlibrary{positioning} +\usetikzlibrary{arrows.meta} +\usetikzlibrary{shapes.misc} +\usetikzlibrary{calc} + +\begin{document} + \begin{tikzpicture}[ + >=latex, + node distance = 2mm, + charge/.style = { + circle, draw = black, thick, + minimum size = 5mm + }, + positive/.style = { fill = red!50 }, + negative/.style = { fill = blue!50 }, + ] + + \node[charge, positive, yshift= 2.5mm] (C1) at ( 60:1.5cm) {}; + \node[charge, negative, yshift= 2.5mm] (C2) at (120:1.5cm) {}; + \node[charge, positive, xshift= 2.5mm] (C3) at (180:1.5cm) {}; + \node[charge, negative, yshift=-2.5mm] (C4) at (240:1.5cm) {}; + \node[charge, positive, yshift=-2.5mm] (C5) at (300:1.5cm) {}; + \node[charge, negative, xshift=-2.5mm] (C6) at (360:1.5cm) {}; + + \draw[black] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1); + % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2); + + \draw[orange, very thick, <-] (C6) to ++(.7,0); + \draw[orange, very thick, <-] (C3) to ++(-.7,0); + + \node[black] (E) {\(\vec{E}_p\)}; + \begin{scope}[node distance = .5mm] + \node[blue!50, right = of E] {\(-\)}; + \node[red!50, left = of E] {\(+\)}; + \end{scope} + % \draw[gray, thick, dotted] (E) to ++(0,2); + % \draw[gray, thick, dotted] (E) to ++(0,-2); + \end{tikzpicture} +\end{document} diff --git a/buch/papers/punktgruppen/tikz/atoms-piezo-force-vertical.tex b/buch/papers/punktgruppen/tikz/atoms-piezo-force-vertical.tex new file mode 100644 index 0000000..892ab42 --- /dev/null +++ b/buch/papers/punktgruppen/tikz/atoms-piezo-force-vertical.tex @@ -0,0 +1,52 @@ +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} + +\usetikzlibrary{arrows} +\usetikzlibrary{intersections} +\usetikzlibrary{math} +\usetikzlibrary{positioning} +\usetikzlibrary{arrows.meta} +\usetikzlibrary{shapes.misc} +\usetikzlibrary{calc} + +\begin{document} + \begin{tikzpicture}[ + >=latex, + node distance = 2mm, + charge/.style = { + circle, draw = black, thick, + minimum size = 5mm + }, + positive/.style = { fill = red!50 }, + negative/.style = { fill = blue!50 }, + ] + + \node[charge, positive, yshift=-2.5mm] (C1) at ( 60:1.5cm) {}; + \node[charge, negative, yshift=-2.5mm] (C2) at (120:1.5cm) {}; + \node[charge, positive, xshift=-2.5mm] (C3) at (180:1.5cm) {}; + \node[charge, negative, yshift= 2.5mm] (C4) at (240:1.5cm) {}; + \node[charge, positive, yshift= 2.5mm] (C5) at (300:1.5cm) {}; + \node[charge, negative, xshift= 2.5mm] (C6) at (360:1.5cm) {}; + + \draw[black] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1); + % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2); + + \foreach \d in {C1, C2} { + \draw[orange, very thick, <-] (\d) to ++(0,.7); + } + + \foreach \d in {C4, C5} { + \draw[orange, very thick, <-] (\d) to ++(0,-.7); + } + + \node[black] (E) {\(\vec{E}_p\)}; + \begin{scope}[node distance = .5mm] + \node[red!50, right = of E] {\(+\)}; + \node[blue!50, left = of E] {\(-\)}; + \end{scope} + % \draw[gray, thick, dotted] (E) to ++(0,2); + % \draw[gray, thick, dotted] (E) to ++(0,-2); + \end{tikzpicture} +\end{document} diff --git a/buch/papers/punktgruppen/tikz/atoms-piezo-still.tex b/buch/papers/punktgruppen/tikz/atoms-piezo-still.tex new file mode 100644 index 0000000..2eb78ba --- /dev/null +++ b/buch/papers/punktgruppen/tikz/atoms-piezo-still.tex @@ -0,0 +1,34 @@ +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} + +\usetikzlibrary{arrows} +\usetikzlibrary{intersections} +\usetikzlibrary{math} +\usetikzlibrary{positioning} +\usetikzlibrary{arrows.meta} +\usetikzlibrary{shapes.misc} +\usetikzlibrary{calc} + +\begin{document} + \begin{tikzpicture}[ + >=latex, + node distance = 2mm, + charge/.style = { + circle, draw = black, thick, + minimum size = 5mm + }, + positive/.style = { fill = red!50 }, + negative/.style = { fill = blue!50 }, + ] + + \foreach \x/\t [count=\i] in {60/positive, 120/negative, 180/positive, 240/negative, 300/positive, 360/negative} { + \node[charge, \t] (C\i) at (\x:1.5cm) {}; + } + + \draw[black] (C1) to (C2) to (C3) to (C4) to (C5) to (C6) to (C1); + \node[circle, draw=gray, fill=gray, outer sep = 0, inner sep = 0, minimum size = 3mm] {}; + % \draw[gray, dashed] (C2) to (C4) to (C6) to (C2); + \end{tikzpicture} +\end{document} diff --git a/buch/papers/punktgruppen/tikz/combine-symmetries.tex b/buch/papers/punktgruppen/tikz/combine-symmetries.tex index 84e0a76..fa669ae 100644 --- a/buch/papers/punktgruppen/tikz/combine-symmetries.tex +++ b/buch/papers/punktgruppen/tikz/combine-symmetries.tex @@ -13,6 +13,7 @@ \begin{document} \begin{tikzpicture}[ + >=latex, dot/.style = { draw, circle, thick, black, fill = gray!40!white, minimum size = 2mm, @@ -45,7 +46,7 @@ (A2) ++(-.5,0) arc (180:60:.5); \draw[red!80!black, dashed, thick, ->] (A2) to (B2); - \draw[yellow!50!orange, thick, ->] + \draw[cyan!40!blue, thick, ->] (B1) to node[above, midway] {\(\vec{Q}'\)} (B2); \draw[gray, dashed, thick] (A1) to (A1 |- B1) node (Xl) {}; diff --git a/buch/papers/punktgruppen/tikz/lattice.tex b/buch/papers/punktgruppen/tikz/lattice.tex index 9c05af3..a6b1876 100644 --- a/buch/papers/punktgruppen/tikz/lattice.tex +++ b/buch/papers/punktgruppen/tikz/lattice.tex @@ -13,23 +13,24 @@ \begin{document} \begin{tikzpicture}[ - dot/.style = { - draw, circle, thick, black, fill = gray!40!white, - minimum size = 2mm, - inner sep = 0pt, - outer sep = 1mm, - }, + >=latex, + dot/.style = { + draw, circle, thick, black, fill = gray!40!white, + minimum size = 2mm, + inner sep = 0pt, + outer sep = 1mm, + }, ] \begin{scope} - \clip (-2,-2) rectangle (3,4); + \clip (-2,-2) rectangle (7,2); \foreach \y in {-7,-6,...,7} { \foreach \x in {-7,-6,...,7} { \node[dot, xshift=3mm*\y] (N\x\y) at (\x, \y) {}; } } \end{scope} - \draw[black, thick] (-2, -2) rectangle (3,4); + \draw[black, thick] (-2, -2) rectangle (7,2); \draw[red!80!black, thick, ->] (N00) to node[midway, below] {\(\vec{a}_1\)} (N10); diff --git a/buch/papers/punktgruppen/tikz/piezo-atoms.tex b/buch/papers/punktgruppen/tikz/piezo-atoms.tex index 82a2710..1811392 100644 --- a/buch/papers/punktgruppen/tikz/piezo-atoms.tex +++ b/buch/papers/punktgruppen/tikz/piezo-atoms.tex @@ -13,6 +13,7 @@ \begin{document} \begin{tikzpicture}[ + >=latex, node distance = 2mm, charge/.style = { circle, draw = black, thick, diff --git a/buch/papers/punktgruppen/tikz/piezo.tex b/buch/papers/punktgruppen/tikz/piezo.tex index 1d16ab7..6542f26 100644 --- a/buch/papers/punktgruppen/tikz/piezo.tex +++ b/buch/papers/punktgruppen/tikz/piezo.tex @@ -12,12 +12,14 @@ \usetikzlibrary{calc} \begin{document} -\begin{tikzpicture} +\begin{tikzpicture}[ + >=latex, + ] \begin{scope}[ node distance = 0cm ] \node[ - rectangle, fill = gray!60!white, + rectangle, fill = gray!20!white, minimum width = 3cm, minimum height = 2cm, ] (body) {\(\vec{E}_p = \vec{0}\)}; @@ -43,9 +45,9 @@ xshift = 7cm ] \node[ - rectangle, fill = gray!40!white, + rectangle, fill = gray!20!white, minimum width = 3cm, minimum height = 1.5cm, - ] (body) {\(\vec{E}_p = \vec{0}\)}; + ] (body) {\(\vec{E}_p \neq \vec{0}\)}; \node[ draw, rectangle, thick, black, fill = red!50, diff --git a/buch/papers/punktgruppen/tikz/projections.tex b/buch/papers/punktgruppen/tikz/projections.tex index a763e77..64ab468 100644 --- a/buch/papers/punktgruppen/tikz/projections.tex +++ b/buch/papers/punktgruppen/tikz/projections.tex @@ -13,6 +13,7 @@ \begin{document} \begin{tikzpicture}[ + >=latex, classcirc/.style = { draw = gray, thick, circle, minimum size = 12mm, diff --git a/buch/papers/punktgruppen/tikz/stereographic-projections.tex b/buch/papers/punktgruppen/tikz/stereographic-projections.tex new file mode 100644 index 0000000..7d612fb --- /dev/null +++ b/buch/papers/punktgruppen/tikz/stereographic-projections.tex @@ -0,0 +1,108 @@ +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{tikz-3dplot} + +\usetikzlibrary{arrows} +\usetikzlibrary{intersections} +\usetikzlibrary{math} +\usetikzlibrary{positioning} +\usetikzlibrary{arrows.meta} +\usetikzlibrary{shapes.misc} +\usetikzlibrary{calc} + +\begin{document} + +\tdplotsetmaincoords{60}{130} +\pgfmathsetmacro{\l}{2} + +\begin{tikzpicture}[ + >=latex, + tdplot_main_coords, + dot/.style = { + black, fill = black, circle, + outer sep = 0, inner sep = 0, + minimum size = 1mm + }, + round/.style = { + draw = orange, thick, circle, + minimum size = 1mm, + inner sep = 0pt, outer sep = 0pt, + }, + cross/.style = { + cross out, draw = magenta, thick, + minimum size = 1mm, + inner sep = 0pt, outer sep = 0pt + }, + ] + + % origin + \coordinate (O) at (0,0,0); + + % poles + \coordinate (NP) at (0,0,\l); + \coordinate (SP) at (0,0,-\l); + + % axis + % \draw[->] (O) -- ++(1.5*\l,0,0); + % \draw[->] (O) -- ++(0,1.5*\l,0); + % \draw[->] (O) -- ++(0,0,1.5*\l); + + % gray unit circle + \tdplotdrawarc[gray, thick]{(O)}{\l}{0}{360}{}{}; + \draw[gray, dotted] (-\l, 0, 0) to (\l, 0, 0); + \draw[gray, dotted] (0, -\l, 0) to (0, \l, 0); + + % meridians + \foreach \phi in {0, 30, 60, ..., 150}{ + \tdplotsetrotatedcoords{\phi}{90}{0}; + \tdplotdrawarc[lightgray, dashed, tdplot_rotated_coords]{(O)}{\l}{0}{360}{}{}; + } + + % dot above and its projection + \pgfmathsetmacro{\phi}{120} + \pgfmathsetmacro{\theta}{60} + + \pgfmathsetmacro{\px}{cos(\phi)*sin(\theta)*\l} + \pgfmathsetmacro{\py}{sin(\phi)*sin(\theta)*\l} + \pgfmathsetmacro{\pz}{cos(\theta)*\l}) + + \coordinate (A) at (\px,\py,\pz); + \coordinate (Aproj) at ({\px * \l / (\l + \pz)}, {\py * \l / (\l + \pz)}, 0); + + % dot below and its projection + \pgfmathsetmacro{\phi}{-60} + \pgfmathsetmacro{\theta}{120} + + \pgfmathsetmacro{\px}{cos(\phi)*sin(\theta)*\l} + \pgfmathsetmacro{\py}{sin(\phi)*sin(\theta)*\l} + \pgfmathsetmacro{\pz}{cos(\theta)*\l}) + + \coordinate (B) at (\px,\py,\pz); + \coordinate (Bproj) at ({\px * \l / (\l - \pz)}, {\py * \l / (\l - \pz)}, 0); + + % projection lines + \draw[gray] (A) to (SP); + \draw[gray] (SP) to (O) to (Aproj); + + \draw[gray] (B) to (NP); + \draw[gray] (NP) to (O) to (Bproj); + + % dots + \draw (O) node[dot] {}; + \draw (SP) node[dot] {}; + \draw (NP) node[dot] {}; + \draw (A) node[dot, fill = magenta, minimum size = 1.5mm] {}; + \draw (B) node[dot, fill = orange, minimum size = 1.5mm] {}; + + % projection markers + \draw[very thick, magenta] + (Aproj) ++(.15,0) to ($(Aproj)+(-.15, 0)$) + (Aproj) ++(0,.15) to ($(Aproj) +(0, -.15)$); + + \tdplotdrawarc[orange, very thick]{(Bproj)}{.1}{0}{360}{}{}; + +\end{tikzpicture} +\end{document} +% vim:ts=2 sw=2 et: diff --git a/buch/papers/punktgruppen/tikz/symmetric-shapes.tex b/buch/papers/punktgruppen/tikz/symmetric-shapes.tex index b2c051f..688fb61 100644 --- a/buch/papers/punktgruppen/tikz/symmetric-shapes.tex +++ b/buch/papers/punktgruppen/tikz/symmetric-shapes.tex @@ -14,6 +14,7 @@ \begin{document} \begin{tikzpicture}[ + >=latex, node distance = 2cm, shapetheme/.style = { very thick, draw = black, fill = magenta!20!white, diff --git a/buch/papers/reedsolomon/Makefile.inc b/buch/papers/reedsolomon/Makefile.inc index 6a676f8..ea51f7a 100644 --- a/buch/papers/reedsolomon/Makefile.inc +++ b/buch/papers/reedsolomon/Makefile.inc @@ -6,9 +6,17 @@ dependencies-reedsolomon = \ papers/reedsolomon/packages.tex \ papers/reedsolomon/main.tex \ - papers/reedsolomon/references.bib \ - papers/reedsolomon/teil0.tex \ - papers/reedsolomon/teil1.tex \ - papers/reedsolomon/teil2.tex \ - papers/reedsolomon/teil3.tex + papers/reedsolomon/einleitung.tex \ + papers/reedsolomon/idee.tex \ + papers/reedsolomon/dtf.tex \ + papers/reedsolomon/endlichekoerper.tex \ + papers/reedsolomon/codebsp.tex \ + papers/reedsolomon/decohnefehler.tex \ + papers/reedsolomon/decmitfehler.tex \ + papers/reedsolomon/rekonstruktion.tex \ + papers/reedsolomon/zusammenfassung.tex \ + papers/reedsolomon/anwendungen.tex \ + papers/reedsolomon/hilfstabellen.tex \ + papers/reedsolomon/references.bib + diff --git a/buch/papers/reedsolomon/RS presentation/images/polynom1 - Kopie.tex b/buch/papers/reedsolomon/RS presentation/images/polynom1 - Kopie.tex new file mode 100644 index 0000000..038e93e --- /dev/null +++ b/buch/papers/reedsolomon/RS presentation/images/polynom1 - Kopie.tex @@ -0,0 +1,33 @@ +% polynome1 +%------------------- +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usetikzlibrary{arrows,intersections,math} +\newcommand{\teiler}{40} +\begin{document} + + +\begin{tikzpicture}[>=latex,thick] + + \begin{axis}[ + axis lines = left, + xlabel = \(x\), + ylabel = {\(f(x)\)}, + ] + %Below the red parabola is defined + \addplot[ + color=blue, + ] + coordinates { + (0,23.1)(10,27.5)(20,32)(30,37.8)(40,44.6)(60,61.8)(80,83.8)(100,114) + }; + %Here the blue parabola is defined + + \end{axis} +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/reedsolomon/anwendungen.tex b/buch/papers/reedsolomon/anwendungen.tex index 83e0f94..b9b1d69 100644 --- a/buch/papers/reedsolomon/anwendungen.tex +++ b/buch/papers/reedsolomon/anwendungen.tex @@ -6,14 +6,38 @@ \section{Anwendungen des Reed-Solomon-Codes \label{reedsolomon:section:anwendung}} \rhead{Anwendungen} -\textcolor{red}{Platzierung der Bilder? Quellenangabe der Bilder?} -In den vorherigen Abschnitten haben wir betrachtet, wie Reed-Solomon-Codes in der Theorie Funktionieren. +In den vorherigen Abschnitten haben wir betrachtet, wie Reed-Solomon-Codes in der Theorie funktionieren. In diesem Abschnitt werden wir einige Anwendungen vorstellen, bei denen ein Reed-Solomon-Code zum Einsatz kommt. + +Dabei teilen all diese Anwendungen das gleiche Problem: Die Daten können nur durch einen (höchst Wahrscheinlichen) fehlerbehafteten Kanal empfangen werden. Es gibt keine andere Methode, an diese Daten zu kommen, als über diesen Kanal. + +In der Netzwerktechnik zum Beispiel ist es üblich, dass bei Paketverluste oder beschädigt empfangene Datenpaketen diese einfach noch einmal innert wenigen Millisekunden angefordert werden können. +In der Raumfahrt ist dies nicht möglich, da aufgrund der beschränkten Speichermöglichkeit die gesammelten Daten so rasch wie möglich zur Erde gesendet werden. +Diese Daten wiederum brauchen aufgrund der grossen Distanz Stunden bis die Daten beim Empfänger ankommen. +Fehlerhafte Daten kann also auf Grund der Zeitverzögerung nicht mehr angefordert werden. + +Bei CDs oder DVDs gibt es zwar kein zeitliches Problem, jedoch erschweren Kratzer, Verschmutzungen oder Produktionsfehler das Lesen einer solchen Disk. +Da vor allem Produktionsfehler und Kratzer irreversibel sind und die Disk nicht nach jedem Kratzer ersetzt werden muss, so wird die korrekte Ausgabe der gespeicherten Information durch die Fehlerkorrektur sichergestellt. + +Einen ähnlichen Ansatz verfolgen QR-Codes, wobei die Information auch dann noch gelesen werden kann wenn der Code nicht mehr vollständig vorhanden ist. + +%Wie man sieht, eignen sich Reed-Solomon-Codes vor allem für Anwendungen, bei der die Informationen nicht auf einen Anderen Weg beschafft werden kann. +% +% +%, bei denen die Wahrscheinlichkeit hoch ist, dass während der Übertragung +% +%Es ist deshalb umso wichtiger die Daten Codiert zu lesen um so gleich die Lesefehler zu korrigieren. +% +% da aufgrund der grossen Distanz Stunden vergehen können bis gesendete Daten auf der Erde empfangen werden kann. +% + Obwohl alle diese Codes nach dem gleichen Prinzip arbeiten gibt es starke Unterschiede in deren Funktionsweise. Dies kommt vor allem daher, da die Codes nur Ressourcen zur Verfügung haben, die von der Hardware bereitstellt wird, auf denen die Codes implementiert wurden. Diese Codes bedienen sich daher verschiedener Tricks und Optimierungen um möglichst effizient zu arbeiten. -% + +Um die Fähigkeit eines verwendeten Reed-Solomon-Codes zu beschreiben verwendet man die Notation ($n$,$k$), wobei $n$ die Grösse des Nachrichtenblocks angibt und $k$ die Anzahl der Stellen, die für Nutzdaten gebraucht werden können. + %Dies kommt vor allem daher, da diese Codes an ihre Hardware gebunden sind, auf denen sie implementiert worden sind. %Deshalb wurden diese Codes stark optimiert damit sie möglichst Effizient arbeiten können. % @@ -45,8 +69,23 @@ Diese Codes bedienen sich daher verschiedener Tricks und Optimierungen um mögli %In den letzten abschnitten haben wir uns ausführlich die Funktionsweise des Reed-Solomon-Codes angeschaut. In diesem Abschnitt möchten wir dem Leser ein paar bekannte beispiele vorstellen, in denen Reed-Solomon-Codes zum einsatz kommen. Es sei jedoch angemerkt, dass diese Anwendungen in der Umsetzung oft ein wenig anderst funktionieren als hier vorgestellt. Dies wurde vor allem wegen technischen optimierungen realisiert. (technische tricks und finessen), von der logik jedoch sehr stark an unserem Beispiel orientieren \subsection{Raumfahrt} -Obwohl Reed-Solomon-Codes bereits in den 1960er entwickelt wurden fanden sie erstmals Anwendung in der Voyager Raumsonde der NASA. Die Daten der zwei im Jahre 1977 gestarteten Sonden werden mit einem RS(255,233)-Code \textcolor{red}{benötigt das weitere erklärungen, wie z.b. 255: grösse nachrichtenblock, 233: anzahl der nutzbaren daten ?} zusammen mit einem konventionellen Faltungscode übertragen. +Obwohl Reed-Solomon-Codes bereits in den 1960er entwickelt wurden fanden sie erstmals Anwendung in der Voyager Raumsonde der NASA. Die Daten der zwei im Jahre 1977 gestarteten Sonden (siehe Abbildung \ref{fig:voyager}) werden mit einem ($255$,$233$)-Code +Codiert. +Der Nachrichtenblock hat somit eine Länge von $255$ Zahlen, wovon $233$ als Nutzlast zur Verfügung stehen. +Damit ist es möglich bis zu $11$ Fehler im Nachrichtenblock zu korrigieren. +Der Codierte Nachrichtenblock wird in kleinere Blöcke aufgeteilt, mit einem Faltungscode erneut Codiert und anschliessend gesendet. +Ein Faltungscode ist wie ein Reed-Solomon-Code in der Lage Fehler zu korrigieren, +Codiert seine Information aber auf eine andere weise. Aus jedem unterteilten Block wird vor dem Versenden ein Paritätsbit erzeugt und dem Block angehängt. Anhand diesem Paritätsbit überprüft der Empfänger, ob bei der Übertragung der Block beschädigt wurde. Ist dies der Fall, wird der Block bei der Decodierung nicht beachtet. Diese so entstandenen ``Lücken'' im Datenstrom werden wiederum vom Reed-Solomon-Code korrigiert. Dieses Zusammenspiel beider Codes garantiert so eine hohe Robustheit gegenüber Übertragungsfeher. +% +% Funktioniert aber nach einem ganz anderen Prinzip. +% +%Durch diese doppelte Codierung wird eine äusserst hohe Übertragungssicherheit garantiert. +% +%Dabei steht die Zahl 255 für grösse des Nachrichtenblocks, der die Anzahl 233 +% +% +% \textcolor{red}{benötigt das weitere Erklärungen, wie z.b. 255: grösse Nachrichtenblock, 233: anzahl der nutzbaren daten ?} zusammen mit einem konventionellen Faltungscode übertragen. Eine von der Sonde gesendete Nachricht hat eine Blockgrösse von 255 Zeichen, wovon 233 für die Nutzdaten gebraucht werden können. Dieser Code ist somit in der Lage 11 Fehler in einem Nachrichtenblock zu korrigieren. % % Die zwei im Jahre 1977 gestarteten Sonden senden Daten mit der Hilfe eines RS(255,233)-Code für die digitalen Bilder sowie einem konventionellen Faltungscode. % @@ -56,14 +95,14 @@ Obwohl Reed-Solomon-Codes bereits in den 1960er entwickelt wurden fanden sie ers \begin{figure} \centering \includegraphics[width=0.5\textwidth]{papers/reedsolomon/images/Voyager_Sonde} - \caption{Voyager Raumsonde} + \caption{Mit einer Entfernung von über 22.8 Milliarden Kilometer ist die Voyager 1 Raumsonde das am weitesten entfernte, von Menschen erschaffene Objekt. Obwohl ihre Schwestersonde Voyager 2 zuerst ins All gestartet wurde befindet Sie sich ``nur'' 19 Milliarden Kilometer weit weg von der Erde. Aufgrund abnehmender Batterieleistung werden die beiden Sonden ihre wissenschaftlichen Aktivitäten etwa 2025 einstellen, bleiben aber bis in die 2030er mit uns in Kontakt.} \label{fig:voyager} \end{figure} \subsection{CD/DVD} Compact discs verwenden sogar zwei ineinander verschachtelte Reed-Solomon-Codes, einen (32,28)-Code und einen (28,24)-Code. -Beide Codes sind in der Lage, Fehler aus dem jeweils anderen gelesenen Block zu korrigieren. Dieses spezielle zusammenspielen dieser beiden Codes werden auch Cross-interleaved Reed-Solomon-Codes (CIRC) genannt. -Diese Vorgehensweise erzielt eine hohe Robustheit gegenüber Produktionsfehler oder Verschmutzung auf der Disc. Bei CD's sind diese in der Lage bis zu 4000 fehlerhafte Bits am Stück (ca. $2.5mm$) zu erkennen und zu korrigieren. +Beide Codes sind in der Lage, Fehler aus dem jeweils anderen gelesenen Block zu korrigieren. Dieses spezielle Zusammenspielen dieser beiden Codes werden auch Cross-interleaved Reed-Solomon-Codes (CIRC) genannt. +Diese Vorgehensweise erzielt eine hohe Robustheit gegenüber Produktionsfehlern oder Verschmutzung auf der Disc. Bei CDs sind diese in der Lage, bis zu 4000 fehlerhafte Bits am Stück (ca. $2.5mm$) zu erkennen und zu korrigieren. Die Digital Video Disc funktioniert nach dem selben Konzept mit grösseren Codeblöcken. Die DVD verwendet einen (208,192)-Code und einen (182,172)-Code. @@ -72,13 +111,25 @@ Die Digital Video Disc funktioniert nach dem selben Konzept mit grösseren Codeb \begin{figure} \centering - \includegraphics[width=0.5\textwidth]{papers/reedsolomon/images/Compact_Disc} - \caption{Compact Disc} + \subfigure[]{ + \includegraphics[width=0.45\textwidth]{papers/reedsolomon/images/Compact_Disc} + } + \subfigure[]{ + \includegraphics[width=0.45\textwidth]{papers/reedsolomon/images/Compact_Disc_zoomed_in} + } + \caption{CDs kamen 1982 auf den Markt. Sie funktioniert durch das Einpressen oder Einbrennen von Punkten und Strichen, die die Daten repräsentieren. Gelesen werden diese wiederum durch die Reflektion eines Lasers an diesen Punkten und Strichen.} \label{fig:cd} \end{figure} \subsection{QR-Codes} -Quick Response Codes funktionieren nach einem sehr ähnlichen Prinzip wie in unserem Beispiel, nur dass QR-Codes in einem $\mathbb{F}_{256}$ Körper arbeiten. Je nach grösse der Codierung ist der QR-Code im Endeffekt robuster gegen Beschädigungen. Bei Low Level Codes können 7\% der Daten Wiederhergestellt werden, beim High Level Code sind das sogar 30\%. +Quick Response Codes oder auch QR-Codes funktionieren nach einem sehr ähnlichen Prinzip wie in unserem Beispiel der Abschnitte \ref{reedsolomon:section:codebsp} - \ref{reedsolomon:section:rekonstruktion} nur das QR-Codes in einem $\mathbb{F}_{256}$ Körper arbeiten. Die physische Grösse eines Codes ist stark abhängig von der Menge an codierten Daten sowie dem verwendeten Fehlerkorrektur-Level. Es ist so auf dem ersten Blick nicht ersichtlich, wie viel Nutzinformationen ein Qr-Code enthält. Die QR-Codes in Abbildung \ref{fig:qr} zeigen jeweils die Gleiche Information mit unterschiedlichem Fehlerkorrektur-Level. Codes mit einem höheren Korrektur-Level können auch für Designer-Codes Zweckentfremdet werden. Dabei wird z.B. das Firmenlogo oder einen Schriftzug über den Qr-Code gelegt, ohne das die Funktion des Codes beeinträchtigt wird. Ein Beispiel dazu ist unter Abbildung \ref{fig:designqr} zu finden. + +% + +%So kann auf den ersten Blick nicht +% +% +% funktionieren nach einem sehr ähnlichen Prinzip wie in unserem Beispiel, nur dass QR-Codes in einem $\mathbb{F}_{256}$ Körper arbeiten. Je nach grösse der Codierung ist der QR-Code im Endeffekt robuster gegen Beschädigungen. Bei Low Level Codes können 7\% der Daten Wiederhergestellt werden, beim High Level Code sind das sogar 30\%. \begin{figure} \centering @@ -88,6 +139,30 @@ Quick Response Codes funktionieren nach einem sehr ähnlichen Prinzip wie in uns \subfigure[]{ \includegraphics[width=0.4\textwidth]{papers/reedsolomon/images/qrcode_l} } - \caption{(a) High Level Code, (b) Low Level Code} +% \subfigure[]{ +% \includegraphics[width=0.4\textwidth]{papers/reedsolomon/images/designer_qrcode_ohnelogo} +% } +% \subfigure[]{ +% \includegraphics[width=0.4\textwidth]{papers/reedsolomon/images/designer_qrcode} +% } + \caption{Anhand der grösse würde man darauf schliessen, dass bei (a) mehr Informationen Codiert sind als bei (b). Tatsächlich aber beinhalten beide Codes die gleiche Information. Das liegt daran, da die Fehlerkorrekturfähigkeit von QR-Codes sich in insgesamt vier Levels aufteilen lassen. Der höchste Fehlerkorrektur-Level, der bei (a) angewendet wurde, ist in der Lage, bis zu 30\% der Daten wiederherzustellen. Der kleinste Level schafft etwa 7\%, der in (b) veranschaulicht wird. Da die Grösse also nichts über die Menge an Daten aussagt, könnte es sich bei (a) auch um einen Code mit viel Nutzdaten und kleinem Fehlerkorrektur-Level handeln. Der Unterschied ist von Auge nicht sichtbar.} \label{fig:qr} \end{figure} + +\begin{figure} + \centering +% \subfigure[]{ +% \includegraphics[width=0.4\textwidth]{papers/reedsolomon/images/qrcode_h} +% } +% \subfigure[]{ +% \includegraphics[width=0.4\textwidth]{papers/reedsolomon/images/qrcode_l} +% } + \subfigure[]{ + \includegraphics[width=0.4\textwidth]{papers/reedsolomon/images/designer_qrcode_ohnelogo} + } + \subfigure[]{ + \includegraphics[width=0.4\textwidth]{papers/reedsolomon/images/designer_qrcode} + } + \caption{Während (a) noch einen unveränderten QR-Code repräsentiert, handelt es sich bei (b) nun um einen Designer-QR-Code. Beide Codes verfügen über einen mittleren Fehlerkorrektur-Level von theoretisch 15\%. Da bei (b) jetzt einen Teil des Codes durch ein Logo verdeckt wird, schränkt sich die Fehlerkorrekturfähigkeit je nach Grösse des verdeckten Teils mehr oder weniger stark ein. Unser Designer-Code in (b) ist nur noch in der Lage etwa 9\% des Codes zu rekonstruieren.} + \label{fig:designqr} +\end{figure}
\ No newline at end of file diff --git a/buch/papers/reedsolomon/dtf.tex b/buch/papers/reedsolomon/dtf.tex index 025be3a..a111527 100644 --- a/buch/papers/reedsolomon/dtf.tex +++ b/buch/papers/reedsolomon/dtf.tex @@ -3,28 +3,53 @@ % % (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil % -\section{Diskrete Fourien Transformation +\section{Diskrete Fourier Transformation \label{reedsolomon:section:dtf}} \rhead{Umwandlung mit DTF} Um die Polynominterpolation zu umgehen, gehen wir nun über in die Fourientransformation. Dies wird weder eine erklärung der Forientransorfmation noch ein genauer gebrauch für den Reed-Solomon-Code. Dieser Abschnitt zeigt nur wie die Fourientransformation auf Fehler reagiert. -wobei sie dann bei späteren Berchnungen ganz nütlich ist. - -\subsection{Übertragungsabfolge -\label{reedsolomon:subsection:Übertragungsabfolge}} -Das Signal.... sind die Daten, Zahlen welche übertragen werden sollen. -Das speziell ist das wir 100 Punkte übertragen und von 64 bis 100, -werden nur Null Punkte übertragen, dies weiss auch unser Empfänger. -Nun wird das Signal in Abbildung... codiert... -Somit wird die Information jedes Punktes auf das ganze spektrum von 0 bis 100 übertragen. -Kommen nuun drei Fehler... hinzu zu diesem codierten Signal sind diese nicht zu erkennen. -Nach dem Empfangen... und decodieren ... erkennt man die fehlerhafte information in den Punkten 64 bis 100. -Filtert man nur diese Punkte heraus und Transformiert sie mit Fourier erhält man die stellen an denen die Fehler sich eingeschlichen haben. +wobei sie dann bei späteren Berchnungen ganz nützlich ist. \subsection{Diskrete Fourientransformation Zusamenhang \label{reedsolomon:subsection:dtfzusamenhang}} Die Diskrete Fourientransformation ist definiert als -.... + \[ + \label{ft_discrete} + \hat{c}_{k} + = \frac{1}{N} \sum_{n=0}^{N-1} + {f}_n \cdot e^{-\frac{2\pi j}{N} \cdot kn} + \] +, wenn man nun + \[ + w = e^{-\frac{2\pi j}{N} k} + \] +ersetzte, und $N$ konstantbleibt, erhält man + \[ + \hat{c}_{k}=\frac{1}{N}( {f}_0 w^0 + {f}_1 w^1 + {f}_2 w^2 + \dots + {f}_{N-1} w^N) + \] +was überaust ähnlich zu unserem Polynomidee ist. +\subsection{Übertragungsabfolge +\label{reedsolomon:subsection:Übertragungsabfolge}} +\begin{enumerate}[1)] +\item Das Signal hat 64 die Daten, Zahlen welche übertragen werden sollen. +Dabei zusätzlich nach 16 Fehler abgesichert, macht insgesamt 96 Übertragungszahlen. +\item Nun wurde mittels der schnellen diskreten Fourientransformation diese 96 codiert. +Das heisst alle information ist in alle Zahlenvorhanden. +\item Nun kommen drei Fehler dazu an den Übertragungsstellen 7, 21 und 75. +\item Dieses wird nun Empfangen und mittels inversen diskreten Fourientransormation, wieder rücktransformiert. +\item Nun sieht man den Fehler im Decodieren in den Übertragungsstellen 64 bis 96. +\item Nimmt man nun nur diese Stellen 64 bis 96, auch Syndrom genannt, und Transformiert diese. +\item Bekommt man die Fehlerstellen im Locator wieder, zwar nichtso genau, dennoch erkkent man wo die Fehler stattgefunden haben. +\end{enumerate} +\begin{figure} + \centering + \resizebox{0.9\textwidth}{!}{ + %\includegraphics[width=0.5\textwidth]{papers/reedsolomon/images/plot.pdf} + \input{papers/reedsolomon/images/plotfft.tex} + } + \caption{Übertragungsabfolge \ref{reedsolomon:subsection:Übertragungsabfolge}} + \label{fig:sendorder} +\end{figure}
\ No newline at end of file diff --git a/buch/papers/reedsolomon/einleitung.tex b/buch/papers/reedsolomon/einleitung.tex index 3d40db1..2b1d878 100644 --- a/buch/papers/reedsolomon/einleitung.tex +++ b/buch/papers/reedsolomon/einleitung.tex @@ -6,13 +6,13 @@ \section{Einleitung \label{reedsolomon:section:einleitung}} \rhead{Einleitung} -Der Reed-Solomon-Code ist entstaden im ... vom .. um, -das Problem der Daten Übertragung zu lösen. -In deiesem Abschnitt wird möglichst verständlich die mathematische Abfolge, Funktion oder Algorithmus erklärt. +Der Reed-Solomon-Code ist entstanden um, +das Problem der Fehler, bei der Datenübertragung, zu lösen. +In diesem Abschnitt wird möglichst verständlich die mathematische Abfolge, Funktion oder Algorithmus erklärt. Es wird jedoch nicht auf die technische Umsetzung oder Implementierung eingegangen. -Um beim Daten Übertragen fehler zu erkennen könnte man die Daten jeweils doppelt senden, +Um beim Datenübertragen Fehler zu erkennen, könnte man die Daten jeweils doppelt senden, und so jeweilige Fehler zu erkennen. -Doch dies braucht schnell unmengen an Daten, wenn man nach vielen Fehler absichern möchte. +Doch nur schon um weinige Fehler zu erkennen werden überproportional viele Daten doppelt und dreifach gesendet. Der Reed-Solomon-Code macht dies auf eine andere, clevere Weise. diff --git a/buch/papers/reedsolomon/experiments/codiert.txt b/buch/papers/reedsolomon/experiments/codiert.txt new file mode 100644 index 0000000..4a481d8 --- /dev/null +++ b/buch/papers/reedsolomon/experiments/codiert.txt @@ -0,0 +1,96 @@ +0,284 +1,131.570790435043 +2,41.9840308053375 +3,12.1189172092243 +4,23.8408857476069 +5,69.1793197789512 +6,24.0186013379153 +7,37.3066577242559 +8,18.2010889773887 +9,12.3214904922455 +10,15.6627133315015 +11,24.5237955316204 +12,32.1114345314062 +13,44.9845039238714 +14,13.5324640263625 +15,10.1736266929292 +16,4.58257569495584 +17,23.217268502288 +18,16.5769107917917 +19,6.89948680823017 +20,4.84567134895776 +21,10.4219666223433 +22,43.6179140616243 +23,35.9073375743642 +24,15.0332963783729 +25,21.7594021268945 +26,23.2496572716993 +27,17.9815599423852 +28,11.3577742151117 +29,38.467599433197 +30,28.3035029562577 +31,9.54321919833388 +32,21.377558326432 +33,17.6292439561917 +34,12.6951848921471 +35,20.0667752354841 +36,22.9097309529208 +37,8.78894645948548 +38,13.360682005498 +39,25.1757616314718 +40,38.0357773686457 +41,18.4633287776253 +42,19.0584505869806 +43,10.8631093309173 +44,12.6147770818983 +45,12.5398140021274 +46,34.901983501949 +47,22.3480442021702 +48,6 +49,22.3480442021702 +50,34.901983501949 +51,12.5398140021274 +52,12.6147770818983 +53,10.8631093309173 +54,19.0584505869806 +55,18.4633287776253 +56,38.0357773686457 +57,25.1757616314718 +58,13.360682005498 +59,8.78894645948548 +60,22.9097309529208 +61,20.0667752354841 +62,12.6951848921471 +63,17.6292439561917 +64,21.377558326432 +65,9.54321919833388 +66,28.3035029562577 +67,38.467599433197 +68,11.3577742151117 +69,17.9815599423852 +70,23.2496572716993 +71,21.7594021268945 +72,15.0332963783729 +73,35.9073375743642 +74,43.6179140616243 +75,10.4219666223433 +76,4.84567134895776 +77,6.89948680823017 +78,16.5769107917917 +79,23.217268502288 +80,4.58257569495584 +81,10.1736266929292 +82,13.5324640263625 +83,44.9845039238714 +84,32.1114345314062 +85,24.5237955316204 +86,15.6627133315015 +87,12.3214904922455 +88,18.2010889773887 +89,37.3066577242559 +90,24.0186013379153 +91,69.1793197789512 +92,23.8408857476069 +93,12.1189172092243 +94,41.9840308053375 +95,131.570790435043 diff --git a/buch/papers/reedsolomon/experiments/decodiert.txt b/buch/papers/reedsolomon/experiments/decodiert.txt new file mode 100644 index 0000000..f6221e6 --- /dev/null +++ b/buch/papers/reedsolomon/experiments/decodiert.txt @@ -0,0 +1,96 @@ +0,6.05208333333333 +1,6.02602539785853 +2,0.0261327016093151 +3,5.98927158561317 +4,4.019445724874 +5,0.0247005083663722 +6,4.97798278395618 +7,1.95246440445439 +8,0.974000110512201 +9,2.00528527696027 +10,1.00071804528155 +11,1.97630907888264 +12,0.0232923747656228 +13,6.01302820392331 +14,3.03567381915226 +15,5.02435590137329 +16,7.00526061008995 +17,5.00739608089369 +18,5.02211514480064 +19,4.02175864806658 +20,1.00236543833726 +21,4.98147315261261 +22,8.97728828610336 +23,8.98481304394618 +24,2.98958333333333 +25,1.98491220960989 +26,5.97728835934715 +27,5.98144124907561 +28,4.00163839998525 +29,2.02176249296313 +30,9.02210713874162 +31,1.00742763919872 +32,1.00557258081044 +33,1.02435888848794 +34,2.03577412756745 +35,6.01302820392331 +36,5.97917574041123 +37,0.976310374034338 +38,9.00062625447998 +39,7.00515849238528 +40,6.97396416790894 +41,0.95256880864368 +42,8.97794719866783 +43,9.01850701506487 +44,10.0194409579917 +45,8.98926601525997 +46,7.9866590265379 +47,5.02603060999077 +48,2.05208333333333 +49,4.02603841132848 +50,0.986882897867895 +51,0.0177592928994285 +52,9.01944131204563 +53,3.0185365665612 +54,2.97803642439316 +55,2.95243072164649 +56,4.97396651395488 +57,6.00516695947321 +58,0.0143895905726619 +59,7.97630812771393 +60,5.97917574041123 +61,9.01298821331865 +62,3.03567381915226 +63,4.02435609145793 +64,0.0275599094902563 +65,0.0115837187254191 +66,0.025877761014238 +67,0.0224618032819697 +68,0.04410594689944 +69,0.0474504002669341 +70,0.0227694695500626 +71,0.0271436638090525 +72,0.0104166666666667 +73,0.0271436638090523 +74,0.0227694695500608 +75,0.0474504002669343 +76,0.0441059468994397 +77,0.0224618032819701 +78,0.0258777610142379 +79,0.0115837187254183 +80,0.027559909490256 +81,0.0245124379481793 +82,0.0499782237195209 +83,0.0401432022864265 +84,0.0232923747656228 +85,0.0237974288564099 +86,0.0143895905726624 +87,0.0271745729691685 +88,0.0275599094902567 +89,0.0515501672184983 +90,0.0358255004834542 +91,0.024700508366373 +92,0.0210194725405171 +93,0.0177592928994296 +94,0.0261327016093158 +95,0.0314909067039411 diff --git a/buch/papers/reedsolomon/experiments/empfangen.txt b/buch/papers/reedsolomon/experiments/empfangen.txt new file mode 100644 index 0000000..38c13b0 --- /dev/null +++ b/buch/papers/reedsolomon/experiments/empfangen.txt @@ -0,0 +1,96 @@ +0,284 +1,131.570790435043 +2,41.9840308053375 +3,12.1189172092243 +4,23.8408857476069 +5,69.1793197789512 +6,23.6290258699579 +7,37.3066577242559 +8,18.2010889773887 +9,12.3214904922455 +10,15.6627133315015 +11,24.5237955316204 +12,32.1114345314062 +13,44.9845039238714 +14,13.5324640263625 +15,10.1736266929292 +16,4.58257569495584 +17,23.217268502288 +18,16.5769107917917 +19,6.89948680823017 +20,5.55320238736303 +21,10.4219666223433 +22,43.6179140616243 +23,35.9073375743642 +24,15.0332963783729 +25,21.7594021268945 +26,23.2496572716993 +27,17.9815599423852 +28,11.3577742151117 +29,38.467599433197 +30,28.3035029562577 +31,9.54321919833388 +32,21.377558326432 +33,17.6292439561917 +34,12.6951848921471 +35,20.0667752354841 +36,22.9097309529208 +37,8.78894645948548 +38,13.360682005498 +39,25.1757616314718 +40,38.0357773686457 +41,18.4633287776253 +42,19.0584505869806 +43,10.8631093309173 +44,12.6147770818983 +45,12.5398140021274 +46,34.901983501949 +47,22.3480442021702 +48,6 +49,22.3480442021702 +50,34.901983501949 +51,12.5398140021274 +52,12.6147770818983 +53,10.8631093309173 +54,19.0584505869806 +55,18.4633287776253 +56,38.0357773686457 +57,25.1757616314718 +58,13.360682005498 +59,8.78894645948548 +60,22.9097309529208 +61,20.0667752354841 +62,12.6951848921471 +63,17.6292439561917 +64,21.377558326432 +65,9.54321919833388 +66,28.3035029562577 +67,38.467599433197 +68,11.3577742151117 +69,17.9815599423852 +70,23.2496572716993 +71,21.7594021268945 +72,15.0332963783729 +73,35.9073375743642 +74,44.6135417384784 +75,10.4219666223433 +76,4.84567134895776 +77,6.89948680823017 +78,16.5769107917917 +79,23.217268502288 +80,4.58257569495584 +81,10.1736266929292 +82,13.5324640263625 +83,44.9845039238714 +84,32.1114345314062 +85,24.5237955316204 +86,15.6627133315015 +87,12.3214904922455 +88,18.2010889773887 +89,37.3066577242559 +90,24.0186013379153 +91,69.1793197789512 +92,23.8408857476069 +93,12.1189172092243 +94,41.9840308053375 +95,131.570790435043 diff --git a/buch/papers/reedsolomon/experiments/f.m b/buch/papers/reedsolomon/experiments/f.m index 6bdc741..bf2587c 100644 --- a/buch/papers/reedsolomon/experiments/f.m +++ b/buch/papers/reedsolomon/experiments/f.m @@ -1,8 +1,8 @@ -# -# f.m -- Reed-Solomon-Visualisierung mit FFT -# -# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule -# +% +% f.m -- Reed-Solomon-Visualisierung mit FFT +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule + N = 64; b = 32; l = N + b; @@ -51,6 +51,7 @@ syndrom(1:N,1) = zeros(N,1) plot(abs(syndrom)); xlim([1, l]); title("Syndrom"); + pause() locator = abs(fft(syndrom)) @@ -59,3 +60,13 @@ plot(locator); xlim([1, l]); title("Locator"); pause() + + +writematrix([transpose(counter), abs(signal)], 'signal.txt') +writematrix([transpose(counter), abs(codiert)], 'codiert.txt') +writematrix([transpose(counter), fehler], 'fehler.txt') +writematrix([transpose(counter), abs(empfangen)], 'empfangen.txt') +writematrix([transpose(counter), abs(decodiert)], 'decodiert.txt') +writematrix([transpose(counter), abs(syndrom)], 'syndrom.txt') +writematrix([transpose(counter), locator], 'locator.txt') + diff --git a/buch/papers/reedsolomon/experiments/fehler.txt b/buch/papers/reedsolomon/experiments/fehler.txt new file mode 100644 index 0000000..23f1a83 --- /dev/null +++ b/buch/papers/reedsolomon/experiments/fehler.txt @@ -0,0 +1,96 @@ +0,0 +1,0 +2,0 +3,0 +4,0 +5,0 +6,2 +7,0 +8,0 +9,0 +10,0 +11,0 +12,0 +13,0 +14,0 +15,0 +16,0 +17,0 +18,0 +19,0 +20,2 +21,0 +22,0 +23,0 +24,0 +25,0 +26,0 +27,0 +28,0 +29,0 +30,0 +31,0 +32,0 +33,0 +34,0 +35,0 +36,0 +37,0 +38,0 +39,0 +40,0 +41,0 +42,0 +43,0 +44,0 +45,0 +46,0 +47,0 +48,0 +49,0 +50,0 +51,0 +52,0 +53,0 +54,0 +55,0 +56,0 +57,0 +58,0 +59,0 +60,0 +61,0 +62,0 +63,0 +64,0 +65,0 +66,0 +67,0 +68,0 +69,0 +70,0 +71,0 +72,0 +73,0 +74,1 +75,0 +76,0 +77,0 +78,0 +79,0 +80,0 +81,0 +82,0 +83,0 +84,0 +85,0 +86,0 +87,0 +88,0 +89,0 +90,0 +91,0 +92,0 +93,0 +94,0 +95,0 diff --git a/buch/papers/reedsolomon/experiments/locator.txt b/buch/papers/reedsolomon/experiments/locator.txt new file mode 100644 index 0000000..b28988c --- /dev/null +++ b/buch/papers/reedsolomon/experiments/locator.txt @@ -0,0 +1,96 @@ +0,0.0301224340567056 +1,0.141653026854885 +2,0.138226631799377 +3,0.0339903276086929 +4,0.310585462557496 +5,0.551427312631385 +6,0.628514858396814 +7,0.51102386251559 +8,0.275861355940449 +9,0.0502396354182268 +10,0.090185502547573 +11,0.110759344849756 +12,0.0684618905063001 +13,0.0362855426992259 +14,0.0697096919781468 +15,0.109288539370248 +16,0.0923187999496653 +17,0.0512198536768088 +18,0.274192386987782 +19,0.51349614953654 +20,0.633154426602466 +21,0.553283743533942 +22,0.307840573214514 +23,0.0341664350328392 +24,0.140270857957 +25,0.138527177682831 +26,0.029637547736156 +27,0.0816962563186052 +28,0.0944383203811073 +29,0.0263932110686261 +30,0.0585881348402056 +31,0.0737117341599984 +32,0.0239973937701886 +33,0.0464215468420038 +34,0.0616218854220964 +35,0.0221963086695009 +36,0.0390764778127646 +37,0.0537637218396934 +38,0.0208333333333332 +39,0.0343107696069045 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+83,0.0309749128751093 +84,0.0315202035072035 +85,0.0627625211892184 +86,0.0360843918243497 +87,0.02794920551495 +88,0.0677921493367236 +89,0.0437167157553067 +90,0.0270640150996317 +91,0.0783380025231622 +92,0.0561293738314281 +93,0.0278742033265809 +94,0.0981443889498639 +95,0.0794543457386548 diff --git a/buch/papers/reedsolomon/experiments/plot.tex b/buch/papers/reedsolomon/experiments/plot.tex new file mode 100644 index 0000000..2196c82 --- /dev/null +++ b/buch/papers/reedsolomon/experiments/plot.tex @@ -0,0 +1,103 @@ +% polynome1 +%------------------- +\documentclass[tikz]{standalone} +\usepackage{amsmath} +\usepackage{times} +\usepackage{txfonts} +\usepackage{pgfplots} +\usepackage{csvsimple} +\usepackage{pgfplotstable} +\usepackage{filecontents} +\usetikzlibrary{arrows,intersections,math} +\newcommand{\x}{10} +\newcommand{\y}{-8} +\begin{document} + +\begin{tikzpicture}[] + + %--------------------------------------------------------------- + %Knote + \matrix[draw = none, column sep=20mm, row sep=4mm]{ + \node(signal) [] { + \begin{tikzpicture} + \begin{axis}[ + title = {\Large {Signal}}, + xlabel={Anzahl Übertragene Zahlen}, + xtick={0,20,40,64,80,98},] + \addplot[blue] table[col sep=comma] {signal.txt}; + \end{axis} + \end{tikzpicture}}; & + + \node(codiert) [] { + \begin{tikzpicture} + \begin{axis}[title = {\Large {Codiert}}] + \addplot[] table[col sep=comma] {codiert.txt}; + \end{axis} + \end{tikzpicture}}; \\ + + &\node(fehler) [] { + \begin{tikzpicture} + \begin{axis}[scale=0.6, title = {\Large {Fehler}}] + \addplot[red] table[col sep=comma] {fehler.txt}; + \end{axis} + \end{tikzpicture}};\\ + + \node(decodiert) [] { + \begin{tikzpicture} + \begin{axis}[title = {\Large {Decodiert}}] + \addplot[blue] table[col sep=comma] {decodiert.txt}; + \end{axis} + \end{tikzpicture}}; & + + \node(empfangen) [] { + \begin{tikzpicture} + \begin{axis}[title = {\Large {Empfangen}}] + \addplot[] table[col sep=comma] {empfangen.txt}; + \end{axis} + \end{tikzpicture}};\\ + + \node(syndrom) [] { + \begin{tikzpicture} + \begin{axis}[title = {\Large {Syndrom}}] + \addplot[blue] table[col sep=comma] {syndrom.txt}; + \end{axis} + \end{tikzpicture}}; & + + \node(locator) [] { + \begin{tikzpicture} + \begin{axis}[title = {\Large {Locator}}] + \addplot[] table[col sep=comma] {locator.txt}; + \end{axis} + \end{tikzpicture}};\\ + }; + %------------------------------------------------------------- + %FFT & IFFT deskription + + \draw[thin,gray,dashed] (0,12) to (0,-12); + \node(IFFT) [scale=0.7] at (0,12.3) {IFFT}; + \draw[<-](IFFT.south west)--(IFFT.south east); + \node(FFT) [scale=0.7, above of=IFFT] {FFT}; + \draw[->](FFT.north west)--(FFT.north east); + + \draw[thick, ->,] (fehler.west)++(-1,0) +(0.05,0.5) -- +(-0.1,-0.1) -- +(0.1,0.1) -- +(0,-0.5); + %Arrows + \draw[ultra thick, ->] (signal.east) to (codiert.west); + \draw[ultra thick, ->] (codiert.south) to (fehler.north); + \draw[ultra thick, ->] (fehler.south) to (empfangen.north); + \draw[ultra thick, ->] (empfangen.west) to (decodiert.east); + \draw[ultra thick, ->] (syndrom.east) to (locator.west); + \draw(decodiert.south east)++(-1.8,1) ellipse (1.3cm and 0.8cm) ++(-1.3,0) coordinate(zoom) ; + \draw[ultra thick, ->] (zoom) to[out=180, in=90] (syndrom.north); + + %item + \node[circle, draw, fill =lightgray] at (signal.north west)+(1,0) {1}; + \node[circle, draw, fill =lightgray] at (codiert.north west) {2}; + \node[circle, draw, fill =lightgray] at (fehler.north west) {3}; + \node[circle, draw, fill =lightgray] at (empfangen.north west) {4}; + \node[circle, draw, fill =lightgray] at (decodiert.north west) {5}; + \node[circle, draw, fill =lightgray] at (syndrom.north west) {6}; + \node[circle, draw, fill =lightgray] at (locator.north west) {7}; + +\end{tikzpicture} +\end{document} + diff --git a/buch/papers/reedsolomon/experiments/signal.txt b/buch/papers/reedsolomon/experiments/signal.txt new file mode 100644 index 0000000..c4fa5f8 --- /dev/null +++ b/buch/papers/reedsolomon/experiments/signal.txt @@ -0,0 +1,96 @@ +0,6 +1,6 +2,0 +3,6 +4,4 +5,0 +6,5 +7,2 +8,1 +9,2 +10,1 +11,2 +12,0 +13,6 +14,3 +15,5 +16,7 +17,5 +18,5 +19,4 +20,1 +21,5 +22,9 +23,9 +24,3 +25,2 +26,6 +27,6 +28,4 +29,2 +30,9 +31,1 +32,1 +33,1 +34,2 +35,6 +36,6 +37,1 +38,9 +39,7 +40,7 +41,1 +42,9 +43,9 +44,10 +45,9 +46,8 +47,5 +48,2 +49,4 +50,1 +51,0 +52,9 +53,3 +54,3 +55,3 +56,5 +57,6 +58,0 +59,8 +60,6 +61,9 +62,3 +63,4 +64,0 +65,0 +66,0 +67,0 +68,0 +69,0 +70,0 +71,0 +72,0 +73,0 +74,0 +75,0 +76,0 +77,0 +78,0 +79,0 +80,0 +81,0 +82,0 +83,0 +84,0 +85,0 +86,0 +87,0 +88,0 +89,0 +90,0 +91,0 +92,0 +93,0 +94,0 +95,0 diff --git a/buch/papers/reedsolomon/experiments/syndrom.txt b/buch/papers/reedsolomon/experiments/syndrom.txt new file mode 100644 index 0000000..8ca9eed --- /dev/null +++ b/buch/papers/reedsolomon/experiments/syndrom.txt @@ -0,0 +1,96 @@ +0,0 +1,0 +2,0 +3,0 +4,0 +5,0 +6,0 +7,0 +8,0 +9,0 +10,0 +11,0 +12,0 +13,0 +14,0 +15,0 +16,0 +17,0 +18,0 +19,0 +20,0 +21,0 +22,0 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a/buch/papers/reedsolomon/idee.tex b/buch/papers/reedsolomon/idee.tex index 4a7716a..39adbbf 100644 --- a/buch/papers/reedsolomon/idee.tex +++ b/buch/papers/reedsolomon/idee.tex @@ -8,51 +8,81 @@ \rhead{Problemstellung} Das Problem liegt darin Informationen, Zahlen, zu Übertragen und Fehler zu erkennen. -Beim Reed-Solomon-Code kann man nicht nur Fehler erkenen, +Beim Reed-Solomon-Code kann man nicht nur Fehler erkennen, man kann sogar einige Fehler korrigieren. \rhead{Polynom-Ansatz} -Eine Idee ist die Daten, -ein Polynom zu bilden und dieses dann mit bestimmten Punkten überträgt. +Eine Idee ist aus den Daten +ein Polynom zu bilden. +Diese Polynomfunktion bei bestimmten Werten, ausrechnet und diese Punkte dann überträgt. Nehmen wir als beisbiel die Zahlen \textcolor{blue}{2}, \textcolor{blue}{1}, \textcolor{blue}{5}, welche uns dann das Polynom \begin{equation} p(x) = -2x^2 + 1x + 5 +\textcolor{blue}{2}x^2 + \textcolor{blue}{1}x + \textcolor{blue}{5} \label{reedsolomon:equation1} \end{equation} ergeben. -Übertragen werden nun die stellen 1, 2, 3\dots 7 dieses Polynomes. -Grafisch sieht man dies dann im Abbild //TODO -Wenn ein Fehler sich in die Übertragung eingeschlichen hatt, muss der Leser/Empfänger erkennen, welches das Richtige Polynom ist. -Der Leser/Empfänger weiss, mit welchem Grad das Polynom entwickelt wurde. +Übertragen werden nun die Werte an den stellen 1, 2, 3\dots 7 dieses Polynomes. +Grafisch sieht man dies dann in Abbildung \ref{fig:polynom}, +mit den Punkten, $p(1),p(2),...,p(7) = (\textcolor{green}{8}, +\textcolor{green}{15}, \textcolor{green}{26}, +\textcolor{green}{41}, \textcolor{green}{60}, +\textcolor{green}{83}, \textcolor{green}{110})$ +Wenn ein Fehler sich in die Übertragung eingeschlichen hatt, muss der Leser/Empfänger diesen erkennen und das Polynom rekonstruieren. +Der Leser/Empfänger weiss, den Grad des Polynoms und dessen Werte übermittelt wurden. + \subsection{Beispiel} -Für das Beispeil aus der Gleichung \ref{reedsolomon:equation1}, +Für das Beispeil aus der Gleichung \eqref{reedsolomon:equation1}, ist ein Polynome zweiten Grades durch drei Punkte eindeutig bestimmbar. -Hat es Fehler in der Übertragunge gegeben, kann man diese erkennen, -da alle Punkte, die korrekt sind, auf dem Polynom liegen müssen. +Hat es Fehler in der Übertragunge gegeben,(Bei Abbildung \ref{fig:polynom}\textcolor{red}{roten Punkte}) kann man diese erkennen, +da alle Punkte, die korrekt sind, auf dem Polynom liegen müssen. +(Bei Abbildung \ref{fig:polynom}\textcolor{green}{grünen Punkte}) Ab wie vielen Fehler ist das Polynom nicht mehr erkennbar beim Übertragen von 7 Punkten? Bei 2 Fehlern kann man noch eindeutig bestimmen, dass das Polynom mit 4 Punkten, -gegenüber dem mit 5 Punkten falsch liegt. -Werden es mehr Fehler kann nur erkennt werden das das Polynom nicht stimmt. -Das Orginale Polynom kann aber nicht mehr gefunden werden. -Dabei sollten mehr Übertragungspunkte gegeben werden. +gegenüber dem mit 5 Punkten falsch liegt.\ref{fig:polynom} +Werden es mehr Fehler kann nur erkennt werden, dass das Polynom nicht stimmt. +Das orginale Polynom kann aber nicht mehr gefunden werden. +Dafür sind mehr übertragene Werte nötig. + +\begin{figure} + \centering + %\includegraphics[width=0.5\textwidth]{papers/reedsolomon/images/polynom2} + \input{papers/reedsolomon/images/polynom2.tex} + \caption{Polynom $p(x)$ \eqref{reedsolomon:equation1}} + \label{fig:polynom} +\end{figure} \section{Fehlerbestimmung \label{reedsolomon:section:Fehlerbestimmmung}} So wird ein Muster indentifiziert, welches genau vorherbestimmen kann, wie gross das Polynom sein muss und wie viele Übertragungspunkte gegeben werden müssen. -Durch ein klein wenig Überlegung ist klar das die anzahl Zahlen (Daten, ab hier verwenden wir das Wort Nutzlast), -die dan Entschlüsselt werden sollen den Grad des Polynoms minus 1 ergeben. +Um zu bestimmen wie viel Fehler erkennt und korriegiert werden können. +Die Anzahl Zahlen (Daten, ab hier verwenden wir das Wort Nutzlast), +die Entschlüsselt werden sollen, brauchen die gleiche Anzahl an Polynomgraden, beginnend bei Grad 0. ( \( k-1 \) ) Für die Anzahl an Übertragungspunkte, muss bestimmt werden wieviel Fehler erkennt und korrigiert werden sollen. -Mit Hilfe der Tabelle.... sieht man das es bei $$t$$ Fehlern und $$k$$ Nutzlast, -für das Übertragen $$k+2t$$ Punkte gegben werden müssen. +Mit Hilfe der Tabelle, sieht man das es bei $t$ Fehlern und $k$ Nutzlast Zahlen, +$k+2t$ Punkte übertragen werden müssen. -Ein toller Nebeneffekt ist das dadurch auch $$2t$$ Fehler erkannt werden. -um zurück auf unser Beispiel zu kommen, -können von den 7 Übertragungspunkten bis zu $$2t = 2*2 = 4 $$ Punkten falsch liegen -und es wird kein eindeutiges Polynom 2ten Grades erkannt, und somit die Nutzlast Daten als fehlerhaft deklariert. +\begin{center} + \begin{tabular}{ c c c } + \hline + Nutzlas & Fehler & Übertragen \\ + \hline + 3 & 2 & 7 Werte eines Polynoms vom Grad 2 \\ + 4 & 2 & 8 Werte eines Polynoms vom Grad 3 \\ + 3 & 3 & 9 Werte eines Polynoms vom Grad 2 \\ + \hline + $k$ & $t$ & $k+2t$ Werte eines Polynoms vom Grad $k-1$ \\ + \hline + \end{tabular} +\end{center} -Ein Polynom durch Punkt mit Polynom Interpolation zu rekonstruieren ist schwierig und Fehleranfällig. +Ein toller Nebeneffekt ist das dadurch auch $2t$ Fehler erkannt werden. +Um zurück auf unser Beispiel zu kommen, +können von den 7 Übertragungspunkten bis zu $2t = 2\cdot2 = 4 $ Punkten falsch liegen +und es wird kein eindeutiges Polynom zweiten Grades erkannt, und somit die Nutzlast Daten als fehlerhaft deklariert. +Um aus den Übertragenen Zahlen wieder die Nutzlastzahlen zu bekommen könnte man eine Polynominterpolation anwenden, +doch die Punkte mit Polynominterpolation zu einem Polynom zu rekonstruieren ist schwierig und Fehleranfällig. diff --git a/buch/papers/reedsolomon/images/Compact_Disc_zoomed_in.png b/buch/papers/reedsolomon/images/Compact_Disc_zoomed_in.png Binary files differnew file mode 100644 index 0000000..69556d0 --- /dev/null +++ b/buch/papers/reedsolomon/images/Compact_Disc_zoomed_in.png diff --git a/buch/papers/reedsolomon/images/codiert.txt b/buch/papers/reedsolomon/images/codiert.txt new file mode 100644 index 0000000..4a481d8 --- /dev/null +++ b/buch/papers/reedsolomon/images/codiert.txt @@ -0,0 +1,96 @@ +0,284 +1,131.570790435043 +2,41.9840308053375 +3,12.1189172092243 +4,23.8408857476069 +5,69.1793197789512 +6,24.0186013379153 +7,37.3066577242559 +8,18.2010889773887 +9,12.3214904922455 +10,15.6627133315015 +11,24.5237955316204 +12,32.1114345314062 +13,44.9845039238714 +14,13.5324640263625 +15,10.1736266929292 +16,4.58257569495584 +17,23.217268502288 +18,16.5769107917917 +19,6.89948680823017 +20,4.84567134895776 +21,10.4219666223433 +22,43.6179140616243 +23,35.9073375743642 +24,15.0332963783729 +25,21.7594021268945 +26,23.2496572716993 +27,17.9815599423852 +28,11.3577742151117 +29,38.467599433197 +30,28.3035029562577 +31,9.54321919833388 +32,21.377558326432 +33,17.6292439561917 +34,12.6951848921471 +35,20.0667752354841 +36,22.9097309529208 +37,8.78894645948548 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--- /dev/null +++ b/buch/papers/reedsolomon/images/designer_qrcode_ohnelogo.png diff --git a/buch/papers/reedsolomon/images/empfangen.txt b/buch/papers/reedsolomon/images/empfangen.txt new file mode 100644 index 0000000..38c13b0 --- /dev/null +++ b/buch/papers/reedsolomon/images/empfangen.txt @@ -0,0 +1,96 @@ +0,284 +1,131.570790435043 +2,41.9840308053375 +3,12.1189172092243 +4,23.8408857476069 +5,69.1793197789512 +6,23.6290258699579 +7,37.3066577242559 +8,18.2010889773887 +9,12.3214904922455 +10,15.6627133315015 +11,24.5237955316204 +12,32.1114345314062 +13,44.9845039238714 +14,13.5324640263625 +15,10.1736266929292 +16,4.58257569495584 +17,23.217268502288 +18,16.5769107917917 +19,6.89948680823017 +20,5.55320238736303 +21,10.4219666223433 +22,43.6179140616243 +23,35.9073375743642 +24,15.0332963783729 +25,21.7594021268945 +26,23.2496572716993 +27,17.9815599423852 +28,11.3577742151117 +29,38.467599433197 +30,28.3035029562577 +31,9.54321919833388 +32,21.377558326432 +33,17.6292439561917 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+82,0 +83,0 +84,0 +85,0 +86,0 +87,0 +88,0 +89,0 +90,0 +91,0 +92,0 +93,0 +94,0 +95,0 diff --git a/buch/papers/reedsolomon/images/locator.txt b/buch/papers/reedsolomon/images/locator.txt new file mode 100644 index 0000000..b28988c --- /dev/null +++ b/buch/papers/reedsolomon/images/locator.txt @@ -0,0 +1,96 @@ +0,0.0301224340567056 +1,0.141653026854885 +2,0.138226631799377 +3,0.0339903276086929 +4,0.310585462557496 +5,0.551427312631385 +6,0.628514858396814 +7,0.51102386251559 +8,0.275861355940449 +9,0.0502396354182268 +10,0.090185502547573 +11,0.110759344849756 +12,0.0684618905063001 +13,0.0362855426992259 +14,0.0697096919781468 +15,0.109288539370248 +16,0.0923187999496653 +17,0.0512198536768088 +18,0.274192386987782 +19,0.51349614953654 +20,0.633154426602466 +21,0.553283743533942 +22,0.307840573214514 +23,0.0341664350328392 +24,0.140270857957 +25,0.138527177682831 +26,0.029637547736156 +27,0.0816962563186052 +28,0.0944383203811073 +29,0.0263932110686261 +30,0.0585881348402056 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+74,0.325309673287599 +75,0.272705389655349 +76,0.149074257381345 +77,0.0247199397628712 +78,0.0680137859566976 +79,0.075388270873485 +80,0.0273637831604903 +81,0.0407867704453274 +82,0.0632964886441949 +83,0.0309749128751093 +84,0.0315202035072035 +85,0.0627625211892184 +86,0.0360843918243497 +87,0.02794920551495 +88,0.0677921493367236 +89,0.0437167157553067 +90,0.0270640150996317 +91,0.0783380025231622 +92,0.0561293738314281 +93,0.0278742033265809 +94,0.0981443889498639 +95,0.0794543457386548 diff --git a/buch/papers/reedsolomon/images/plotfft.tex b/buch/papers/reedsolomon/images/plotfft.tex new file mode 100644 index 0000000..83a89eb --- /dev/null +++ b/buch/papers/reedsolomon/images/plotfft.tex @@ -0,0 +1,89 @@ +% +% Plot der Übertrangungsabfolge ins FFT und zurück mit IFFT +% +\begin{tikzpicture}[] + +%--------------------------------------------------------------- + %Knote +\matrix[draw = none, column sep=25mm, row sep=2mm]{ + \node(signal) [] { + \begin{tikzpicture} + \begin{axis} + [title = {\Large {Signal}}, + xlabel={Anzahl Übertragene Zahlen}, + xtick={0,20,40,64,80,98},] + \addplot[blue] table[col sep=comma] {papers/reedsolomon/images/signal.txt}; + \end{axis} + \end{tikzpicture}}; & + + \node(codiert) [] { + \begin{tikzpicture} + \begin{axis}[title = {\Large {Codiert}}] + \addplot[] table[col sep=comma] {papers/reedsolomon/images/codiert.txt}; + \end{axis} + \end{tikzpicture}}; \\ + + &\node(fehler) [] { + \begin{tikzpicture} + \begin{axis}[scale=0.6, title = {\Large {Fehler}}, + xtick={7,21,75}] + \addplot[red] table[col sep=comma] {papers/reedsolomon/images/fehler.txt}; + \end{axis} + \end{tikzpicture}};\\ + + \node(decodiert) [] { + \begin{tikzpicture} + \begin{axis}[title = {\Large {Decodiert}}] + \addplot[blue] table[col sep=comma] {papers/reedsolomon/images/decodiert.txt}; + \end{axis} + \end{tikzpicture}}; & + + \node(empfangen) [] { + \begin{tikzpicture} + \begin{axis}[title = {\Large {Empfangen}}] + \addplot[] table[col sep=comma] {papers/reedsolomon/images/empfangen.txt}; + \end{axis} + \end{tikzpicture}};\\ + + \node(syndrom) [] { + \begin{tikzpicture} + \begin{axis}[title = {\Large {Syndrom}}] + \addplot[blue] table[col sep=comma] {papers/reedsolomon/images/syndrom.txt}; + \end{axis} + \end{tikzpicture}}; & + + \node(locator) [] { + \begin{tikzpicture} + \begin{axis}[title = {\Large {Locator}}] + \addplot[] table[col sep=comma] {papers/reedsolomon/images/locator.txt}; + \end{axis} + \end{tikzpicture}};\\ +}; +%------------------------------------------------------------- + %FFT & IFFT deskription + +\draw[thin,gray,dashed] (0,12) to (0,-12); +\node(IFFT) [scale=0.7] at (0,12.3) {IFFT}; +\draw[<-](IFFT.south west)--(IFFT.south east); +\node(FFT) [scale=0.7, above of=IFFT] {FFT}; +\draw[->](FFT.north west)--(FFT.north east); + +\draw[thick, ->,] (fehler.west)++(-1,0) +(0.05,0.5) -- +(-0.1,-0.1) -- +(0.1,0.1) -- +(0,-0.5); +%Arrows +\draw[ultra thick, ->] (signal.east) to (codiert.west); +\draw[ultra thick, ->] (codiert.south) to (fehler.north); +\draw[ultra thick, ->] (fehler.south) to (empfangen.north); +\draw[ultra thick, ->] (empfangen.west) to (decodiert.east); +\draw[ultra thick, ->] (syndrom.east) to (locator.west); +\draw(decodiert.south east)++(-1.8,1) ellipse (1.3cm and 0.8cm) ++(-1.3,0) coordinate(zoom) ; +\draw[ultra thick, ->] (zoom) to[out=180, in=90] (syndrom.north); + +%item +\node[circle, draw, fill =lightgray] at (signal.north west) {1}; +\node[circle, draw, fill =lightgray] at (codiert.north west) {2}; +\node[circle, draw, fill =lightgray] at (fehler.north west) {3}; +\node[circle, draw, fill =lightgray] at (empfangen.north west) {4}; +\node[circle, draw, fill =lightgray] at (decodiert.north west) {5}; +\node[circle, draw, fill =lightgray] at (syndrom.north west) {6}; +\node[circle, draw, fill =lightgray] at (locator.north west) {7}; +\end{tikzpicture}
\ No newline at end of file diff --git a/buch/papers/reedsolomon/images/polynom2.tex b/buch/papers/reedsolomon/images/polynom2.tex new file mode 100644 index 0000000..288b51c --- /dev/null +++ b/buch/papers/reedsolomon/images/polynom2.tex @@ -0,0 +1,49 @@ +% polynome +%------------------- +% Teiler für das Skalieren der Grafik /40 +\newcommand{\teiler}{40} + + +%////////////////////////////////////// + +\begin{tikzpicture}[>=latex,thick] + \draw[color=blue, line width=1.4pt] + plot[domain=0:8, samples=100] + ({\x},{(2*\x^2+1*\x+5)/\teiler}); + + \draw[->] (-0.2,0) -- (8,0) coordinate[label={$x$}]; + \draw[->] (0,-0.2) -- (0,150/\teiler) coordinate[label={right:$p(x)$}]; + + \def\punkt#1{ + \fill[color=green] #1 circle[radius=0.08]; + \draw #1 circle[radius=0.07]; + } + + \def\hellpunkt#1{ + \fill[color=lightgray] #1 circle[radius=0.08]; + \draw #1 circle[radius=0.07]; + } + + \punkt{(1,8/\teiler)} + \hellpunkt{(2,15/\teiler)} + \hellpunkt{(3,26/\teiler)} + \punkt{(4,41/\teiler)} + \punkt{(5,60/\teiler)} + \punkt{(6,83/\teiler)} + \punkt{(7,110/\teiler)} + + \draw[color=gray,line width=1pt,dashed] + plot[domain=0.5:7, samples=100] + ({\x},{(7.832*\x^2-51.5*\x+121.668)/\teiler}); + + \def\erpunkt#1{ + \fill[color=red] #1 circle[radius=0.08]; + \draw #1 circle[radius=0.07]; + } + \erpunkt{(2,50/\teiler)} + \erpunkt{(3,37.66/\teiler)} + + \draw(0,100/\teiler) -- (-0.1,100/\teiler) coordinate[label={left:$100$}]; + \draw(1,0) -- (1,-0.1) coordinate[label={below:$1$}]; +\end{tikzpicture} +%\end{document} diff --git a/buch/papers/reedsolomon/images/signal.txt b/buch/papers/reedsolomon/images/signal.txt new file mode 100644 index 0000000..c4fa5f8 --- /dev/null +++ b/buch/papers/reedsolomon/images/signal.txt @@ -0,0 +1,96 @@ +0,6 +1,6 +2,0 +3,6 +4,4 +5,0 +6,5 +7,2 +8,1 +9,2 +10,1 +11,2 +12,0 +13,6 +14,3 +15,5 +16,7 +17,5 +18,5 +19,4 +20,1 +21,5 +22,9 +23,9 +24,3 +25,2 +26,6 +27,6 +28,4 +29,2 +30,9 +31,1 +32,1 +33,1 +34,2 +35,6 +36,6 +37,1 +38,9 +39,7 +40,7 +41,1 +42,9 +43,9 +44,10 +45,9 +46,8 +47,5 +48,2 +49,4 +50,1 +51,0 +52,9 +53,3 +54,3 +55,3 +56,5 +57,6 +58,0 +59,8 +60,6 +61,9 +62,3 +63,4 +64,0 +65,0 +66,0 +67,0 +68,0 +69,0 +70,0 +71,0 +72,0 +73,0 +74,0 +75,0 +76,0 +77,0 +78,0 +79,0 +80,0 +81,0 +82,0 +83,0 +84,0 +85,0 +86,0 +87,0 +88,0 +89,0 +90,0 +91,0 +92,0 +93,0 +94,0 +95,0 diff --git a/buch/papers/reedsolomon/images/syndrom.txt b/buch/papers/reedsolomon/images/syndrom.txt new file mode 100644 index 0000000..8ca9eed --- /dev/null +++ b/buch/papers/reedsolomon/images/syndrom.txt @@ -0,0 +1,96 @@ +0,0 +1,0 +2,0 +3,0 +4,0 +5,0 +6,0 +7,0 +8,0 +9,0 +10,0 +11,0 +12,0 +13,0 +14,0 +15,0 +16,0 +17,0 +18,0 +19,0 +20,0 +21,0 +22,0 +23,0 +24,0 +25,0 +26,0 +27,0 +28,0 +29,0 +30,0 +31,0 +32,0 +33,0 +34,0 +35,0 +36,0 +37,0 +38,0 +39,0 +40,0 +41,0 +42,0 +43,0 +44,0 +45,0 +46,0 +47,0 +48,0 +49,0 +50,0 +51,0 +52,0 +53,0 +54,0 +55,0 +56,0 +57,0 +58,0 +59,0 +60,0 +61,0 +62,0 +63,0 +64,0.0275599094902563 +65,0.0115837187254191 +66,0.025877761014238 +67,0.0224618032819697 +68,0.04410594689944 +69,0.0474504002669341 +70,0.0227694695500626 +71,0.0271436638090525 +72,0.0104166666666667 +73,0.0271436638090523 +74,0.0227694695500608 +75,0.0474504002669343 +76,0.0441059468994397 +77,0.0224618032819701 +78,0.0258777610142379 +79,0.0115837187254183 +80,0.027559909490256 +81,0.0245124379481793 +82,0.0499782237195209 +83,0.0401432022864265 +84,0.0232923747656228 +85,0.0237974288564099 +86,0.0143895905726624 +87,0.0271745729691685 +88,0.0275599094902567 +89,0.0515501672184983 +90,0.0358255004834542 +91,0.024700508366373 +92,0.0210194725405171 +93,0.0177592928994296 +94,0.0261327016093158 +95,0.0314909067039411 diff --git a/buch/papers/reedsolomon/main.tex b/buch/papers/reedsolomon/main.tex index 6bd04f2..ab4e4be 100644 --- a/buch/papers/reedsolomon/main.tex +++ b/buch/papers/reedsolomon/main.tex @@ -30,7 +30,7 @@ Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren % Joshua \input{papers/reedsolomon/einleitung.tex} \input{papers/reedsolomon/idee.tex} -\input{papers/reedsolomon/teil2.tex} +%\input{papers/reedsolomon/teil2.tex} \input{papers/reedsolomon/dtf.tex} % Michael @@ -45,6 +45,13 @@ Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren \nocite{reedsolomon:weitz} \nocite{reedsolomon:informationkommunikation} +\nocite{reedsolomon:voyager_programm} +\nocite{reedsolomon:voyager} +\nocite{reedsolomon:cd_wiki} +\nocite{reedsolomon:cd} +\nocite{reedsolomon:strichepunkte} +\nocite{reedsolomon:qr_wiki} +\nocite{reedsolomon:qr} %\nocite{reedsolomon:mendezmueller} \printbibliography[heading=subbibliography] diff --git a/buch/papers/reedsolomon/packages.tex b/buch/papers/reedsolomon/packages.tex index 3643731..b84e228 100644 --- a/buch/papers/reedsolomon/packages.tex +++ b/buch/papers/reedsolomon/packages.tex @@ -8,3 +8,5 @@ % following example %\usepackage{packagename} +\usepackage{pgfplots} +\usepackage{filecontents} diff --git a/buch/papers/reedsolomon/references.bib b/buch/papers/reedsolomon/references.bib index 731bd35..b84b5a4 100644 --- a/buch/papers/reedsolomon/references.bib +++ b/buch/papers/reedsolomon/references.bib @@ -23,3 +23,65 @@ volume = {1} } +@online{reedsolomon:voyager_programm, + title = {Information über das Voyager Programm}, + url = {https://de.wikipedia.org/wiki/Voyager-Programm}, + date = {2021-07-19}, + year = {2021}, + month = {7}, + day = {19} +} + +@online{reedsolomon:voyager, + title = {Bild der Voyager Raumsonde}, + url = {https://en.wikipedia.org/wiki/Voyager_1}, + date = {2021-07-19}, + year = {2021}, + month = {7}, + day = {19} +} + +@online{reedsolomon:cd_wiki, + title = {Alles über die CD}, + url = {https://de.wikipedia.org/wiki/Compact_Disc}, + date = {2021-07-19}, + year = {2021}, + month = {7}, + day = {19} +} + +@online{reedsolomon:cd, + title = {Abbildung einer CD}, + url = {https://www.stickpng.com/img/electronics/compact-discs/stack-compact-disc}, + date = {2021-07-19}, + year = {2021}, + month = {7}, + day = {19} +} + +@online{reedsolomon:strichepunkte, + title = {Abbildung der Striche und Punkte einer CD}, + url = {https://www.researchgate.net/figure/The-readable-area-of-a-CD-is-magnified-in-order- to-see-the-pit-and-land-sizing-The_fig7_303401629}, + date = {2021-07-26}, + year = {2021}, + month = {7}, + day = {26} +} + +@online{reedsolomon:qr_wiki, + title = {Funktionsweise des QR-Codes}, + url = {https://de.wikipedia.org/wiki/QR-Code}, + date = {2021-07-19}, + year = {2021}, + month = {7}, + day = {19} +} + +@online{reedsolomon:qr, + title = {Tool zum erstellen von QR-Codes}, + url = {https://www.qrcode-generator.ch}, + date = {2021-07-19}, + year = {2021}, + month = {7}, + day = {19} +}
\ No newline at end of file diff --git a/buch/papers/reedsolomon/teil2.tex b/buch/papers/reedsolomon/teil2.tex deleted file mode 100644 index b2adc9f..0000000 --- a/buch/papers/reedsolomon/teil2.tex +++ /dev/null @@ -1,40 +0,0 @@ -% -% teil2.tex -- Beispiel-File für teil2 -% -% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil -% -\section{Teil 2 -\label{reedsolomon: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? - -\subsection{De finibus bonorum et malorum -\label{reedsolomon: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. - - diff --git a/buch/papers/spannung/Einleitung.tex b/buch/papers/spannung/Einleitung.tex index b1588ff..8e0d36d 100644 --- a/buch/papers/spannung/Einleitung.tex +++ b/buch/papers/spannung/Einleitung.tex @@ -1,17 +1,18 @@ \section{Einleitung\label{spannung:section:Einleitung}} \rhead{Einleitung} Das Hook'sche Gesetz beschreibt die Beziehung von Spannung und Dehnung von linear-elastischen Materialien im Eindimensionalen. -In diesem Kapitel geht es darum das Hook'sche Gesetz im Dreidimensionalen zu beschreiben. +In diesem Kapitel geht es darum, das Hook'sche Gesetz im Dreidimensionalen zu beschreiben. Durch variable Krafteinwirkungen entstehen in jedem Punkt des Materials eine Vielzahl an unterschiedlichen Spannungen. In jedem erdenklichen Punkt im Dreidimensionalen herrscht daher ein entsprechender individueller Spannungszustand. Um das Hook'sche Gesetz für den 3D Spannungszustand formulieren zu können, reichen Skalare nicht aus. -Darum werden Vektoren, Matrizen und Tensoren zur Hilfe gezogen. +Darum werden Vektoren, Matrizen und Tensoren zu Hilfe gezogen. Mit diesen lässt sich eine Spannungsformel für den 3D Spannungszustand bilden. Diese Spannungsformel ist Grundlage für Computerprogramme und geotechnische Versuche, wie der Oedometer-Versuch. -Um die mathematische Untersuchung vorzunehmen, beschäftigt man sich zuerst mit den spezifischen Gegebenheiten und Voraussetzungen. -Ebenfalls gilt es ein paar wichtige Begriffe und deren mathematischen Zeichen einzuführen. -In diesem Kapitel gehen wir auch auf die Zusammenhänge von Spannung, Dehnungen und Verformungen an elastischen Materialien ein, +Um die mathematischen und physikalischen Berechnungen anwenden zu können, +müssen vorerst ein paar spezifische Bedingungen vorausgesetzt und Annahmen getroffen werden. +Ebenfalls gilt es, ein paar wichtige Begriffe und deren mathematischen Zeichen einzuführen. +In diesem Kapitel gehen wir auch auf die Zusammenhänge von Spannungen, Dehnungen und Verformungen an elastischen Materialien ein, wie sie in gängigen Lehrbüchern der Mechanik oder der Geotechnik behandelt werden, z.~B.~\cite{spannung:Grundlagen-der-Geotechnik}. \section{Spannungsausbreitung\label{spannung:section:Spannungsausbreitung}} @@ -29,7 +30,7 @@ Belastet man den Boden mit einer Spannung so wird diese in den Boden geleitet und von diesem kompensiert. Im Boden entstehen unterschiedlich hohe Zusatzspannungen. Diese Zusatzspannung breitet sich räumlich im Boden aus. -Im Falle einer konstanten Flächenlast $\sigma$ siehe Abbildung~\ref{spannung:Bild4} breitet sich die Zusatzspannung zwiebelartig aus. +Im Falle einer konstanten Flächenlast $\sigma$ siehe Abbildung~\ref{fig:Bild4} breitet sich die Zusatzspannung zwiebelartig aus. \begin{figure} \centering @@ -38,11 +39,11 @@ Im Falle einer konstanten Flächenlast $\sigma$ siehe Abbildung~\ref{spannung:Bi \label{fig:Bild4} \end{figure} -Mit der Tiefe $t$ nimmt diese permanent ab (siehe Abbildung~\ref{spannung:Bild5}). -Wie diese Geometrie der Ausbreitung ist, kann durch viele Modelle und Ansätze näherungsweise beschrieben werden. +Mit der Tiefe $t$ nimmt diese permanent ab (siehe Abbildung~\ref{fig:Bild5}). +Wie diese Geometrie der Ausbreitung aussieht, kann durch viele Modelle und Ansätze näherungsweise beschrieben werden. Diese Zusatzspannung $\sigma$ ist im Wesentlichen abhängig von $(x,y,t)$. Je nach Modell werden noch andere Parameter berücksichtigt. -Das können beispielsweise jenste Bodenkennwerte oder auch der Wassergehalt sein. +Das können beispielsweise verschiedene Bodenkennwerte oder auch der Wassergehalt sein. \begin{figure} \centering @@ -72,18 +73,18 @@ berechnet werden mit: t &= \text{Tiefe [\si{\meter}]} \\ s &= \text{Setzung, Absenkung [m].} \end{align*} -Diese Zusammenhänge sind wie erwähnt unter anderem im Lehrbuch [\cite{spannung:Grundlagen-der-Geotechnik}] beschrieben. +Diese Zusammenhänge sind wie erwähnt unter anderem im Lehrbuch \cite{spannung:Grundlagen-der-Geotechnik} beschrieben. In der praktischen Geotechnik wird man allerdings weitaus schwierigere Situationen antreffen. -Ein Beispiel wäre eine Baugrube mit einem Baugrubenabschluss, wo ein Teil des Bodens abgetragen ist (siehe Abbildung~\ref{spannung:Bild3}). +Ein Beispiel wäre eine Baugrube mit einem Baugrubenabschluss, wo ein Teil des Bodens abgetragen ist (siehe Abbildung~\ref{fig:Bild3}). Die Ausbreitung der Zusatzspannung $\sigma(x,y,t)$ würde hier deutlich komplizierter ausfallen. Dies bedeutet auch eine komplexere Setzung der Bodenoberfläche infolge einer Flächenlast $\sigma$. Aus allen zusätzlichen Spannungen müssen die adäquaten Dehnungen mit Hilfe einer Spannungsgleichung berechnet werden. Diese beruht auf Annahmen nach Hooke auf einem linear-elastischen Boden. -Generell wird im Ingenieurwesen versucht Phänomene möglichst nach dem Hook'schen Gesetz abbilden zu können. +Generell wird im Bauingenieurwesen oder auch im Maschinenbau versucht, manche Phänomene möglichst nach dem Hook'schen Gesetz abbilden zu können. \begin{figure} \centering \includegraphics[width=0.45\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild3.png} - \caption{Beispiel eines Lastauftrags auf den Boden bei einer komplexeren Situation, welches kompliziertere Spannungsausbreitung zur Folge hat} + \caption{Beispiel eines Lastauftrags auf den Boden bei einer komplexeren Situation, welche kompliziertere Spannungsausbreitung zur Folge hat} \label{fig:Bild3} \end{figure} diff --git a/buch/papers/spannung/main.tex b/buch/papers/spannung/main.tex index bbdf730..d2aeda9 100644 --- a/buch/papers/spannung/main.tex +++ b/buch/papers/spannung/main.tex @@ -3,7 +3,7 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:spannung}} +\chapter{Dreidimensionaler Spannungszustand\label{chapter:spannung}} \lhead{Dreiachsiger Spannungszustand} \begin{refsection} \chapterauthor{Adrian Schuler und Thomas Reichlin} diff --git a/buch/papers/spannung/teil0.tex b/buch/papers/spannung/teil0.tex index 7647252..089c28e 100644 --- a/buch/papers/spannung/teil0.tex +++ b/buch/papers/spannung/teil0.tex @@ -1,9 +1,10 @@ \section{Der Spannungszustand\label{spannung:section:Der Spannungsustand}} \rhead{Der Spannungszustand} -Ein Spannungszustand ist durch alle Spannungen, welche in einem beliebigen Punkt im Körper wirken, definiert (siehe Abbildung~\ref{spannung:Bild2}). +Ein Spannungszustand ist durch alle Spannungen, welche in einem beliebigen Punkt im Körper wirken, definiert (siehe Abbildung~\ref{fig:Bild2}). Änderungen der äusseren Kräfte verändern die inneren Spannungszustände im Material. -Um alle Spannungen eines Punktes darstellen zu können, wird ein infinitesimales Bodenelement in Form eines Würfels modellhaft vorgestellt. -Man spricht auch von einem Elementarwürfel, da dieser elementar klein ist. +Um alle Spannungen eines Punktes darstellen zu können, +stellt man sich modellhaft ein infinitesimales Bodenelement in Form eines Würfels vor. +Man spricht auch von einem Elementarwürfel. \begin{figure} \centering @@ -15,19 +16,19 @@ Man spricht auch von einem Elementarwürfel, da dieser elementar klein ist. Es werden jeweils drei Seiten dieses Würfels betrachtet, wobei die drei gegenüberliegenden Seiten im Betrag die selben Spannungen aufweisen, sodass der Elementarwürfel im Gleichgewicht ist. Wäre dieses Gleichgewicht nicht vorhanden, käme es zu Verschiebungen und Drehungen. -Das infinitesimale Bodenteilchen hat die Koordinaten $1$, $2$, $3$. +Das infinitesimale Bodenteilchen hat die Koordinatenachsen $1$, $2$, $3$. Veränderungen der Normalspannungen können durch Schubspannungen kompensiert werden und umgekehrt. -So sind insgesamt neun verschiedene Spannungen möglich, wobei drei Normal- und sechs Schubspannungen sind. +So sind insgesamt neun verschiedene Spannungen möglich, konkret sind dies drei Normal- und sechs Schubspannungen. Normalspannungen wirken normal (mit rechtem Winkel) zur angreifenden Fläche und Schubspannungen parallel zur angreifenden Fläche. Alle Beträge dieser neun Spannungen am Elementarwürfel bilden den Spannungszustand. -Daraus können die äquivalenten Dehnungen $\varepsilon$ mit Hilfe des Hook'schen Gesetz berechnet werden. +Daraus können die äquivalenten Dehnungen $\varepsilon$ mit Hilfe des Hook'schen Gesetzes berechnet werden. Daher gibt es auch den entsprechenden Dehnungszustand. \section{Spannungszustand\label{spannung:section:Spannungsustand}} \rhead{Spannungszustand} -Im einachsigen Spannungszustand herrscht nur die Normalspannung $\sigma_{11}$ (siehe Abbildung~\ref{spannung:Bild1}). +Im einachsigen Spannungszustand herrscht nur die Normalspannung $\sigma_{11}$ (siehe Abbildung~\ref{fig:Bild1}). Das Hook'sche Gesetz beschreibt genau diesen 1D Spannungszustand. Nach Hooke gilt: \[ @@ -59,7 +60,7 @@ mit A &= \text{Fläche [\si{\meter\squared}].} \end{align*} Diese Beziehung gilt bei linear-elastischen Materialien, welche reversible Verformungen zulassen. -Es ist praktisch die relative Dehnung $\varepsilon$ anzugeben und nicht eine absolute Längenänderung $\Delta l$. +Es ist praktisch, die relative Dehnung $\varepsilon$ anzugeben und nicht eine absolute Längenänderung $\Delta l$. \begin{figure} \centering \includegraphics[width=0.35\linewidth,keepaspectratio]{papers/spannung/Grafiken/Bild1.png} @@ -73,10 +74,10 @@ Mithilfe vom Elastizitätsmodul $E$ als Proportionalitätskonstante lässt sich E\cdot\varepsilon \] beschreiben. -Im Falle, dass $E$ nicht konstant ist, kann dieser näherungsweise durch +Im Falle, dass $E$ nicht konstant ist, wird dieser durch \[ E = -\frac{\Delta\sigma}{\Delta\varepsilon} +\frac{\text{d}\sigma}{\text{d}\varepsilon} \] -ausgedrückt werden.
\ No newline at end of file +ausgedrückt.
\ No newline at end of file diff --git a/buch/papers/spannung/teil1.tex b/buch/papers/spannung/teil1.tex index 74516c1..647b452 100644 --- a/buch/papers/spannung/teil1.tex +++ b/buch/papers/spannung/teil1.tex @@ -1,8 +1,8 @@ \section{Skalare, Vektoren, Matrizen und Tensoren\label{spannung:section:Skalare,_Vektoren,_Matrizen_und_Tensoren}} \rhead{Skalare, Vektoren, Matrizen und Tensoren} -Der Begriff Tensor kann als Überbegriff, der mathematischen Objekte Skalar, Vektor und Matrix, betrachtet werden. +Der Begriff Tensor kann als Überbegriff der mathematischen Objekte Skalar, Vektor und Matrix, betrachtet werden. Allerdings sind noch höhere Stufen dieser Objekte beinhaltet. -Ein Skalar, ein Vektor oder eine Matrix ist daher auch ein Tensor. +Skalare, Vektoren oder Matrizen sind daher auch Tensoren. Ein Skalar ist ein Tensor 0. Stufe. Mit einem Vektor können mehrere Skalare auf einmal beschrieben werden. Ein Vektor hat daher die Stufe 1 und ist höherstufig als ein Skalar. @@ -14,11 +14,10 @@ Jede Stufe von Tensoren verlangt andere Rechenregeln. So zeigt sich auch der Nachteil von Tensoren mit Stufen höher als 2. Man ist also bestrebt höherstufige Tensoren mit Skalaren, Vektoren oder Matrizen zu beschreiben. -Der Begriff Tensor wurde 1840 von Rowan Hamilton in die Mathematik eingeführt. +In den 40er Jahren vom 19. Jahrhundert wurde der Begriff Tensor von Rowan Hamilton in die Mathematik eingeführt. James Clerk Maxwell hat bereits mit Tensoren operiert, ohne den Begriff Tensor gekannt zu haben. Erst Woldemar Voigt hat den Begriff in die moderne Bedeutung von Skalar, Matrix und Vektor verallgemeinert. Er hat in der Elastizitätstheorie als erstes Tensoren eingesetzt und beschrieben. Auch Albert Einstein hat solche Tensoren eingesetzt, um in der Relativitätstheorie die Änderung der 4D Raumzeit beschreiben zu können. \cite{spannung:Tensor} -\cite{spannung:Voigtsche-Notation} diff --git a/buch/papers/spannung/teil2.tex b/buch/papers/spannung/teil2.tex index 6326eab..8620afe 100644 --- a/buch/papers/spannung/teil2.tex +++ b/buch/papers/spannung/teil2.tex @@ -3,7 +3,7 @@ Durch komplexe Spannungsausbreitungen im Boden entstehen im 3D Spannungszustand unterschiedliche Normal- und Schubspannungen. \begin{figure} \centering - \includegraphics[width=0.4\linewidth,keepaspectratio]{papers/spannung/Grafiken/infinitesimalerWuerfel.png} + \includegraphics[width=0.30\linewidth,keepaspectratio]{papers/spannung/Grafiken/infinitesimalerWuerfel.png} \caption{Beispiel eines Spannungszustandes; Vergrösserung eines infinitesimalen Bodenteilchen} \label{fig:infinitesimalerWuerfel} \end{figure} @@ -49,7 +49,7 @@ Der Dehnungstensor ist ebenfalls ein Tensor 2. Stufe und kann somit auch als $3\ dargestellt werden und beschreibt den gesamten Dehnungszustand. Der Spannungs- und Dehnungstensor 2. Stufe kann je in einen Tensor 1. Stufe überführt werden, welches ein Spaltenvektor ist. -Gemäss der Hadamard-Algebra dürfen Zeile um Zeile in eine Spalte notiert werden, sodass es einen Spaltenvektor ergibt. +Man darf Zeile um Zeile in eine Spalte notieren, sodass es einen Spaltenvektor ergibt. So ergibt sich der Spannungsvektor \[ @@ -79,7 +79,7 @@ So ergibt sich der Spannungsvektor \sigma_{33} \end{pmatrix} \] -und Dehnungsvektor +und der Dehnungsvektor \[ \overline{\varepsilon} = @@ -140,14 +140,6 @@ C_{3311} & C_{3312} & C_{3313} & C_{3321} & C_{3322} & C_{3323} & C_{3331} & C_{ \end{pmatrix} \] geschrieben werden kann. -Dieser Elastizitätstensor muss für isotrope Materialien zwingend symmetrisch sein. -Folglich gilt: -\[ -\overline{\overline{C}} -= -\overline{\overline{C}}~^{T} -. -\] Die allgemeine Spannungsgleichung lautet nun: \[ \vec\sigma @@ -155,8 +147,7 @@ Die allgemeine Spannungsgleichung lautet nun: \overline{\overline{C}}\cdot\vec{\varepsilon} . \] - -Als Indexnotation +Sie kann ebenfalls als Indexnotation \[ \sigma_{ij} = @@ -164,7 +155,15 @@ Als Indexnotation \sum_{l=1}^3 C_{ijkl}\cdot\varepsilon_{kl} \] -kann dies ebenfalls geschrieben werden. +geschrieben werden. +Der Elastizitätstensor muss für isotrope Materialien zwingend symmetrisch sein. +Folglich gilt: +\[ +\overline{\overline{C}} += +\overline{\overline{C}}~^{T} +. +\] Die Konstanten $C$ werden nun nach dem Hook'schen Gesetz mit Hilfe des Elastizitätsmoduls $E$ definiert. Da dieser Modul durch die eindimensionale Betrachtung definiert ist, @@ -221,7 +220,7 @@ definiert ist. Trägt man die Konstanten in die Matrix ein, ergibt sich \end{pmatrix} . \] -Die Normalspannung $\sigma_{22}$ lässt sich exemplarisch als +Die Normalspannung $\sigma_{22}$ lässt sich zum Beispiel als \[ \sigma_{22} = @@ -229,11 +228,13 @@ Die Normalspannung $\sigma_{22}$ lässt sich exemplarisch als \] berechnen. +Reduzierte Spannungs- und Dehnungsgleichungen + Man betrachte nun die Eigenschaften des Elastizitätstensors. Dieser ist quadratisch und symmetrisch, die verschiedenen Einträge wechseln sich aber miteinander ab. Es ergeben sich keine Blöcke mit einheitlichen Einträgen. -Allerdings weiss man, dass im isotropen Boden der Spannungs-, Dehnungs- und daher auch Elastizitätstensor symmetrisch sind. +Allerdings weiss man, dass im isotropen Boden der Spannungs-, Dehnungs- und daher auch der Elastizitätstensor symmetrisch sind. Wäre dem nicht so, würde sich das Material je nach Richtung unterschiedlich elastisch verhalten. Diese Symmetrie setzt daher voraus, dass \[ @@ -399,7 +400,7 @@ Somit lässt sich die reduzierte allgemeine Spannungsgleichung mit \] beschreiben. Die Konstanten $C$ werden wieder nach dem Hook'schen Gesetz definiert. -Dies ergibt die Spannungsformel, welche weit möglichst vereinfacht ist: +Dies ergibt die Spannungsgleichung, welche weit möglichst vereinfacht ist: \begin{equation} \begin{pmatrix} \sigma_{11}\\ @@ -433,7 +434,7 @@ Dies ergibt die Spannungsformel, welche weit möglichst vereinfacht ist: Im Elastizitätstensor fallen zwei $3\times3$ Blöcke auf, welche nur Einträge mit $0$ haben. Der Tensor besagt also, dass diese jeweiligen Dehnungen keinen Einfluss auf unsere Spannung haben. -Man sieht nun auch ganz gut, dass sich im Vergleich zu der allgemeinen Spannungsgleichung, die Einträge verschoben haben. +Man sieht nun auch ganz gut, dass sich im Vergleich zu der allgemeinen Spannungsgleichung die Einträge verschoben haben. Da nach Voigt zuerst die Normalspannungen und anschliessend die Schubspannungen notiert worden sind, ergeben sich die $3\times3$ Blöcke. Man betrachte als Beispiel die Berechnung von $\sigma_{33}$. @@ -441,8 +442,8 @@ Es ist ersichtlich, dass die Schubdehnungen keinen Einfluss auf $\sigma_{33}$ ha Der Einfluss der zu $\sigma_{33}$ äquivalenten Dehnung $\varepsilon_{33}$ hat den grössten Einfluss. Die anderen Normalspannungen $\sigma_{11}$ und $\sigma_{22}$ haben einen unter anderem mit $\nu$ korrigierten Einfluss. -Von $\overline{\overline{C}}$ bildet man noch die inverse Matrix $\overline{\overline{C}}\mathstrut^{-1}$ um die Gleichung umstellen zu können. -Dadurch erhält man die Dehnungsgleichung: +Von $\overline{\overline{C}}$ bildet man die inverse Matrix $\overline{\overline{C}}\mathstrut^{-1}$, mithilfe des Gauss - Jordan Algorithmus, um die Gleichung umstellen zu können. +Durch einige Berechnungsschritte erhält man die Dehnungsgleichung: \[ \vec{\varepsilon} diff --git a/buch/papers/spannung/teil3.tex b/buch/papers/spannung/teil3.tex index 3e456c3..a9080ea 100644 --- a/buch/papers/spannung/teil3.tex +++ b/buch/papers/spannung/teil3.tex @@ -30,7 +30,7 @@ q \label{spannung:Invariante_q} . \end{equation} -Diese Zusammenhänge werden im Skript [\cite{spannung:Stoffgesetze-und-numerische-Modellierung-in-der-Geotechnik}] aufgezeigt. +Diese Zusammenhänge werden im Skript \cite{spannung:Stoffgesetze-und-numerische-Modellierung-in-der-Geotechnik} aufgezeigt. Die hydrostatische Spannung $p$ kann gemäss Gleichung \eqref{spannung:Invariante_p} als \[ p @@ -38,28 +38,28 @@ p \frac{\sigma_{11}+2\sigma_{33}}{3} \] vereinfacht werden. -Die deviatorische Spannung $q$ wird gemäss Gleichung \eqref{spannung:Invariante_q}als +Die deviatorische Spannung $q$ wird gemäss Gleichung \eqref{spannung:Invariante_q} als \[ q = \sigma_{11}-\sigma_{33} \] -vereinfacht. Man kann $p$ als Isotrop und $q$ als Schub betrachten. +vereinfacht. Man kann $p$ als Druck und $q$ als Schub betrachten. -Die Invarianten können mit der Spannungsformel \eqref{spannung:Spannungsgleichung} berechnet werden. +Die Invarianten $p$ und $q$ können mit der Spannungsgleichung \eqref{spannung:Spannungsgleichung} berechnet werden. Durch geschickte Umformung dieser Gleichung, lassen sich die Module als Faktor separieren. Dabei entstehen spezielle Faktoren mit den Dehnungskomponenten. So ergibt sich \[ -\overbrace{\frac{\sigma_{11}+2\sigma_{33}}{3}}^{p} +\overbrace{\frac{\sigma_{11}+2\sigma_{33}}{3}}^{\displaystyle{p}} = -\frac{E}{3(1-2\nu)} \overbrace{(\varepsilon_{11} - 2\varepsilon_{33})}^{\varepsilon_{v}} +\frac{E}{3(1-2\nu)} \overbrace{(\varepsilon_{11} - 2\varepsilon_{33})}^{\displaystyle{{\varepsilon_{v}}}} \] und \[ -\overbrace{\sigma_{11}-\sigma_{33}}^{q} +\overbrace{\sigma_{11}-\sigma_{33}}^{\displaystyle{q}} = -\frac{3E}{2(1+\nu)} \overbrace{\frac{2}{3}(\varepsilon_{11} - \varepsilon_{33})}^{\varepsilon_{s}} +\frac{3E}{2(1+\nu)} \overbrace{\frac{2}{3}(\varepsilon_{11} - \varepsilon_{33})}^{\displaystyle{\varepsilon_{s}}} . \] Die Faktoren mit den Dehnungskomponenten können so mit @@ -79,8 +79,8 @@ eingeführt werden, mit \varepsilon_{v} &= \text{Hydrostatische Dehnung [-]} \\ \varepsilon_{s} &= \text{Deviatorische Dehnung [-].} \end{align*} -Die hydrostatische Dehnung $\varepsilon_{v}$ kann mit einer Kompression verglichen werden. -Die deviatorische Dehnung $\varepsilon_{s}$ kann mit einer Verzerrung verglichen werden. +Die hydrostatische Dehnung $\varepsilon_{v}$ kann mit einer Kompression und +die deviatorische Dehnung $\varepsilon_{s}$ mit einer Verzerrung verglichen werden. Diese zwei Gleichungen kann man durch die Matrixschreibweise \begin{equation} @@ -90,8 +90,8 @@ Diese zwei Gleichungen kann man durch die Matrixschreibweise \end{pmatrix} = \begin{pmatrix} - \frac{3E}{2(1+\nu)} & 0 \\ - 0 & \frac{E}{3(1-2\nu)} + \displaystyle{\frac{3E}{2(1+\nu)}} & 0 \\ + 0 & \displaystyle{\frac{E}{3(1-2\nu)}} \end{pmatrix} \begin{pmatrix} \varepsilon_{s}\\ @@ -100,9 +100,11 @@ Diese zwei Gleichungen kann man durch die Matrixschreibweise \label{spannung:Matrixschreibweise} \end{equation} vereinfachen. -Man hat so eine Matrix multipliziert mit einem Vektor und erhält einen Vektor. -Änderungen des Spannungszustandes können mit dieser Gleichung vollumfänglich erfasst werden. +Änderungen des Spannungszustandes können mit diesen Gleichungen vollumfänglich erfasst werden. +Diese Spannungsgleichung mit den zwei Einträgen ($p$ und $q$) ist gleichwertig +wie die ursprüngliche Spannungsgleichung mit den neun Einträgen +($\sigma_{11}$, $\sigma_{12}$, $\sigma_{13}$, $\sigma_{21}$, $\sigma_{22}$, $\sigma_{23}$, $\sigma_{31}$, $\sigma_{32}$, $\sigma_{33}$). Mit dieser Formel \eqref{spannung:Matrixschreibweise} lassen sich verschieden Ergebnisse von Versuchen analysieren und berechnen. -Ein solcher Versuch, den oft in der Geotechnik durchgeführt wird, ist der Oedometer-Versuch. +Ein solcher Versuch, der oft in der Geotechnik durchgeführt wird, ist der Oedometer-Versuch. Im nächsten Kapitel wird die Anwendung der Matrix an diesem Versuch beschrieben. diff --git a/buch/papers/spannung/teil4.tex b/buch/papers/spannung/teil4.tex index 2f2e4ce..00b2d4f 100644 --- a/buch/papers/spannung/teil4.tex +++ b/buch/papers/spannung/teil4.tex @@ -1,6 +1,6 @@ -\section{Oedometer-Versuch\label{spannung:section:Oedometer-Versuch}} -\rhead{Oedometer-Versuch} -Mit dem Oedometer-Versuch kann der oedometrische Elastizitätsmodul $E_{OED}$ bestimmt werden. +\section{Oedometrischer Elastizitätsmodul\label{spannung:section:Oedometrischer Elastizitätsmodul}} +\rhead{Oedometrischer Elastizitätsmodul} +Mit dem Oedometer-Versuch kann der oedometrische Elastizitätsmodul $E_{\text{OED}}$ bestimmt werden. Dieser beschreibt ebenfalls das Verhältnis zwischen Spannung und Dehnung, allerdings unter anderen Bedingungen. Diese Bedingung ist das Verhindern der seitlichen Verformung, sprich der Dehnung in Richtung $1$ und $2$. Es wird ein Probeelement mit immer grösseren Gewichten belastet, welche gleichmässig auf das Material drücken. @@ -43,8 +43,8 @@ Diese lautet nun: \end{pmatrix} = \begin{pmatrix} - \frac{E_{OED}}{(1+\nu)} & 0 \\ - 0 & \frac{E_{OED}}{3(1-2\nu)} + \displaystyle{\frac{E_{\text{OED}}}{(1+\nu)}} & 0 \\ + 0 & \displaystyle{\frac{E_{\text{OED}}}{3(1-2\nu)}} \end{pmatrix} \begin{pmatrix} \varepsilon_{11}\\ @@ -52,28 +52,28 @@ Diese lautet nun: \end{pmatrix} . \] -Daraus lässt sich bei jedem Setzungsgrad der oedometrische Elastitzitätsmodul $E_{OED}$ und die seitlichen Spannungen $\sigma_{33}$ mit den 2 Gleichungen +Daraus lässt sich bei jedem Setzungsgrad der oedometrische Elastitzitätsmodul $E_{\text{OED}}$ und die seitlichen Spannungen $\sigma_{33}$ mit den zwei Gleichungen \[ \sigma_{11}-\sigma_{33} = -\frac{E_{OED}}{(1+\nu)}\cdot\varepsilon_{11} +\frac{E_{\text{OED}}}{(1+\nu)}\cdot\varepsilon_{11} \] und \[ \sigma_{11}+2\sigma_{33} = -\frac{E_{OED}}{3(1-2\nu)}\cdot\varepsilon_{11} +\frac{E_{\text{OED}}}{3(1-2\nu)}\cdot\varepsilon_{11} \] berechnen. -Mit diesen Gleichungen hat man das Gleichungssystem um $E_{OED}$ und $\sigma_{33}$ zu berechnen. +Mit diesen Gleichungen hat man das Gleichungssystem um $E_{\text{OED}}$ und $\sigma_{33}$ zu berechnen. Die Poisson-Zahl muss als Kennwert gemäss der Bodenklasse gewählt werden. -Den Versuch kann man auf einem $\sigma$-$\varepsilon$-Diagramm abtragen (siehe Abbildung~\ref{spannung:DiagrammOedometer-Versuch}). +Den Versuch kann man auf einem $\sigma$-$\varepsilon$-Diagramm abtragen (siehe Abbildung~\ref{fig:DiagrammOedometer-Versuch}). Durch die Komprimierung nimmt der Boden mehr Spannung auf, und verformt sich zugleich weniger stark. -Mit diesem ermittelten $E_{OED}$ kann man nun weitere Berechnungen für die Geotechnik durchführen. +Mit diesem ermittelten $E_{\text{OED}}$ kann man nun weitere Berechnungen für die Geotechnik durchführen. \begin{figure} \centering - \includegraphics[width=0.5\linewidth,keepaspectratio]{papers/spannung/Grafiken/DiagrammOedometer-Versuch.png} + \includegraphics[width=0.45\linewidth,keepaspectratio]{papers/spannung/Grafiken/DiagrammOedometer-Versuch.png} \caption{Diagramm Charakteristik verschiedener Elastizitätsmodule bei gleichem Material} \label{fig:DiagrammOedometer-Versuch} \end{figure}
\ No newline at end of file diff --git a/buch/papers/verkehr/main.tex b/buch/papers/verkehr/main.tex index 6348993..98d0581 100644 --- a/buch/papers/verkehr/main.tex +++ b/buch/papers/verkehr/main.tex @@ -3,8 +3,7 @@ % % (c) 2020 Hochschule Rapperswil % -\chapter{Thema\label{chapter:verkehr}} -\lhead{Verkehrsfluss und Verkehrsnetze} +\chapter{Verkehrsfluss und Verkehrsnetze\label{chapter:verkehr}} \begin{refsection} \chapterauthor{Pascal Andreas Schmid und Robine Luchsinger} diff --git a/buch/papers/verkehr/section1.tex b/buch/papers/verkehr/section1.tex index d96d450..416e311 100644 --- a/buch/papers/verkehr/section1.tex +++ b/buch/papers/verkehr/section1.tex @@ -1,99 +1,75 @@ -\section{Einführung} \label{section:verkehr/einfuehrung} -\subsection{Verkehrsnetze} Das Verkehrsnetz besteht aus allen Anlagen, auf oder unter der Erdoberfläche, auf denen eine räumliche Fortbewegung von Personen oder auch Gütern stattfindet. Verkehrsnetze sind ein Bestandteil der Verkehrsinfrastruktur, die auf topografischen Karten festgehalten werden. Sie umfassen den Schienenverkehr, alle Strassen und Wege, wie auch Flugplätze und alle dazugehörigen Bauwerke. Aus verkehrsgeografischer Sicht besteht das Verkehrsnetz aus Kanten, Knotenpunkten und dem Hinterland. Die Knotenpunkte werden auch hier durch die Kanten verbunden, die den Verkehrsstrom aufnehmen, wobei das Hinterland durch einzelne Knoten versorgt wird. Die Aufteilung in Kanten und Knotenpunkte ermöglicht eine Vereinfachung komplexer Verkehrsnetze, damit sie mittels der Graphentheorie untersucht werden können. Grundsätzlich können kurze Wege zwischen den Knotenpunkten das Ziel beim Aufbau eines Verkehrsnetzes sein. Es kann aber auch versucht werden, die Bau- und Unterhaltskosten des Verkehrsnetzes in einem gewissen Rahmen zu halten. Aus diesen Vorgaben ergibt sich dann, je nach dem was gewünscht wird, eine grob- oder feinmaschige Struktur des Netzes. Ziel ist aber ein möglichst wirtschaftliches und optimales Verkehrsnetz. -\subsection{Suchalgorithmen} +\section{Suchalgorithmen} +Inbesondere bei Graphen in Form von Verkehrsnetzen ist das Finden eines kürzesten Weges von Interesse. Mathematisch betrachtet handelt es sich hierbei um ein Optimierungsproblem, bei dem die Summe der Kantengewichte zwischen zwei Knoten minimiert werden soll. Zu diesem Zweck existieren verschiedene Suchalgorithmen. In den folgenden Abschnitten wird auf eine Auswahl davon eingegangen. Zuvor ist es jedoch notwendig, einige Begriffe und Eigenschaften von Suchalgorithmen zu definieren. -\subsubsection{Dijkstra-Algorithmus} -Der Algorithmus von Dijkstra ist benannt nach seinem Erfinder dem Mathematik- und Infomratikprofessor Edsger Dijkstra. Den Algorithmus hat er im Jahr 1959 erfunden. -Der Algorithmus von Dijkstra ist ein Greedy-Algorithmus (gieriger Algorithmus), der schrittweise einen Folgezustand auswählt, damit beim Zeitpunkt der Wahl der grösste Gewinn bzw. das beste Ergebnis erzielt werden kann. -Trotz der Schnelligkeit der Greedy-Algorithmen, können viele Probleme nicht optimal gelöst werden. -Vereinfacht wird beim Dijkstra-Algorithmus, ausgehend von einem Startknoten so lange dem kürzesten Pfad gefolgt, bis der Zielknoten erreicht wird. Dabei muss für jeden besuchten Knoten die Kostenfunktion als auch der Pfad dahin (vorheriger Knoten) gespeichert werden. -Dadurch wird hingegen garantiert, dass, wenn der Zielknoten erreicht wird, auch der kürzeste Pfad gefunden wurde. -Grundlegende Voraussetzung für den Dijkstra-Algorithmus ist die strikte Positivität der Kantengewichte. Andernfalls würde ein wiederholtes Ablaufen einer Kante mit negativem Gewicht zu einer stetigen Reduktion der Kostenfunktion führen, was zu einer unendlichen Schlaufe führen würde. +Einerseits wird zwischen optimalen und nicht-optimalen Algorithmen unterschieden. Ein Suchalgorithmus gilt als optimal, falls er einen günstigsten Pfad zwischen zwei Knoten findet. Es gilt zu beachten, dass im Falle des Vorhandenseins von mehrerern Pfaden mit identischer, minimaler Summe der Kantengewichte zwischen zwei Knoten, mindestens einer dieser Pfade gefunden wird. -Gegeben sei ein Netzwerk mit $n$ Knoten und dem Startknoten $a$. -Alle Kanten sind mit $k(i, j)$ bewertet. -Gesucht wird der kürzeste Pfad zwischen dem Startknoten und allen übrigen Knoten im Netz. -$D(i)$ ist die kürzeste Distanz vom Startknoten $a$ zum Knoten $i, V(i)$ ist der unmittelbare Vorgängerknoten vom Knoten $i$ auf dem kürzesten Weg vom Startknoten $a$ zum Konten $i$ und die Menge $M$ ist die Menge einer bestimmten Auswahl an Knoten. +Weiter wird zwischen informierten und uninformierten Algorithmen differenziert. Während uninformierte Suchalgorithmen den Suchraum schematisch auf Basis der Eigenschaften des Graphen absuchen, bis eine günstigste Lösung gefunden wurde, verwenden informierte Suchalgorithmen eine Heuristik zur Abschätzung der Suchrichtung. Oftmals wird bei informierten Algorithmen ein Verlust der Optimalität zugunsten einer verbesserten Rechenzeit in Kauf genommen. Es exisitieren jedoch auch Heurstiken, die eine optimale Lösung gewährleisten. -Dabei gilt -\begin{equation}M={a}\end{equation} -\begin{equation}D(a)=0\end{equation} wobei -\begin{equation}D(i)=\infty\end{equation} und -\begin{equation}i \neq a \end{equation} -Ausserdem gilt \begin{equation}V(i)=(-) \text{für alle Knoten $i$}\end{equation}\\ +Eine besondere Art von Suchalgorithmen stellen die sogenannten Greedy-Algorithmen, zu deutsch gierige Algorithmen, dar. Sie zeichnen sich dadurch aus, dass sie stets den zurzeit günstigsten Folgezustand auswählen. Dadurch sind sie in der Regel äusserst effizient, garantieren bei vielen Problemstellungen jedoch keine optimale Lösung. -%THEORIE... -Iteration +\subsection{Dijkstra-Algorithmus} +Der Algorithmus von Dijkstra ist benannt nach seinem Erfinder dem Mathematik- und Informatikprofessor Edsger Dijkstra. Er gehört zur Klasse der uninformierten Greedy-Algorithmen. Zudem ist die Optimalität bei strikt positiven Kantengewichten gewährleistet. +Vorteilhaft ist die einfache Implementierung. Abhängig von der Programmiersprache sind zwischen 30 und 40 Zeilen an Code ausreichend, damit er den kürzesten Pfad zwischen einem Startknoten $a$ und Zielknoten $b$ finden kann. -1. Auswahl eines Knotens \begin{equation} K\in M \text{mit} D(K)=D(i);i\in M\end{equation} +Die für dieses Paper verwendete programmierte Funktion (MATLAB) verwendet eine abgewandelte Form der gewichteten Adjazenz-Matrix $A$, für welche gilt: +Der Matrix-Eintrag $A_{i,j}$ enthält das Kantengewicht der Kante von Knoten $j$ nach $i$ auf. Falls keine Kante zwischen $j$ und $i$ vorhanden ist, beträgt der Eintrag $\infty$. Dies vereinfacht die Implementierung zur Bestimmung des nächst-günstigsten Pfades. +Zudem werden zwei Hilfs-Vektoren $\vec{d}$ und $\vec{b}$ der Länge $n$ eingeführt, wobei $n$ die Anzahl Knoten des Graphen ist. Im Vektoreintrag $\vec{d}(i)$ wird das kummulierte Kantengewicht zur Erreichung von Knoten $i$ vom Startknoten $a$ gespeichert. Der Eintrag $\vec{d}(a)$ beträgt somit $0$. Im Vektor $\vec{b}$ wird zudem vermerkt, falls ein Knoten bereits als Ziel eines kürzesten Pfads gefunden wurde und somit für die weitere Suche nicht mehr berücksichtigt werden muss ($\vec{b}(i)=1$, sonst $\vec{b}(i)=0$). -2. Für alle Nachfolger $N(j)$ vom Knoten $K$ gilt: -\begin{equation}D(K) + k_Kj < D(j)\end{equation} dann wird \begin{equation}D(j) = D(K) + k_Kj, V(j) = K\end{equation} gesetzt und somit wird der Knoten $j$ in die Menge $M$ aufgenommen. +Ausgehend vom Startknoten $a$ wird nun anhand der Matrix $A$ in der Spalte $a$ nach dem kleinsten Eintrag gesucht. Somit wird der Folgeknoten $c$ gefunden. Dieser Vorgang wird nun wiederholt, wobei jedoch sämtliche von Knoten $a$ und $c$ erreichbaren Knoten berücksichtigt werden, die noch nicht besucht wurden. In anderen Worten alle nicht verschwindenden Einträge $i$ der Spalten $a$ und $c$ der Matrix $A$, für welche gilt $\vec{b}(i)=0$. Ausschlaggebend für die folgende Auswahl ist die Summe der kummulierten Kantengewichte und des Kantengewichts des nächsten Knotens. Als Beispiel zur Erreichung von Knoten $k$ über Knoten $j$: +\begin{equation} +\vec{d}(k)=\vec{d}(j)+A(k,j) +\end{equation} +Diese Iteration wird solange durchgeführt, bis der Folgeknoten dem Zielknoten entspricht. -3. Der ausgewählte Knoten \begin{equation}K\in M\text{wird aus der Menge herausgelöscht}\end{equation}\\ -Diese drei Schritte werden so lange wiederholt bis gilt -\begin{equation}M=\{\}\end{equation} +\subsection{A*-Algorithmus} +Der A*-Algorithmus basiert auf dem Dijkstra-Algorithmus, verwendet jedoch eine Heuristik zur Abschätzung der günstigsten Suchrichtung. Somit handelt es sich um einen informierten Greedy-Algorithmus, der abhängig von der verwendeten Heuristik auch optimal sein kann. Er wurde von Peter Hart, Nils Nilsson und Bertram Raphael entwickelt. -\subsubsection{A*-Algorithmus} -Suchalgorithmen werden nach einfachen (uninformierte) und heuristischen (informierten) Algorithmen unterschieden. Während einfache Algorithmen den Suchraum intuitiv durchsuchen, beziehen heuristische Algorithmen Wissen über den Suchraum mit ein. -Der A*-Algorithmus geht auf seine Erfinder Peter Hart, Nils Nilsson und Bertram Raphael zurück, die den Algorithmus erstmals im Jahr 1968 beschrieben. -Der A*-Algorithmus ist ein heuristischer Suchalgorithmus, der den kürzesten Pfad zwischen zwei Knoten in einem Graphen mit positiven Kantengewichten berechnet. -Im Gegensatz zu einfachen Suchalgorithmen, wird beim A*-Algorithmus eine Schätzfunktion, die sogenannte Heuristik, verwendet. Dies ermöglicht ein zielgerichtetes Suchen und gleichzeitig wird die Laufzeit verringert. -Ausserdem findet der A*-Algorithmus immer eine optimale Lösung, sofern eine vorhanden ist. -Der A*-Algorithmus wird als Verallgemeinerung gehandhabt und gilt als Erweiterung des Dijkstra-Algorithmus. +\subsection{Anwendung A*-Algorithmus} +Wie oben erwähnt basiert der A*-Algorithmus auf dem Shortest-Path-Algorithmus von Dijkstra. Gemäss dem Algorihtmus von Dijkstra werden von einem Startknoten aus die jeweiligen Nachbarknoten, die Nachbarknoten der Nachbarknoten usw. verarbeitet. Die Kantengewichte werden dabei aufsummiert und die Priorität wird auf die Kante gelegt, die das geringste Gewicht aufweist. Mit diesem Verfahren wird sichergestellt, dass die erste gefundene Lösung auch eine optimale Lösung darstellt.\\ -\subsubsection{Anwendung A*-Algorithmus} -Wie oben erwähnt basiert der A*-Algorithmus auf dem Shortest-Path-Algorithmus von Dijkstra. Gemäss dem Algorihtmus von Dijkstra werden von einem Startknoten aus die jeweiligen Nachbarknoten, die Nachbarknoten der Nachbarknoten usw. verarbeitet. Die Kantengewichte werden dabei aufsummiert und die Priorität wird auf die Kante gelegt, die das geringste Gewicht aufweist. Mit diesem Verfahren wird sichergestellt, dass die erste gefundene Lösung auch die optimalste Lösung darstellt.\\ +Der A*-Algorithmus unterscheidet sich vom Dijkstra-Algorithmus dahingehend, dass bei der Auswahl des Folgeknotens, nicht nur die Summe der Kantengewichte $\vec{d}(j)+A(k,j)$, sondern zusätzlich die für jeden Knoten definierte Abschätzfunktion $f(k)$ hinzuaddiert wird. Dies passiert jedoch nur bei der \emph{Auswahl} des Folgeknotens. Der Wert von $f(k)$ wird nicht im Eintrag $\vec{d}(k)$ gespeichert. Somit wird gewährleistet, dass der gefundene Pfad, der Summe der Kantengewichte entspricht. Ein Beispiel dafür, wie eine Abschätzfunktion gebildet werden kann findet sich in Abschnitt \ref{sec:verkehr/euklidische} -Die Kantengewichte werden für jeden Knoten in Form einer Funktion dargestellt -\begin{equation}f(n)=g(n)\end{equation} mit -\begin{equation}g(n)=\text{Summe aller Kantengewichte vom Startknoten bis n}\end{equation}\\ -Der A*-Algorithmus erweitert die Vorgehensweise des Algorithmus von Dijkstra um die Heuristik $h(n)$, die für jeden Knoten $n$ die geschätzte Entfernung zum Zielknoten beschreibt. -Somit gilt: -\begin{equation}f(n)=g(n)+h(n)\end{equation}\\ -Wie auch der Algorithmus von Dijkstra findet der A*-Algorithmus die optimalste Lösung. +\subsection{Euklidische Heuristik} +\label{sec:verkehr/euklidische} +Bei Verkehrsnetzen ist die euklidische Distanz eine gängige und zuverlässige Heurstik. Dabei wird zu den effektiven Reisekosten zum aktuellen Knoten die euklidische Distanz bis zum Zielknoten hinzuaddiert. Dadurch wird die Kostenfunktion konsequent nie überschätzt. Dies stellt eine Voraussetzung an eine zulässige Heuristik dar. Unter Verwendung dieser Heuristik gilt der A*-Algorithmus als optimal. -\subsubsection{Floyd-Warshall-Algorithmus} -Der Floyd-Warshall-Algorithmus, auch Tripel-Algorithmus genannt, wurde erstmals im Jahr 1962 von seinen Namensgebern Robert Floyd und Stephen Warshall vorgestellt. -Der Floyd-Warshall-Algorithmus sucht kürzeste Wege innerhalb eines Graphen. Er ermittelt aber nicht nur die Distanz zwischen zwei Knoten, sondern berechnet die kürzesten Wege zwischen allen Knotenpaaren eines gewichteten Graphen. Somit werden die kürzesten , beziehungsweise die optimalsten Wege zwischen allen Paaren von Knoten berechnet. Der Floyd-Warhshall-Algrithmus kann ausserdem mit negativen Kantengewichten umgehen, sofern der Graph aber keinen negativen Kreis (Zyklus) aufweist. Ist dies der Fall, führt der Algorithmus zu einem falschen Ergebnis. -Ein Kreis (Zyklus) in einem Graphen ist ein Weg, bei dem Start- und Endpunkt den gleichen Knoten aufweisen. Dieser wird negativ, wenn die Summe der gewichteten Kanten kleiner als Null wird.\\ -Der Floyd-Warshall-Algorithmus besteht grundsätzlich aus Floyd's Berechnung der kürzesten Distanzen zwischen zwei Knoten und Warshall's Konstruktion der kürzesten Wege. Werden diese beiden Teilgebiete zusammengefügt, ergibt sich der Floyd-Warshall-Algorithmus. +Bei der euklidischen Heuristik wird die Abschätzfunktion $f(k)$ für jeden Knoten $k$ durch euklidische Distanz zum Zielknoten $b$ gebildet. +\begin{equation} +f(k)=\sqrt{(x_k-x_b)^2+(y_k-y_b)^2} +\end{equation} + +Was bei einem physischen Verkehrsnetz einfach zu bewältigen ist, da Koordinaten von Verkehrsnetzen zur Berechnung der Distanz verwendet werden können, ist bei virtuellen Netzwerken (z.B. Servernetzen) entweder nicht möglich, oder nicht relevant. Hier können hingegen andere Eigenschaften des Netzwerks verwendet werden, auf welche in diesem Paper nicht weiter eingegangen wird. -\subsubsection{Anwendung Floyd-Warshall-Algorithmus} +\subsection{Floyd-Warshall-Algorithmus} +Der Floyd-Warshall-Algorithmus, auch Tripel-Algorithmus genannt, wurde erstmals im Jahr 1962 von seinen Namensgebern Robert Floyd und Stephen Warshall vorgestellt. +Der Floyd-Warshall-Algorithmus sucht kürzeste Wege innerhalb eines Graphen. Er ermittelt aber nicht nur die Distanz zwischen zwei Knoten, sondern berechnet die kürzesten Wege zwischen allen Knotenpaaren eines gewichteten Graphen. Somit werden die günstigsten Wege zwischen allen Paaren von Knoten berechnet. Der Floyd-Warhshall-Algrithmus kann ausserdem mit negativen Kantengewichten umgehen, sofern der Graph keinen negativen Kreis (Zyklus) aufweist. Ein Kreis, sprich ein Weg mit identischem Start- und Zielknoten, ist negativ, falls die Summe der Kantengewichte des Weges kleiner als null ist. Ist dies der Fall, führt der Algorithmus zu einem falschen Ergebnis. -Wie oben erwähnt, besteht der Floyd-Warshall-Algorithmus aus dem Teil von Floyd zur Berechnung der kürzesten Pfade und dem Teil von Warshall zur Konstruktion der kürzesten Pfade. +\subsection{Anwendung Floyd-Warshall-Algorithmus} %THEORIE... -Als erstes wird eine Gewichtsmatrix $W$ mit den Matrixeinträgen $W[i, j]$ erstellt. +In einem ersten Schritt wird eine Gewichtsmatrix $W$ mit den Matrixeinträgen $W[i, j]$ erstellt. Der Algorithmus berechnet danach in einer Hauptschleife alle Knoten $k$ von 1 bis $n$. Dabei versucht er in jeder Iteration alle Wege von $i$ nach $j$ durch die Wege $(i, k)$ und $(k, j)$ zu verbessern. -Falls dieser mögliche Umweg zu einer Verbesserung führt, wird der Algorithmus aktualisiert. +Falls dieser mögliche Umweg zu einer Verbesserung führt, wird der entsprechende Eintrag aktualisiert. Die aktuelle Gewichtung der Pfade wird mit -\begin{equation}d[i, j]=min[d[i,j], d[i,k] + d[k,i]]\end{equation} +\begin{equation}d[i, j]=\min[d[i,j], d[i,k] + d[k,i]]\end{equation} ermittelt. -\subsubsection{Euklidische Heuristik} -Bei Verkehrsnetzen ist die euklidische Distanz eine gängige und zuverlässige Heurstik. Dabei wird zu den effektiven Reisekosten zum aktuellen Knoten die euklidische Distanz bis zum Zielknoten hinzuaddiert. Dadurch wird die Kostenfunktion konsequent nie überschätzt. Dies stellt eine Voraussetzung an eine zulässige Heuristik dar. -Was bei einem physischen Verkehrsnetz einfach zu bewältigen ist, da Koordinaten von Verkehrsnetzen zur Berechnung der Distanz verwendet werden können, ist bei virtuellen Netzwerken (z.B. Servernetzen) entweder nicht möglich, oder nicht relevant. -\subsection{PageRank-Algorithmus} -Der PageRank-Algorithmus wurde von den Gründern von Google, Larry Page und Sergey Brin im Jahr 1996 entwickelt und zum Patent angemeldet. Zwei Jahre später gründeten sie ihr Unternehmen Google Inc.. -Beim PageRank-Algorithmus handelt es sich um den Algorithmus von Google, aus dem die Google-Matrix abgeleitet wird. -Die Google-Matrix ist eine immens grosse Matrix mit Millionen Zeilen und Spalten, die für die schnelle und vor allem exakte Bestimmung der PageRanks (Gewichtung) eine grosse Bedeutung hat. -Der PageRank-Algorithmus analysiert und gewichtet beispielsweise die Verlinkungsstruktur verschiedener Websites des World Wide Web anhand ihrer Struktur. -Der PageRank wird umso höher, je mehr hochwertige Links auf eine Webseite verweisen und je höher die Gewichtung einer Webseite ist, desto grösser ist der Effekt.\\ -Dabei handelt es sich um einen iterativen Prozess. Ausgegangen wird von der Adjazenz-Matrix $A$, für welche gilt. -%THEORIE... -Grundsätzlich setzt sich der PageRank Algorithmus mit der Fragestellung auseinander, wie eine Suchmaschine wie Google Suchresultate bewertet und somit sortieren soll. Öfters aufgerufene Resultate sollen schliesslich höher gewichtet werden. Dabei wird angenommen, dass eine Website populärer ist, je mehr andere Websites darauf verweisen. +\section{PageRank-Algorithmus} +Der PageRank-Algorithmus wurde von den Gründern von Google, Larry Page und Sergey Brin im Jahr 1996 entwickelt und zum Patent angemeldet. Zwei Jahre später gründeten sie ihr Unternehmen Google Inc. +Beim PageRank-Algorithmus handelt es sich nicht um einen Suchalgorithmus, stattdessen werden Knoten aufgrund der Vernetzung des vorliegenden Graphen bewertet. +Verwendet wird er beispielsweise um die Verlinkungsstruktur verschiedener Websites des World Wide Web anhand ihrer Struktur zu bewerten und relevante Suchergebnisse zu ermittteln. Der PageRank wird umso höher, je mehr hochwertige Links auf eine Webseite verweisen und je höher die Gewichtung einer Webseite ist, desto grösser ist der Effekt.\\ +Dabei handelt es sich um einen iterativen Prozess. Ausgegangen wird von der Adjazenz-Matrix $A$, für welche folgendes gilt: \begin{equation} A_{i,j}=\left\{ \begin{matrix} @@ -103,16 +79,20 @@ A_{i,j}=\left\{ \begin{matrix} \label{verkehr:Adja} \end{equation} +%THEORIE... +Grundsätzlich setzt sich der PageRank Algorithmus mit der Fragestellung auseinander, wie eine Suchmaschine wie Google Suchresultate bewertet und somit sortieren soll. Öfters aufgerufene Resultate sollen schliesslich höher gewichtet werden. Dabei wird angenommen, dass eine Website populärer ist, je mehr andere Websites darauf verweisen. + + -Für ungerichtete Graphen mit $n$ Knoten gilt \begin{equation}A_{i,j}=A_{j,i}\end{equation} und weiter \begin{equation}A_{i,i}=0\quad\forall i\in \left\{1...n\right\}\end{equation} +Für ungerichtete Graphen mit $n$ Knoten gilt \begin{equation}A_{i,j}=A_{j,i}\end{equation} und weiter \begin{equation}A_{i,i}=0\quad\forall i\in \left\{1\dot n\right\}\end{equation} Beim PageRank-Algorithmus wird eine abgewandelte Form der Adjazenz-Matrix verwendet. Dabei werden die Matrix-Einträge spaltenweise durch die jeweilige Spaltensumme geteilt. -\begin{equation} P_{i,j}=\frac{A_{i,j}}{\sum_{i=1}^{n}A_{i,j}} \end{equation} +\[ P_{i,j}=\frac{A_{i,j}}{\sum_{i=1}^{n}A_{i,j}} \] Anschliessend multipliziert man diese Matrix $P$ mit einem Spaltenvektor $\Vec{r_0}$ mit $n$ Einträgen, für welchen gilt: -\begin{equation} \Vec{r_0}(i) = \frac{1}{n} \quad\forall i\in \left\{1...n\right\} \end{equation} +\[ \Vec{r_0}(i) = \frac{1}{n} \quad\forall i\in \left\{1\dot n\right\} \] Dieser Vektor stellt ein neutrales Ranking dar. Alle Knoten werden gleich gewichtet. -Dadurch erhält man wiederum einen $n$-zeiligen Spaltenvektor $\Vec{r_1}$, der das "erste" Ranking darstellt. Durch Multiplikation der ursprünglichen Matrix $P$ mit dem 1. Ranking-Vektor $\Vec{r_1}$ wird auf Basis des ersten Rankings ein zweites erstellt. -\begin{equation} \Vec{r_2} = P\cdot\Vec{r_1} = P\cdot(P\cdot\Vec{r_0}) = P^2\cdot\Vec{r_0}\end{equation} +Dadurch erhält man wiederum einen $n$-zeiligen Spaltenvektor $\Vec{r_1}$, der das ``erste" Ranking darstellt. Durch Multiplikation der ursprünglichen Matrix $P$ mit dem 1. Ranking-Vektor $\Vec{r_1}$ wird auf Basis des ersten Rankings ein zweites erstellt. +\[ \Vec{r_2} = P\cdot\Vec{r_1} = P\cdot(P\cdot\Vec{r_0}) = P^2\cdot\Vec{r_0}\] somit \begin{equation} \Vec{r_i} = P^i\cdot\Vec{r_0}\end{equation} -Der Vektor $\Vec{r_i}$ konvergiert zu einem Eigenvektor von $P$ und stellt das abschliessende Ranking dar. +Der Vektor $\Vec{r_i}$ konvergiert zu einem Eigenvektor von $P$ der das abschliessende Ranking darstellt. diff --git a/buch/papers/verkehr/section2.tex b/buch/papers/verkehr/section2.tex index 638d9dd..527885e 100644 --- a/buch/papers/verkehr/section2.tex +++ b/buch/papers/verkehr/section2.tex @@ -1,12 +1,12 @@ \section{Versuchsreihe} \label{section:verkehr/versuchsreihe} -Um zwei der vorgestellten Suchalgorithmen zu vergleichen, wurden zwei Versuchsreihen erstellt. Dazu wurden in einem ersten Schritt zufällige Netzwerke generiert und anschliessend der \emph{Dijkstra}-, sowie der \emph{$A^*$}-Algorithmus auf das Netzwerk angewandt. -Dieser Vorgang wurde für die zufällig generierten Netzwerke mit einer Knotenzahl von 10, 20 50, 100, 200, 500 und 1000 je zehnmal repetiert. -Die Anzahl der Knoten im abgesuchten Netzwerk wirkt sich direkt auf die Rechenzeit aus. Der \emph{Dijkstra}-Algorithmus weist eine Zeitkomplexität von $\mathcal{O}(E\log{}V)$ auf, wobei $E$ die Anzahl Kanten (engl. \emph{edges}) und $V$ die Anzahl Knoten (engl. \emph{vertices}) darstellt. -Für den \emph{A*}-Algorithmus ist die Zeitkomplexität einerseits abhängig von der verwendeten Heuristik, andererseits aber auch vom vorliegenden Netzwerk selbst. Aus diesem Grund lässt sich keine defintive Angabe zu $\mathcal{O}$ machen. +Um zwei der vorgestellten Suchalgorithmen zu vergleichen, wurden zwei Versuchsreihen erstellt. Dazu wurden in einem ersten Schritt zufällige Netzwerke generiert und anschliessend der Dijkstra- und der A*-Algorithmus auf das Netzwerk angewandt. +Dieser Vorgang wurde für die zufällig generierten Netzwerke mit einer Knotenzahl von 10, 20 50, 100, 200, 500 und 1000 je zehnmal wiederholt. +Die Anzahl der Knoten im abgesuchten Netzwerk wirkt sich direkt auf die Rechenzeit aus. Der \emph{Dijkstra}-Algorithmus weist eine Zeitkomplexität von $\mathcal{O}(|E|\log{}|V|)$ auf, wobei $E$ die Menge der Kanten (engl. \emph{edges}) und $V$ die Menge der Knoten (engl. \emph{vertices}) des Graphen $G$ darstellt. +Für den A*-Algorithmus ist die Zeitkomplexität einerseits abhängig von der verwendeten Heuristik, andererseits aber auch vom vorliegenden Netzwerk selbst. Aus diesem Grund lässt sich keine definitive Angabe zur Zeitkomplexität machen. -Die beiden Versuchsreihen unterscheiden sich zudem dahingehend, dass der Start- und Zielknoten bei der ersten Versuchsreihe im Netzwerk diametral gegenüber liegen. Dadurch gehen viele Knoten verloren, welcher \emph{Dijkstra} als uninformierter Suchalgorithmus absuchen würde. In der zweiten Veruschsreihe werden hingegen Start- un Zielpunkt zufällig im Netzwerk ausgewählt. Es wird deshalb erwwartet, dass die Unterschiede in der Rechenzeit der beiden Algorithmen in der zweiten Versuchsreihe deutlich ausgeprägter sind. +Die beiden Versuchsreihen unterscheiden sich zudem dahingehend, dass der Start- und Zielknoten bei der ersten Versuchsreihe im Netzwerk diametral gegenüber liegen. Dadurch gehen viele Knoten verloren, welcher \emph{Dijkstra} als uninformierter Suchalgorithmus absuchen würde. In der zweiten Veruschsreihe werden hingegen Start- un Zielpunkt zufällig im Netzwerk ausgewählt. Es wird deshalb erwartet, dass die Unterschiede in der Rechenzeit der beiden Algorithmen in der zweiten Versuchsreihe deutlich ausgeprägter sind. \subsection{Einfluss der Knotenzahl auf die Rechenzeit} \label{verkehr:Knotenzahl} @@ -19,9 +19,9 @@ Die beiden Versuchsreihen unterscheiden sich zudem dahingehend, dass der Start- \label{verkehr:Vr1} \end{figure} -In \ref{verkehr:Vr1} ist ersichtlich, dass der Unterschied in der Rechenzeit zwischen \emph{Dijkstra} und \emph{A*} erst aber einer Knotenzahl von ca. $n=500$ merklich ansteigt. Dieses etwas überraschende Resultat ist darauf zurückzuführen, dass bei steigender Knotenzahl die Abweichung des effektiven kürzesten Pfades von der Distanz der Luftlinie abnimmt. +In \ref{verkehr:Vr1} ist ersichtlich, dass der Unterschied in der Rechenzeit zwischen Dijkstra und A* erst ab einer Knotenzahl von ca. $n=500$ merklich ansteigt. Dieses etwas überraschende Resultat ist darauf zurückzuführen, dass bei steigender Knotenzahl die Abweichung des effektiven kürzesten Pfades von der Distanz der Luftlinie abnimmt. Die Effektivität von \emph{A*} mit euklidischer Heuristik ist wiederum grösser, wenn die Abweichung des kürzesten Pfads von der Luftlinie minimal ist. -Bei Betrachtung von \ref{verkehr:pathDifference} wird dies ersichtlich, wobei die relative Abweichung erstaunlicherweise bei einer Knotenzahl von $n=100$ maximal ist und nach $n=500$ nur noch marginal abnimmt. +Abbildung \ref{verkehr:pathDifference} illustriert dies, wobei die relative Abweichung erstaunlicherweise bei einer Knotenzahl von $n=100$ maximal ist und nach $n=500$ nur noch marginal abnimmt. \begin{figure} \centering @@ -36,13 +36,13 @@ Bei Betrachtung von \ref{verkehr:pathDifference} wird dies ersichtlich, wobei di \begin{figure} \centering -\includegraphics[width=12cm]{papers/verkehr/figures/chart_Vr2.png}\\ +\includegraphics[width=12cm]{papers/verkehr/figures/chart_Vr2.png} \caption{Gemessene Rechenzeiten der zweiten Versuchsreihe in Abhängigkeit der Knotenzahl.} \label{verkehr:Vr2} \end{figure} -Zum Vergleich der Resultate in \ref{verkehr:Knotenzahl} zeigt \ref{verkehr:Vr2} die Rechenzeiten der zweiten Versuchsreihe, in welcher die Start- und Zielknoten zufällig im Netzwerk ausgewählt wurden. Einerseits ist eine reduzierte durchschnittliche Rechenzeit festzustellen, was schlicht daran liegt, dass die zufällige Wahl der Knoten dazu führt, dass diese tendenziell weniger weit auseinander liegen.\\ -Des weiteren ist festzustellen, dass sich die Unterschiede der Rechenzeiten zwischen \emph{Dijkstra} und \emph{A*} deutlich früher abzeichnen. Dieses Phänomen lässt sich leicht durch die zielgerichtete Suche des \emph{A*}-Algorithmus erklären. +Zum Vergleich der Resultate in Abschnitt \ref{verkehr:Knotenzahl} zeigt Abbildung \ref{verkehr:Vr2} die Rechenzeiten der zweiten Versuchsreihe, in welcher die Start- und Zielknoten zufällig im Netzwerk ausgewählt wurden. Einerseits ist eine reduzierte durchschnittliche Rechenzeit festzustellen, was daran liegt, dass die zufällige Wahl der Knoten dazu führt, dass diese tendenziell weniger weit auseinander liegen. +Des weiteren ist festzustellen, dass sich die Unterschiede der Rechenzeiten zwischen Dijkstra und A* deutlich früher abzeichnen. Dieses Phänomen lässt sich leicht durch die zielgerichtete Suche des A*-Algorithmus erklären. \begin{figure} \centering @@ -52,4 +52,4 @@ Des weiteren ist festzustellen, dass sich die Unterschiede der Rechenzeiten zwis \label{verkehr:Comparison} \end{figure} -In \ref{verkehr:Comparison} ist ersichtlich, dass bei einem im Netzwerk liegenden Startknoten die zielgerichtete Suche von \emph{A*} deutlich ausgeprägter zum Zuge kommt, als wenn dieser am Rand des Netzwerks liegen würde. +In Abbildung \ref{verkehr:Comparison} ist ersichtlich, dass bei einem im Netzwerk liegenden Startknoten die zielgerichtete Suche von \emph{A*} deutlich ausgeprägter zum Zuge kommt, als wenn dieser am Rand des Netzwerks liegen würde. diff --git a/buch/papers/verkehr/section3.tex b/buch/papers/verkehr/section3.tex index 99a0d92..9aa8ae4 100644 --- a/buch/papers/verkehr/section3.tex +++ b/buch/papers/verkehr/section3.tex @@ -1,8 +1,9 @@ \section{Ausblick} \subsection{Optimierungsprobleme bei Graphen} -Das Finden eines kürzesten Pfades, sprich die Minimierung der Summe der Kantengewichte, ist nur eines der Optimierungsprobleme, die sich im Bereich von Grafen aufstellen lassen. Verschiedene, ähnliche Problemstellungen lassen sich teilweise mit denselben Algorithmen lösen.\\ -Im Bereich vom Computernetzwerken könnte zum Beispiel die Minimierung der Knotenzahl zur Datenübbertragung von Interesse sein. Dabei lässt sich dieses Problem einfach dadurch lösen, dass dem \emph{Dijkstra}, oder dem \emph{A*}-Algorithmus anstelle der Graph-Matrix (mit Kantengewichten als Einträgen) die Adjazenz-Matrix als Argument übergeben wird. Der gefundene kürzeste Pfad enstpricht der Anzahl benutzter Kanten, bzw. der Anzahl besuchter Knoten. +Das Finden eines kürzesten Pfades, sprich die Minimierung der Summe der Kantengewichte, ist nur eines der Optimierungsprobleme, die sich im Bereich von Graphen aufstellen lassen. Verschiedene, ähnliche Problemstellungen lassen sich teilweise mit denselben Algorithmen lösen. + +Im Bereich vom Computernetzwerken könnte zum Beispiel die Minimierung der Knotenzahl zur Datenübbertragung von Interesse sein. Dabei lässt sich dieses Problem einfach dadurch lösen, dass dem Dijkstra- oder dem A*-Algorithmus anstelle der gewichteten Adjazenz-Matrix (mit Kantengewichten als Einträgen) die ungewichtet Adjazenz-Matrix als Argument übergeben wird. Der gefundene kürzeste Pfad enstpricht der Anzahl benutzter Kanten, bzw. der Anzahl besuchter Knoten. \subsection{Wahl der Heuristik} -Ein grundlegendes Problem bei der Anwendung des \emph{A*} oder ähnlicher informierter Suchalgorithmen ist die Wahl der Heurstik. Bei einem physischen Verkehrsnetz kann bspw. die euklidische Distanz problems ermittelt werde. Bei einem regionalen Netzwerk ist die Annahme eines orthogonalen X-Y-Koordinatenetzes absolut ausreichend. Dies gilt z.B. auch für das Vernessungsnetz der Schweiz\footnote{Die aktuelle Schweizer Referenzsystem LV95 benutzt ein E/N-Koordinatennetz, wobei aufgrund zunehmender Abweichung vom Referenzellipsoid bei grosser Entfernung vom Nullpunkt ein Korrekturfaktor für die Höhe angebracht werden muss.} Bei überregionalen Netzwerken (Beispiel: Flugverbindungen) ist hingegen eine Berechnung im dreidimensionalen Raum, oder vereinfacht als Projektion auf das Geoid notwendig. Anonsten ist der Ablauf bei der Ausführung des Algorithmus allerdings identisch.\\ +Ein grundlegendes Problem bei der Anwendung des A* oder ähnlicher informierter Suchalgorithmen ist die Wahl der Heurstik. Bei einem physischen Verkehrsnetz kann bspw. die euklidische Distanz problems ermittelt werde. Bei einem regionalen Netzwerk ist die Annahme eines orthogonalen X-Y-Koordinatenetzes absolut ausreichend. Dies gilt z.B. auch für das Vernessungsnetz der Schweiz\footnote{Die aktuelle Schweizer Referenzsystem LV95 benutzt ein E/N-Koordinatennetz, wobei aufgrund zunehmender Abweichung vom Referenzellipsoid bei grosser Entfernung vom Nullpunkt ein Korrekturfaktor für die Höhe angebracht werden muss.} Bei überregionalen Netzwerken (Beispiel: Flugverbindungen) ist hingegen eine Berechnung im dreidimensionalen Raum, oder vereinfacht als Projektion auf das Geoid notwendig. Anonsten ist der Ablauf bei der Ausführung des Algorithmus allerdings identisch. In nicht-physischen Netzwerken stellt sich jedoch eine zweite Problematik. Da eine physische Distanz entweder nicht ermittelt werden kann, oder aber nicht ausschlaggebend ist, sind andere Netzwerk-Eigenschaften zur Beurteilung beizuziehen. Die Zuverlässigkeit ist dabei aber in den meisten Fällen nicht vergleichbar hoch, wie bei der euklidischen Heuristik. Oftmals werden deshalb bei derartigen Problem auch Algorithmen angewendet, die eine deutlich optimierte Zeitkomplexität aufweisen, dafür aber nicht mit Sicherheit den effizienstesten Pfad finden. |
