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authorAndreas Müller <andreas.mueller@ost.ch>2021-07-28 08:07:03 +0200
committerGitHub <noreply@github.com>2021-07-28 08:07:03 +0200
commit9a8dcc1cf9aa0ddd918008e6f2421b48797c38ec (patch)
treeb5113260e190dfc7a94e4298bf6eb5ae21c08344 /buch/papers/multiplikation/presentation/slides
parentMerge pull request #50 from paschost/patch-1 (diff)
parentadded first part of paper and code (diff)
downloadSeminarMatrizen-9a8dcc1cf9aa0ddd918008e6f2421b48797c38ec.tar.gz
SeminarMatrizen-9a8dcc1cf9aa0ddd918008e6f2421b48797c38ec.zip
Merge pull request #52 from Nunigan/master
Multiplikation #1
Diffstat (limited to 'buch/papers/multiplikation/presentation/slides')
-rw-r--r--buch/papers/multiplikation/presentation/slides/algo.tex111
-rw-r--r--buch/papers/multiplikation/presentation/slides/bigO.tex251
-rw-r--r--buch/papers/multiplikation/presentation/slides/blas.tex18
-rw-r--r--buch/papers/multiplikation/presentation/slides/conclusuion.tex0
-rw-r--r--buch/papers/multiplikation/presentation/slides/logo.pdfbin0 -> 8987 bytes
-rw-r--r--buch/papers/multiplikation/presentation/slides/meas.tex42
-rw-r--r--buch/papers/multiplikation/presentation/slides/nn.tex97
-rw-r--r--buch/papers/multiplikation/presentation/slides/parcomp.tex66
-rw-r--r--buch/papers/multiplikation/presentation/slides/slides.tex15
-rw-r--r--buch/papers/multiplikation/presentation/slides/strassen.tex429
10 files changed, 1029 insertions, 0 deletions
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
new file mode 100644
index 0000000..d78ca88
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/slides/logo.pdf
Binary files differ
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 corners=0pt, inner sep=-1pt, fill=green, fit=(X4-3-3)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X4-4-4)] {};
+
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M11-4-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M11-1-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M11-4-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M11-1-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M14-1-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M14-2-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M15-4-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M15-4-2)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M17-2-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M17-4-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M17-2-2)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M17-4-2)] {};
+
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M23-3-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M23-4-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M25-4-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M25-4-2)] {};
+
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M32-1-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M32-1-3)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M34-1-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M34-2-4)] {};
+
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M41-4-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M41-1-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M41-4-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M41-1-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M42-1-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M42-1-3)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M43-3-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M43-4-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M46-1-3)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M46-1-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M46-3-3)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M46-3-1)] {};
+\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}