Merge pull request #52 from Nunigan/master

Multiplikation #1

93 files changed, 49671 insertions, 235 deletions

diff --git a/buch/chapters/10-vektorenmatrizen/linear.tex b/buch/chapters/10-vektorenmatrizen/linear.tex
index e368364..3ad51f1 100644..100755
--- a/buch/chapters/10-vektorenmatrizen/linear.tex
+++ b/buch/chapters/10-vektorenmatrizen/linear.tex
@@ -33,7 +33,7 @@ aber mit Punkten kann man trotzdem noch nicht rechnen.
 Ein Vektor fasst die Koordinaten eines Punktes in einem Objekt zusammen,
 mit dem man auch rechnen und zum Beispiel Parallelverschiebungen
 algebraisieren kann.
-Um auch Streckungen ausdrücken zu können, wird auch eine Menge von 
+Um auch Streckungen ausdrücken zu können, wird auch eine Menge von
 Streckungsfaktoren benötigt, mit denen alle Komponenten eines Vektors
 multipliziert werden können.
 Sie heissen auch {\em Skalare} und liegen in $\Bbbk$.
@@ -73,7 +73,7 @@ a+b
 =
 \begin{pmatrix}\lambda a_1\\\vdots\\\lambda a_n\end{pmatrix}.
 \]
-Die üblichen Rechenregeln sind erfüllt, nämlich 
+Die üblichen Rechenregeln sind erfüllt, nämlich
 \begin{equation}
 \begin{aligned}
 &\text{Kommutativität:}
@@ -149,7 +149,7 @@ kann als (abstrakter) Vektor betrachtet werden.
 \begin{definition}
 Eine Menge $V$ von Objekten, auf der zwei Operationen definiert,
 nämlich die Addition, geschrieben $a+b$ für $a,b\in V$ und die
-Multiplikation mit Skalaren, geschrieben $\lambda a$ für $a\in V$ und 
+Multiplikation mit Skalaren, geschrieben $\lambda a$ für $a\in V$ und
 $\lambda\in \Bbbk$, heisst ein {\em $\Bbbk$-Vektorraum} oder {\em Vektorraum
 über $\Bbbk$} (oder
 einfach nur {\em Vektorraum}, wenn $\Bbbk$ aus dem Kontext klar sind),
@@ -172,7 +172,7 @@ $\mathbb{C}$ ein Vektorraum über $\mathbb{R}$.
 \end{beispiel}
 
 \begin{beispiel}
-Die Menge $C([a,b])$ der stetigen Funktionen $[a,b]\to\mathbb{Re}$ 
+Die Menge $C([a,b])$ der stetigen Funktionen $[a,b]\to\mathbb{Re}$
 bildet ein Vektorraum.
 Funktionen können addiert und mit reellen Zahlen multipliziert werden:
 \[
@@ -188,7 +188,7 @@ Die Vektorraum-Rechenregeln
 \end{beispiel}
 
 Die Beispiele zeigen, dass der Begriff des Vektorraums die algebraischen
-Eigenschaften eine grosse Zahl sehr verschiedenartiger mathematischer 
+Eigenschaften eine grosse Zahl sehr verschiedenartiger mathematischer
 Objekte beschreiben kann.
 Alle Erkenntnisse, die man ausschliesslich aus Vekotorraumeigenschaften
 gewonnen hat, sind auf alle diese Objekte übertragbar.
@@ -300,7 +300,7 @@ folgt, dass alle $\lambda_1,\dots,\lambda_n=0$ sind.
 Lineare Abhängigkeit der Vektoren $a_1,\dots,a_n$ bedeutet auch, dass
 man einzelne der Vektoren durch andere ausdrücken kann.
 Hat man nämlich eine
-Linearkombination~\eqref{buch:vektoren-und-matrizen:eqn:linabhdef} und 
+Linearkombination~\eqref{buch:vektoren-und-matrizen:eqn:linabhdef} und
 ist der Koeffizient $\lambda_k\ne 0$, dann kann man nach $a_k$ auflösen:
 \[
 a_k = -\frac{1}{\lambda_k}(\lambda_1a_1+\dots+\widehat{\lambda_ka_k}+\dots+\lambda_na_n).
@@ -323,7 +323,7 @@ offenbar eine besondere Bedeutung.
 Eine linear unabhängig Menge von Vektoren
 $\mathcal{B}=\{a_1,\dots,a_n\}\subset V$
 heisst {\em Basis} von $V$.
-Die maximale Anzahl linear unabhängiger Vektoren in $V$ heisst 
+Die maximale Anzahl linear unabhängiger Vektoren in $V$ heisst
 {\em Dimension} von $V$.
 \end{definition}
 
@@ -331,7 +331,7 @@ Die Standardbasisvektoren bilden eine Basis von $V=\Bbbk^n$.
 
 \subsubsection{Unterräume}
 Die Mengen $\langle a_1,\dots,a_n\rangle$ sind Teilmengen
-von $V$, in denen die Addition von Vektoren und die Multiplikation mit 
+von $V$, in denen die Addition von Vektoren und die Multiplikation mit
 Skalaren immer noch möglich ist.
 
 \begin{definition}
@@ -352,7 +352,7 @@ gilt.
 %
 \subsection{Matrizen
 \label{buch:grundlagen:subsection:matrizen}}
-Die Koeffizienten eines linearen Gleichungssystems finden in einem 
+Die Koeffizienten eines linearen Gleichungssystems finden in einem
 Zeilen- oder Spaltenvektor nicht Platz.
 Wir erweitern das Konzept daher in einer Art, dass Zeilen- und
 Spaltenvektoren Spezialfälle sind.
@@ -378,14 +378,14 @@ M_{m\times n}(\Bbbk) = \{ A\;|\; \text{$A$ ist eine $m\times n$-Matrix}\}.
 \]
 Falls $m=n$ gilt, heisst die Matrix $A$ auch {\em quadratisch}
 \index{quadratische Matrix}%
-Man kürzt die Menge der quadratischen Matrizen als 
+Man kürzt die Menge der quadratischen Matrizen als
 $M_n(\Bbbk) = M_{n\times n}(\Bbbk)$ ab.
 \end{definition}
 
-Die $m$-dimensionalen Spaltenvektoren $v\in \Bbbk^m$ sind $m\times 1$-Matrizen 
+Die $m$-dimensionalen Spaltenvektoren $v\in \Bbbk^m$ sind $m\times 1$-Matrizen
 $v\in M_{n\times 1}(\Bbbk)$, die $n$-dimensionalen Zeilenvetoren $u\in\Bbbk^n$
 sind $1\times n$-Matrizen $v\in M_{1\times n}(\Bbbk)$.
-Eine $m\times n$-Matrix $A$ mit den Koeffizienten $a_{ij}$ besteht aus 
+Eine $m\times n$-Matrix $A$ mit den Koeffizienten $a_{ij}$ besteht aus
 den $n$ Spaltenvektoren
 \[
 a_1 = \begin{pmatrix} a_{11} \\ a_{21} \\ \vdots \\ a_{m1} \end{pmatrix},\quad
@@ -435,7 +435,7 @@ werden kann.
 \begin{definition}
 Eine $m\times n$-Matrix $A\in M_{m\times n}(\Bbbk)$ und eine
 $n\times l$-Matrix $B\in M_{n\times l}(\Bbbk)$ haben als Produkt
-eine $n\times l$-Matrix $C=AB\in M_{n\times l}(\Bbbk)$ mit den
+eine $m\times l$-Matrix $C=AB\in M_{m\times l}(\Bbbk)$ mit den
 Koeffizienten
 \begin{equation}
 c_{ij} = \sum_{k=1}^n a_{ik} b_{kj}.
@@ -483,7 +483,7 @@ I
 1     &0     &\dots &0     \\
 0     &1     &\dots &0     \\[-2pt]
 \vdots&\vdots&\ddots&\vdots\\
-0     &0     &\dots &1  
+0     &0     &\dots &1
 \end{pmatrix}.
 \]
 
@@ -521,10 +521,10 @@ Ein Gleichungssystem mit $0$ auf der rechten Seite ist also bereits
 ausreichend um zu entscheiden, ob die Lösung eindeutig ist.
 Ein Gleichungssystem mit rechter Seite $0$ heisst {\em homogen}.
 \index{homogenes Gleichungssystem}%
-Zu jedem {\em inhomogenen} Gleichungssystem $Ax=b$ mit $b\ne 0$ 
+Zu jedem {\em inhomogenen} Gleichungssystem $Ax=b$ mit $b\ne 0$
 ist $Ax=0$ das zugehörige homogene Gleichungssystem.
 
-Ein homogenes Gleichungssytem $Ax=0$ hat immer mindestens die 
+Ein homogenes Gleichungssytem $Ax=0$ hat immer mindestens die
 Lösung $x=0$, man nennt sie auch die {\em triviale} Lösung.
 Eine Lösung $x\ne 0$ heisst auch eine nichttriviale Lösung.
 Die Lösungen eines inhomgenen Gleichungssystem $Ax=b$ ist also nur dann
@@ -535,7 +535,7 @@ Lösung hat.
 Der Gauss-Algorithmus oder genauer Gausssche Eliminations-Algorithmus
 löst ein lineare Gleichungssystem der
 Form~\eqref{buch:vektoren-und-matrizen:eqn:vektorform}.
-Die Koeffizienten werden dazu in das Tableau 
+Die Koeffizienten werden dazu in das Tableau
 \[
 \begin{tabular}{|>{$}c<{$}>{$}c<{$}>{$}c<{$}|>{$}c<{$}|}
 \hline
@@ -552,7 +552,7 @@ Der Algorithmus is so gestaltet, dass er nicht mehr Speicher als
 das Tableau benötigt, alle Schritte operieren direkt auf den Daten
 des Tableaus.
 
-In jedem Schritt des Algorithmus wird zunächst eine Zeile $i$ und 
+In jedem Schritt des Algorithmus wird zunächst eine Zeile $i$ und
 Spalte $j$ ausgewählt, das Elemente $a_{ij}$ heisst das Pivotelement.
 \index{Pivotelement}%
 Die {\em Pivotdivision}
@@ -646,7 +646,7 @@ In der Phase der {\em Vorwärtsreduktion} werden Pivotelemente von links
 nach rechts möglichst auf der Diagonale gewählt und mit Zeilensubtraktionen
 die darunterliegenden Spalten freigeräumt.
 \index{Vorwärtsreduktion}%
-Während des Rückwärtseinsetzens werden die gleichen Pivotelemente von 
+Während des Rückwärtseinsetzens werden die gleichen Pivotelemente von
 rechts nach links genutzt, um mit Zeilensubtraktionen auch die
 Spalten über den Pivotelemnten frei zu räumen.
 \index{Rückwärtseinsetzen}%
@@ -800,7 +800,7 @@ $x = b_1c_1+b_2c_2+\dots+b_nc_n$ konstruieren.
 Tatsächlich gilt
 \begin{align*}
 Ax
-&= 
+&=
 A( b_1c_1+b_2c_2+\dots+b_nc_n)
 \\
 &=
@@ -851,10 +851,10 @@ für eine Gleichungssystem mit quadratischer Koeffizientenmatrix $A$
 heisst die Determinante $\det(A)$ der Matrix $A$.
 \end{definition}
 
-Aus den Regeln für die Durchführung des Gauss-Algorithmus kann man die 
+Aus den Regeln für die Durchführung des Gauss-Algorithmus kann man die
 folgenden Regeln für die Determinante ableiten.
 Wir stellen die Eigenschaften hier nur zusammen, detaillierte Herleitungen
-kann man in jedem Kurs zur linearen Algebra finden, zum Beispiel im 
+kann man in jedem Kurs zur linearen Algebra finden, zum Beispiel im
 Kapitel~2 des Skripts \cite{buch:linalg}.
 \begin{enumerate}
 \item
@@ -877,11 +877,11 @@ wird auch der Wert der Determinanten mit $\lambda$ multipliziert.
 \item
 \label{buch:linear:determinante:asymetrisch}
 Die Determinante ist eine lineare Funktion der Zeilen von $A$.
-Zusammen mit der Eigeschaft~\ref{buch:linear:determinante:vorzeichen} 
+Zusammen mit der Eigeschaft~\ref{buch:linear:determinante:vorzeichen}
 folgt, dass die Determinante eine antisymmetrische lineare Funktion
 der Zeilen ist.
 \item
-Die Determinante ist durch die Eigenschaften 
+Die Determinante ist durch die Eigenschaften
 \ref{buch:linear:determinante:einheitsmatrix}
 und
 \ref{buch:linear:determinante:asymetrisch}
@@ -895,7 +895,7 @@ Die Determinante der $n\times n$-Matrix $A$ kann mit der Formel
 =
 \sum_{i=1}^n (-1)^{i+j} a_{ij} \cdot \det(A_{ij})
 \end{equation}
-wobei die $(n-1)\times(n-1)$-Matrix $A_{ij}$ die Matrix $A$ ist, aus der 
+wobei die $(n-1)\times(n-1)$-Matrix $A_{ij}$ die Matrix $A$ ist, aus der
 man Zeile $i$ und Spalte $j$ entfernt hat.
 $A_{ij}$ heisst ein {\em Minor} der Matrix $A$.
 \index{Minor einer Matrix}%
@@ -949,7 +949,7 @@ der rechten Seiten ersetzt worden ist.
 \end{satz}
 
 Die Cramersche Formel ist besonders nützlich, wenn die Abhängigkeit
-einer Lösungsvariablen von den Einträgen der Koeffizientenmatrix 
+einer Lösungsvariablen von den Einträgen der Koeffizientenmatrix
 untersucht werden soll.
 Für die Details der Herleitung sei wieder auf \cite{buch:linalg}
 verwiesen.
@@ -993,7 +993,7 @@ heisst die {\em Adjunkte} $\operatorname{adj}A$ von $A$.
 \end{satz}
 
 Der Satz~\ref{buch:linalg:inverse:adjoint} liefert eine algebraische
-Formel für die Elemente der inversen Matrix. 
+Formel für die Elemente der inversen Matrix.
 Für kleine Matrizen wie im nachfolgenden Beispiel ist die
 Formel~\eqref{buch:linalg:inverse:formel} oft einfachter anzuwenden.
 Besonders einfach wird die Formel für eine $2\times 2$-Matrix,
@@ -1035,7 +1035,7 @@ Die Adjunkte ist
 \begin{pmatrix*}[r]
  \det A_{11} & -\det A_{21} &  \det A_{31} \\
 -\det A_{12} &  \det A_{22} & -\det A_{32} \\
- \det A_{13} & -\det A_{23} &  \det A_{33} 
+ \det A_{13} & -\det A_{23} &  \det A_{33}
 \end{pmatrix*}
 \intertext{und damit ist die inverse Matrix}
 A^{-1}
@@ -1084,7 +1084,7 @@ A^{-1}
 \end{pmatrix}.
 \label{buch:vektoren-und-matrizen:abeispiel:eqn2}
 \end{equation}
-für die Inverse einer Matrix der Form 
+für die Inverse einer Matrix der Form
 \eqref{buch:vektoren-und-matrizen:abeispiel:eqn1}.
 \end{beispiel}
 
@@ -1118,7 +1118,7 @@ Eine Abbildung $f\colon V\to U$ zwischen Vektorräumen $V$ und $U$
 heisst linear, wenn
 \[
 \begin{aligned}
-f(v+w) &= f(v) + f(w)&&\forall v,w\in V 
+f(v+w) &= f(v) + f(w)&&\forall v,w\in V
 \\
 f(\lambda v) &= \lambda f(v) &&\forall v\in V,\lambda \in \Bbbk
 \end{aligned}
@@ -1129,16 +1129,16 @@ gilt.
 Lineare Abbildungen sind in der Mathematik sehr verbreitet.
 
 \begin{beispiel}
-Sie $V=C^1([a,b])$ die Menge der stetig differenzierbaren Funktionen 
+Sie $V=C^1([a,b])$ die Menge der stetig differenzierbaren Funktionen
 auf dem Intervall $[a,b]$ und $U=C([a,b])$ die Menge der
-stetigen Funktion aif $[a,b]$. 
+stetigen Funktion aif $[a,b]$.
 Die Ableitung $\frac{d}{dx}$ macht aus einer Funktion $f(x)$ die
 Ableitung $f'(x)$.
-Die Rechenregeln für die Ableitung stellen sicher, dass 
+Die Rechenregeln für die Ableitung stellen sicher, dass
 \[
 \frac{d}{dx}
 \colon
-C^1([a,b]) \to  C([a,b]) 
+C^1([a,b]) \to  C([a,b])
 :
 f \mapsto f'
 \]
@@ -1157,7 +1157,7 @@ eine lineare Abbildung.
 \end{beispiel}
 
 \subsubsection{Matrix}
-Um mit linearen Abbildungen rechnen zu können, ist eine Darstellung 
+Um mit linearen Abbildungen rechnen zu können, ist eine Darstellung
 mit Hilfe von Matrizen nötig.
 Sei also $\mathcal{B}=\{b_1,\dots,b_n\}$ eine Basis von $V$ und
 $\mathcal{C} = \{ c_1,\dots,c_m\}$ eine Basis von $U$.
@@ -1165,12 +1165,12 @@ Das Bild des Basisvektors $b_i$ kann als Linearkombination der
 Vektoren $c_1,\dots,c_m$ dargestellt werden.
 Wir verwenden die Bezeichnung
 \[
-f(b_i) 
+f(b_i)
 =
 a_{1i} c_1 + \dots + a_{mi} c_m.
 \]
 Die lineare Abbildung $f$ bildet den Vektor $x$ mit Koordinaten
-$x_1,\dots,x_n$ ab auf 
+$x_1,\dots,x_n$ ab auf
 \begin{align*}
 f(x)
 &=
@@ -1193,7 +1193,7 @@ x_n(a_{1n} c_1 + \dots + a_{mn} c_m)
 +
 ( a_{m1} x_1 + \dots + a_{mn} x_n ) c_m
 \end{align*}
-Die Koordinaten von $f(x)$ in der Basis $\mathcal{C}$ in $U$ sind 
+Die Koordinaten von $f(x)$ in der Basis $\mathcal{C}$ in $U$ sind
 also gegeben durch das Matrizenprodukt $Ax$, wenn $x$ der Spaltenvektor
 aus den Koordinaten in der Basis $\mathcal{B}$ in $V$ ist.
 
@@ -1231,7 +1231,7 @@ b_{m1}x_1&+& \dots &+&b_{mn}x_n&=&b_{m1}'x_1'&+& \dots &+&b_{mn}'x_n'
 \end{linsys}
 \]
 Dieses Gleichungssystem kann man mit Hilfe eines Gauss-Tableaus lösen.
-Wir schreiben die zugehörigen Variablen 
+Wir schreiben die zugehörigen Variablen
 \[
 \renewcommand{\arraystretch}{1.1}
 \begin{tabular}{|>{$}c<{$} >{$}c<{$} >{$}c<{$}|>{$}c<{$}>{$}c<{$}>{$}c<{$}|}
@@ -1277,7 +1277,7 @@ Für zwei Vektoren $u$ und $w$ in $U$ gibt es daher Vektoren $a=g(u)$
 und $b=g(w)$ in $V$ derart, dass $f(a)=u$ und $f(b)=w$.
 Weil $f$ linear ist, folgt daraus $f(a+b)=u+w$ und $f(\lambda a)=\lambda a$
 für jedes $\lambda\in\Bbbk$.
-Damit kann man jetzt 
+Damit kann man jetzt
 \begin{align*}
 g(u+w)&=g(f(a)+f(b)) = g(f(a+b)) = a+b = g(u)+g(w)
 \\
@@ -1315,7 +1315,7 @@ Der Kern oder Nullraum der Matrix $A$ ist die Menge
 \]
 \end{definition}
 
-Der Kern ist ein Unterraum, denn für zwei Vektoren $u,w\in \ker f$ 
+Der Kern ist ein Unterraum, denn für zwei Vektoren $u,w\in \ker f$
 \[
 \begin{aligned}
 f(u+v)&=f(u) + f(v) = 0+0 = 0 &&\Rightarrow& u+v&\in\ker f\\
@@ -1331,7 +1331,7 @@ Wir definieren daher das Bild einer linearen Abbildung oder Matrix.
 
 \begin{definition}
 Ist $f\colon V\to U$ eine lineare Abbildung dann ist das Bild von $f$
-der Unterraum 
+der Unterraum
 \[
 \operatorname{im}f = \{ f(v)\;|\;v\in V\} \subset U
 \]
@@ -1375,7 +1375,7 @@ $\operatorname{def}A=\dim\ker A$.
 \end{definition}
 
 Da der Kern mit Hilfe des Gauss-Algorithmus bestimmt werden kann,
-können Rang und Defekt aus dem Schlusstableau 
+können Rang und Defekt aus dem Schlusstableau
 eines homogenen Gleichungssystems mit $A$ als Koeffizientenmatrix
 abgelesen werden.
 
@@ -1391,8 +1391,3 @@ n-\operatorname{def}A.
 
 \subsubsection{Quotient}
 TODO: $\operatorname{im} A \simeq \Bbbk^m/\ker A$
-
-
-
-
-
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
new 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
new 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
new 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
new file mode 100644
index 0000000..95b68b5
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_1024.pdf
diff --git a/buch/papers/multiplikation/code/c_meas_128.pdf b/buch/papers/multiplikation/code/c_meas_128.pdf
new file mode 100644
index 0000000..56b9200
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_128.pdf
diff --git a/buch/papers/multiplikation/code/c_meas_16.pdf b/buch/papers/multiplikation/code/c_meas_16.pdf
new file mode 100644
index 0000000..2edc82d
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_16.pdf
diff --git a/buch/papers/multiplikation/code/c_meas_2048.pdf b/buch/papers/multiplikation/code/c_meas_2048.pdf
new file mode 100644
index 0000000..caba698
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_2048.pdf
diff --git a/buch/papers/multiplikation/code/c_meas_256.pdf b/buch/papers/multiplikation/code/c_meas_256.pdf
new file mode 100644
index 0000000..383ae86
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_256.pdf
diff --git a/buch/papers/multiplikation/code/c_meas_32.pdf b/buch/papers/multiplikation/code/c_meas_32.pdf
new file mode 100644
index 0000000..180fd22
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_32.pdf
diff --git a/buch/papers/multiplikation/code/c_meas_4096.pdf b/buch/papers/multiplikation/code/c_meas_4096.pdf
new file mode 100644
index 0000000..547d794
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_4096.pdf
diff --git a/buch/papers/multiplikation/code/c_meas_512.pdf b/buch/papers/multiplikation/code/c_meas_512.pdf
new file mode 100644
index 0000000..5e8894e
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_512.pdf
diff --git a/buch/papers/multiplikation/code/c_meas_64.pdf b/buch/papers/multiplikation/code/c_meas_64.pdf
new file mode 100644
index 0000000..8ff905c
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_64.pdf
diff --git a/buch/papers/multiplikation/code/c_meas_8.pdf b/buch/papers/multiplikation/code/c_meas_8.pdf
new 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
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diff --git a/buch/papers/multiplikation/code/meas/test/blas.txt b/buch/papers/multiplikation/code/meas/test/blas.txt
new file mode 100644
index 0000000..7b0a9d1
--- /dev/null
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diff --git a/buch/papers/multiplikation/code/meas/test/winograd.txt b/buch/papers/multiplikation/code/meas/test/winograd.txt
new file mode 100644
index 0000000..d01fefd
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/test/winograd.txt
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index 0000000..3a4d88b
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diff --git a/buch/papers/multiplikation/code/meas_512.pdf b/buch/papers/multiplikation/code/meas_512.pdf
new file mode 100644
index 0000000..4d8f04b
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas_512.pdf
diff --git a/buch/papers/multiplikation/code/meas_512.txt b/buch/papers/multiplikation/code/meas_512.txt
new file mode 100644
index 0000000..1b2089d
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diff --git a/buch/papers/multiplikation/code/meas_64.pdf b/buch/papers/multiplikation/code/meas_64.pdf
new file mode 100644
index 0000000..3a90949
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas_64.pdf
diff --git a/buch/papers/multiplikation/code/meas_64.txt b/buch/papers/multiplikation/code/meas_64.txt
new file mode 100644
index 0000000..ae6ff9b
--- /dev/null
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diff --git a/buch/papers/multiplikation/code/meas_8.pdf b/buch/papers/multiplikation/code/meas_8.pdf
new file mode 100644
index 0000000..16d177d
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas_8.pdf
diff --git a/buch/papers/multiplikation/code/meas_8.txt b/buch/papers/multiplikation/code/meas_8.txt
new file mode 100644
index 0000000..6cf6515
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas_8.txt
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diff --git a/buch/papers/multiplikation/code/test.tex b/buch/papers/multiplikation/code/test.tex
new file mode 100644
index 0000000..40ea239
--- /dev/null
+++ b/buch/papers/multiplikation/code/test.tex
@@ -0,0 +1,92 @@
+% This file was created by tikzplotlib v0.9.8.
+\begin{tikzpicture}
+
+\definecolor{color0}{rgb}{0.886274509803922,0.290196078431373,0.2}
+\definecolor{color1}{rgb}{0.203921568627451,0.541176470588235,0.741176470588235}
+\definecolor{color2}{rgb}{0.596078431372549,0.556862745098039,0.835294117647059}
+\definecolor{color3}{rgb}{0.984313725490196,0.756862745098039,0.368627450980392}
+
+\begin{axis}[
+axis background/.style={fill=white!89.8039215686275!black},
+axis line style={white},
+legend cell align={left},
+legend style={
+  fill opacity=0.8,
+  draw opacity=1,
+  text opacity=1,
+  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 0.000133275985717773
+8 0.000703096389770508
+16 0.00453472137451172
+32 0.0282893180847168
+64 0.181003332138062
+128 1.40816903114319
+};
+\addlegendentry{Strassen MM}
+\addplot [line width=2pt, white!46.6666666666667!black]
+table {%
+2 2.19345092773438e-05
+4 9.01222229003906e-05
+8 0.000406503677368164
+16 0.00258469581604004
+32 0.0171687602996826
+64 0.126588344573975
+128 1.02698183059692
+};
+\addlegendentry{Winograd MM}
+\addplot [line width=2pt, color3]
+table {%
+2 1.45435333251953e-05
+4 1.1444091796875e-05
+8 7.39097595214844e-06
+16 1.28746032714844e-05
+32 2.83718109130859e-05
+64 0.000111103057861328
+128 0.000159025192260742
+};
+\addlegendentry{np MM}
+\end{axis}
+
+\end{tikzpicture}
diff --git a/buch/papers/multiplikation/einlteung.tex b/buch/papers/multiplikation/einlteung.tex
new file mode 100755
index 0000000..bc4bfcf
--- /dev/null
+++ b/buch/papers/multiplikation/einlteung.tex
@@ -0,0 +1,52 @@
+%
+% einleitung.tex -- Beispiel-File für die Einleitung
+%
+% (c) 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
new 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
new 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
new 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 (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)] {};
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diff --git a/buch/papers/multiplikation/loesungsmethoden.tex b/buch/papers/multiplikation/loesungsmethoden.tex
new file mode 100755
index 0000000..83be814
--- /dev/null
+++ b/buch/papers/multiplikation/loesungsmethoden.tex
@@ -0,0 +1,309 @@
+%
+% teil2.tex -- Beispiel-File für teil2
+%
+% (c) 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
new 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
new file mode 100755
index 0000000..b647fc0
--- /dev/null
+++ b/buch/papers/multiplikation/papers/Strassen_original_1969.pdf
diff --git a/buch/papers/multiplikation/papers/assay_fast_MM.pdf b/buch/papers/multiplikation/papers/assay_fast_MM.pdf
new file mode 100755
index 0000000..3cd6b63
--- /dev/null
+++ b/buch/papers/multiplikation/papers/assay_fast_MM.pdf
diff --git a/buch/papers/multiplikation/papers/strassen_video.txt b/buch/papers/multiplikation/papers/strassen_video.txt
new file mode 100755
index 0000000..f84122c
--- /dev/null
+++ b/buch/papers/multiplikation/papers/strassen_video.txt
@@ -0,0 +1 @@
+https://www.youtube.com/watch?v=0oJyNmEbS4w
diff --git a/buch/papers/multiplikation/papers/winograd_original.pdf b/buch/papers/multiplikation/papers/winograd_original.pdf
new file mode 100755
index 0000000..a7aba36
--- /dev/null
+++ b/buch/papers/multiplikation/papers/winograd_original.pdf
diff --git a/buch/papers/multiplikation/presentation/common.tex b/buch/papers/multiplikation/presentation/common.tex
new 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}}
+\headcommand {\beamer@framepages {9}{10}}
+\headcommand {\slideentry {1}{0}{5}{11/12}{}{0}}
+\headcommand {\beamer@framepages {11}{12}}
+\headcommand {\slideentry {1}{0}{6}{13/13}{}{0}}
+\headcommand {\beamer@framepages {13}{13}}
+\headcommand {\slideentry {1}{0}{7}{14/14}{}{0}}
+\headcommand {\beamer@framepages {14}{14}}
+\headcommand {\beamer@sectionpages {2}{14}}
+\headcommand {\beamer@subsectionpages {2}{14}}
+\headcommand {\sectionentry {2}{Strassen's Algorithm}{15}{Strassen's Algorithm}{0}}
+\headcommand {\slideentry {2}{0}{1}{15/15}{}{0}}
+\headcommand {\beamer@framepages {15}{15}}
+\headcommand {\slideentry {2}{0}{2}{16/18}{}{0}}
+\headcommand {\beamer@framepages {16}{18}}
+\headcommand {\slideentry {2}{0}{3}{19/19}{}{0}}
+\headcommand {\beamer@framepages {19}{19}}
+\headcommand {\slideentry {2}{0}{4}{20/20}{}{0}}
+\headcommand {\beamer@framepages {20}{20}}
+\headcommand {\slideentry {2}{0}{5}{21/23}{}{0}}
+\headcommand {\beamer@framepages {21}{23}}
+\headcommand {\slideentry {2}{0}{6}{24/24}{}{0}}
+\headcommand {\beamer@framepages {24}{24}}
+\headcommand {\slideentry {2}{0}{7}{25/25}{}{0}}
+\headcommand {\beamer@framepages {25}{25}}
+\headcommand {\slideentry {2}{0}{8}{26/26}{}{0}}
+\headcommand {\beamer@framepages {26}{26}}
+\headcommand {\slideentry {2}{0}{9}{27/29}{}{0}}
+\headcommand {\beamer@framepages {27}{29}}
+\headcommand {\slideentry {2}{0}{10}{30/32}{}{0}}
+\headcommand {\beamer@framepages {30}{32}}
+\headcommand {\beamer@sectionpages {15}{32}}
+\headcommand {\beamer@subsectionpages {15}{32}}
+\headcommand {\sectionentry {3}{Measurements}{33}{Measurements}{0}}
+\headcommand {\slideentry {3}{0}{1}{33/40}{}{0}}
+\headcommand {\beamer@framepages {33}{40}}
+\headcommand {\slideentry {3}{0}{2}{41/49}{}{0}}
+\headcommand {\beamer@framepages {41}{49}}
+\headcommand {\beamer@sectionpages {33}{49}}
+\headcommand {\beamer@subsectionpages {33}{49}}
+\headcommand {\sectionentry {4}{How To Matrix Multiply}{50}{How To Matrix Multiply}{0}}
+\headcommand {\slideentry {4}{0}{1}{50/50}{}{0}}
+\headcommand {\beamer@framepages {50}{50}}
+\headcommand {\beamer@partpages {1}{50}}
+\headcommand {\beamer@subsectionpages {50}{50}}
+\headcommand {\beamer@sectionpages {50}{50}}
+\headcommand {\beamer@documentpages {50}}
+\headcommand {\gdef \inserttotalframenumber {21}}
diff --git a/buch/papers/multiplikation/presentation/presentation.pdf b/buch/papers/multiplikation/presentation/presentation.pdf
new file mode 100644
index 0000000..842e68c
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/presentation.pdf
diff --git a/buch/papers/multiplikation/presentation/presentation.snm b/buch/papers/multiplikation/presentation/presentation.snm
new file mode 100644
index 0000000..e69de29
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/presentation.snm
diff --git a/buch/papers/multiplikation/presentation/presentation.tex b/buch/papers/multiplikation/presentation/presentation.tex
new file mode 100644
index 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
new 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 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}
diff --git a/buch/papers/multiplikation/presentation/tikz/algo.pdf b/buch/papers/multiplikation/presentation/tikz/algo.pdf
new 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. 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/tikz_formulas/algo.fdb_latexmk b/buch/papers/multiplikation/tikz_formulas/algo.fdb_latexmk
new file mode 100644
index 0000000..5f14129
--- /dev/null
+++ b/buch/papers/multiplikation/tikz_formulas/algo.fdb_latexmk
@@ -0,0 +1,254 @@
+# Fdb version 3
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diff --git a/buch/papers/multiplikation/tikz_formulas/algo.pdf b/buch/papers/multiplikation/tikz_formulas/algo.pdf
new file mode 100644
index 0000000..f711224
--- /dev/null
+++ b/buch/papers/multiplikation/tikz_formulas/algo.pdf
diff --git a/buch/papers/multiplikation/tikz_formulas/algo.tex b/buch/papers/multiplikation/tikz_formulas/algo.tex
new file mode 100755
index 0000000..1e437c2
--- /dev/null
+++ b/buch/papers/multiplikation/tikz_formulas/algo.tex
@@ -0,0 +1,131 @@
+\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{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 @@
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diff --git a/buch/papers/multiplikation/tikz_formulas/algo_graph.pdf b/buch/papers/multiplikation/tikz_formulas/algo_graph.pdf
new file mode 100755
index 0000000..7f5a984
--- /dev/null
+++ b/buch/papers/multiplikation/tikz_formulas/algo_graph.pdf
diff --git a/buch/papers/multiplikation/tikz_formulas/algo_graph.tex b/buch/papers/multiplikation/tikz_formulas/algo_graph.tex
new file mode 100755
index 0000000..ad4228b
--- /dev/null
+++ b/buch/papers/multiplikation/tikz_formulas/algo_graph.tex
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+\usepackage{amsfonts}
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+\usepackage{lipsum}
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+\usepackage{pdfpages}
+\usepackage{cite}
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