Merge branch 'AndreasFMueller:master' into master

41 files changed, 1171 insertions, 346 deletions

diff --git a/buch/papers/mceliece/aufbau.tex b/buch/papers/mceliece/aufbau.tex
index 521488d..200cb7b 100644
--- a/buch/papers/mceliece/aufbau.tex
+++ b/buch/papers/mceliece/aufbau.tex
@@ -10,7 +10,8 @@ Das McEliece-Kryptosystem besteht aus folgenden Elementen:
 \subsection{Datenvektor $d_k$
 \label{mceliece:subsection:d_k}}
 In diesem Vektor der Länge $k$ sind die zu verschlüsselnden Daten enthalten.
-Beispielsweise
+
+Beispiel:
 \[d_4=
 \begin{pmatrix}
     1\\
@@ -25,7 +26,7 @@ Beispielsweise
 $S_k$ ist eine Binäre Zufallsmatrix der Grösse $k \times k$.
 Auch muss diese Matrix in $\mathbb{F}_2$ invertierbar sein.
 Für kleine Matrizen kann durchaus jedes Matrizenelement zufällig generiert werden,
-wobei danach mithilfe des Gauss-Algorythmusses deren Inverse bestimmt werden kann.
+wobei danach mithilfe des Gauss-Algorithmus deren Inverse bestimmt werden kann.
 Da eine solche Matrix möglicherweise singulär ist, muss in diesem Fall eine neue Zufallsmatrix erzeugt werden.
 Für grössere Matrizen existieren bessere Methoden, auf welche hier nicht weiter eingegangen wird \cite{mceliece:GenerationRandMatrix}.
 
@@ -53,9 +54,9 @@ Beispiel:
 \label{mceliece:subsection:g_nk}}
 Das wichtigste Element des McEliece-Systems ist ein fehlerkorrigierender Code,
 der in der Lage ist, $t$ Fehler zu korrigieren.
-Im Zusammenhang mit McEliece werden dabei meist Goppa-Codes verwendet,
-es können prinzipiell auch andere Codes wie beispielsweise Reed-Solomin verwendet werden,
-jedoch besitzen einige Codes Schwachstellen \cite{mceliece:lorenz}.
+Im Zusammenhang mit McEliece werden dabei meist binäre Goppa-Codes \cite{mceliece:goppa} verwendet,
+es können prinzipiell auch andere Codes wie beispielsweise Reed-Solomon verwendet werden,
+jedoch besitzen einige (unter anderem auch Reed-Solomon) Codes Schwachstellen \cite{mceliece:lorenz}.
 Das Codieren mit diesem linearen Code kann mithilfe dessen Generatormatrix $G_{n,k}$ erfolgen.
 Da es sich um einen fehlerkorrigierenden Code handelt,
 wird das Codewort länger als das Datenwort,
diff --git a/buch/papers/mceliece/fazit.tex b/buch/papers/mceliece/fazit.tex
index d618993..186708b 100644
--- a/buch/papers/mceliece/fazit.tex
+++ b/buch/papers/mceliece/fazit.tex
@@ -35,12 +35,12 @@ Grosse unterschiede zwischen den beiden Kryptosystemen gibt es jedoch bei der Si
 Der Kern der RSA-Verschlüsselung beruht auf dem Problem, eine grosse Zahl in ihre beiden Primfaktoren zu zerlegen.
 Bei genügend grossen Zahlen ist diese Zerlegung auch mit den heute besten verfügbaren Computern kaum innerhalb vernünftiger Zeit zu lösen.
 Weiter ist aber bekannt,
-dass mithilfe des sogenannten Shor-Algorithmuses \cite{mceliece:shor} und einem Quantencomputer auch diese Zerlegung zügig realisiert werden könnte,
+dass mithilfe des sogenannten Shor-Algorithmus \cite{mceliece:shor} und einem Quantencomputer auch diese Zerlegung zügig realisiert werden könnte,
 was zur Folge hätte, dass die Verschlüsselung von RSA unwirksam würde.
 Zurzeit sind die Quantencomputer jedoch noch bei weitem nicht in der Lage, grosse Zahlen mithilfe dieses Algorithmuses zu zerlegen.
-Das McEliece-System hingegen beruht auf dem Problem des ``Syndrome decoding'' (Korrektur von Bitfehlern eines Codewortes, das mit dem entsprechenden Linearcode codiert wurde).
+Das McEliece-System hingegen beruht auf dem Problem des ``Syndrome decoding'' (Korrektur von Bitfehlern eines Codewortes, das mit einem entsprechenden Linearcode codiert wurde).
 Für das ``Syndrome decoding'' sind bis heute keine Methoden bekannt,
-welche nennenswerte Vorteile gegenüber dem durchprobieren (brute-force) bringen,
+welche nennenswerte Vorteile gegenüber dem Durchprobieren (brute-force) bringen,
 auch nicht mithilfe eines Quantencomputers.
 \begin{center}
 \begin{tabular}{c|c|c}
diff --git a/buch/papers/mceliece/funktionsweise.tex b/buch/papers/mceliece/funktionsweise.tex
index 93bb1c7..7c69b13 100644
--- a/buch/papers/mceliece/funktionsweise.tex
+++ b/buch/papers/mceliece/funktionsweise.tex
@@ -18,7 +18,7 @@ Dazu erstellt sie die einzelnen Matrizen $S_k$, $G_{n,k}$ und $P_n$.
 Diese drei einzelnen Matrizen bilden den privaten Schlüssel von Alice
 und sollen geheim bleiben.
 Der öffentliche Schlüssel $K_{n,k}$ hingegen berechnet sich
-aus der Multiplikation der privaten Matrizen\ref{mceliece:subsection:k_nk}
+aus der Multiplikation der privaten Matrizen (Abschnitt \ref{mceliece:subsection:k_nk})
 und wird anschliessend Bob zugestellt.
 
 \subsection{Verschlüsselung
@@ -61,7 +61,7 @@ können nun die Bitfehler, verursacht durch den Fehlervektor $e'_n$,
 entfernt werden.
 Da es sich bei diesem Schritt nicht um eine einfache Matrixmultiplikation handelt,
 wird die Operation durch eine Funktion dargestellt.
-Wie dieser Decoder genau aufgebaut ist ist,
+Wie dieser Decoder genau aufgebaut ist,
 hängt vom verwendeten Linearcode ab.
 \begin{align*}
     c_{k}'\,&=\text{Linear-Code-Decoder($c''_n$)}\\
diff --git a/buch/papers/mceliece/references.bib b/buch/papers/mceliece/references.bib
index 52aa166..0388ff4 100644
--- a/buch/papers/mceliece/references.bib
+++ b/buch/papers/mceliece/references.bib
@@ -37,4 +37,12 @@
 	year = {2021},
 	month = {8},
 	day = {9}
+}
+
+@online{mceliece:goppa,
+	title = {Binary Goppa code},
+	url = {https://en.m.wikipedia.org/wiki/Binary_Goppa_code},
+	year = {2021},
+	month = {8},
+	day = {10}
 }
\ No newline at end of file
diff --git a/buch/papers/multiplikation/code/MM b/buch/papers/multiplikation/code/MM
deleted file mode 100755
index d52dda4..0000000
--- a/buch/papers/multiplikation/code/MM
+++ /dev/null
diff --git a/buch/papers/multiplikation/code/MM.c b/buch/papers/multiplikation/code/MM.c
index a897d4f..2588262 100755
--- a/buch/papers/multiplikation/code/MM.c
+++ b/buch/papers/multiplikation/code/MM.c
@@ -28,11 +28,12 @@ int main() {
 	// 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);

+ 	run_algo(winograd, "winograd", 0);

+ 	run_algo_cblas(0);

 

 	return 0;

 }

@@ -414,12 +415,12 @@ void run_algo(void (*algo)(), char alog_name[], int print)
 

 	for(int i=0; i<n_arrays; ++i)

 	{

-		for(int j = 0; j<1; ++j)

+		for(int j = 0; j<10; ++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;

+	    		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]);

 

@@ -428,7 +429,7 @@ void run_algo(void (*algo)(), char alog_name[], int print)
 				printMatrix((int*)C, n[i]);

 			}

 			free(C);

-  }

+ 		 }

 	}

 	fclose(fptr);

 

@@ -442,7 +443,7 @@ void run_algo_cblas(int print)
 	fptr = fopen("meas/blas.txt", "w");

 	for(int i=0; i<n_arrays; ++i)

 	{

-		for(int j = 0; j<1; ++j)

+		for(int j = 0; j<10; ++j)

 		{

 			double *dC = (double*) malloc(n[i] * n[i] * sizeof(double));

 			double dtime = omp_get_wtime();

diff --git a/buch/papers/multiplikation/code/MM.py b/buch/papers/multiplikation/code/MM.py
index 7220ae1..8057850 100644
--- a/buch/papers/multiplikation/code/MM.py
+++ b/buch/papers/multiplikation/code/MM.py
@@ -5,6 +5,7 @@ Created on Fri Mar 19 07:31:29 2021
 
 @author: nunigan
 """
+import scipy.stats
 import numpy as np
 import time
 import matplotlib.pyplot as plt
@@ -133,9 +134,6 @@ def winograd2(A, B):
         
 def test_perfomance(n):
     
-    import mkl
-    mkl.set_num_threads(1)
-    
     t_mm = []
     t_mm_dc = []
     t_mm_strassen = []
@@ -148,21 +146,21 @@ def test_perfomance(n):
         # 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()
+        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()
+        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()
+        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()
+        C4 = winograd2(A, B)
+        t_wino.append(time.time() - start)
 
         start = time.time()
         C = A@B
@@ -173,10 +171,10 @@ def test_perfomance(n):
     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_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.xscale('log', base=2)
     plt.legend()
@@ -186,9 +184,9 @@ def test_perfomance(n):
     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)    
+    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 t_np
 
 
@@ -249,6 +247,8 @@ def plot_c_res(ave, num):
     # 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
 
@@ -261,11 +261,11 @@ def plot_c_res(ave, num):
     plt.rc('axes', labelsize=23)
     plt.rc('xtick', labelsize=23)
     plt.rc('ytick', labelsize=23)
-    plt.loglog(MM_n, MM_t, label='3 For Loops', lw=5)
-    plt.loglog(winograd_n, winograd_t, label='Winograd MM', lw=5)
-    plt.loglog(blas_n, blas_t, label='Blas', lw=5)
-    plt.loglog(strassen_n, strassen_t, label='Strassen', lw=5)
-    plt.loglog(MM_dc_n, MM_dc_t, label='Divide and Conquer', lw=5)
+    plt.loglog(MM_n, MM_t, '.', label='3 For Loops', lw=5)
+    plt.loglog(winograd_n, winograd_t, '.', label='Winograd MM', lw=5)
+    plt.loglog(blas_n, blas_t, '.', label='Blas', lw=5)
+    plt.loglog(strassen_n, strassen_t, '.', label='Strassen', lw=5)
+    plt.loglog(MM_dc_n, MM_dc_t, '.', label='Divide and Conquer', lw=5)
     plt.xlabel("n")
     # plt.yscale('log', base=10)
     # plt.xscale('log', base=2)
@@ -281,16 +281,33 @@ def plot_c_res(ave, num):
   
     plt.legend()
     # return [MM_n,winograd_n,blas_n,strassen_n,MM_dc_n]
+    
+    
     return [MM_t,winograd_t,blas_t,strassen_t,MM_dc_t]
 
 
+def mean_confidence_interval(data, confidence=0.95):
+    a = 1.0 * np.array(data)
+    n = len(a)
+    m, se = np.mean(a), scipy.stats.sem(a)
+    h = se * scipy.stats.t.ppf((1 + confidence) / 2., n-1)
+    return m, h
+
 # test%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 if __name__ == '__main__':
-    # A = plot_c_res(1, 4096)
-
- 
-    arr = plot(1024)    
+    # A = plot_c_res(10, 4096)
+    # name = ['MM', 'Wino', 'blas', 'strassen', 'dc']
+    # for i in range(5):
+    #     ci_inner = []
+    #     print(name[i])
+    #     for j in range(11):
+    #         m,h=mean_confidence_interval(A[i][j*10:(j+1)*10])
+    #         print("({},{})".format(2**(j+1),m))
+    #     np.savetxt('meas/ci/' + name[i]+'.txt',ci_inner)
+
+    arr = plot(4096)
     # n = np.logspace(1,12,12,base=2,dtype=(np.int))
+    # n=[2048,4096]
     # n = np.arange(1,50,2)  
     # A = np.random.randint(-10, 6, (5,3))
     # B = np.random.randint(-10, 6, (3,5))
diff --git a/buch/papers/multiplikation/code/c_matrix.h b/buch/papers/multiplikation/code/c_matrix.h
index 14389fc..63d5390 100644
--- a/buch/papers/multiplikation/code/c_matrix.h
+++ b/buch/papers/multiplikation/code/c_matrix.h
@@ -1,101 +1,177 @@
-/* Seminar Matrizen, autogenerated File, Michael Schmid, 02/08/2021, 22:48:43 */ 
+/* Seminar Matrizen, autogenerated File, Michael Schmid, 10/08/2021, 05:46:32 */ 
  
 #include <stdint.h> 
 const int A0[][2] = 
   {
-    {75,47},
-    {-41,-24}
+    {60,-84},
+    {-66,-1}
   };
 const int B0[][2] = 
   {
-    {-53,-95},
-    {-93,30}
+    {-45,87},
+    {-38,-73}
   };
 const double dB0[][2] = 
   {
-    {-53,-95},
-    {-93,30}
+    {-45,87},
+    {-38,-73}
   };
 const double dA0[][2] = 
   {
-    {75,47},
-    {-41,-24}
+    {60,-84},
+    {-66,-1}
   };
 const int A1[][4] = 
   {
-    {47,11,-66,8},
-    {36,98,39,82},
-    {-32,12,40,-79},
-    {61,-20,-85,-98}
+    {-72,-19,-91,62},
+    {-36,-74,-44,-47},
+    {-39,-31,50,-93},
+    {-81,2,-17,-86}
   };
 const int B1[][4] = 
   {
-    {37,75,-53,9},
-    {37,-33,-67,38},
-    {70,39,-93,43},
-    {43,41,23,-4}
+    {-66,39,-23,52},
+    {-88,-13,13,-13},
+    {-45,-70,28,-20},
+    {96,5,88,96}
   };
 const double dB1[][4] = 
   {
-    {37,75,-53,9},
-    {37,-33,-67,38},
-    {70,39,-93,43},
-    {43,41,23,-4}
+    {-66,39,-23,52},
+    {-88,-13,13,-13},
+    {-45,-70,28,-20},
+    {96,5,88,96}
   };
 const double dA1[][4] = 
   {
-    {47,11,-66,8},
-    {36,98,39,82},
-    {-32,12,40,-79},
-    {61,-20,-85,-98}
+    {-72,-19,-91,62},
+    {-36,-74,-44,-47},
+    {-39,-31,50,-93},
+    {-81,2,-17,-86}
   };
 const int A2[][8] = 
   {
-    {-54,-87,87,69,52,-21,-86,55},
-    {19,-75,-61,-50,-55,-23,66,-92},
-    {-73,-67,-36,19,84,-11,24,46},
-    {-98,62,-76,57,-100,6,-23,-51},
-    {62,46,1,-64,42,-9,85,-12},
-    {35,-59,-17,-47,78,86,-50,74},
-    {-15,45,33,-59,-9,-81,49,96},
-    {-57,22,-43,7,-30,-45,-5,13}
+    {-36,-2,-58,-32,34,-89,49,-55},
+    {-68,-73,52,-3,-51,-37,-31,70},
+    {73,-90,-21,-79,-15,96,-99,12},
+    {68,-25,38,-73,-60,35,-99,72},
+    {-43,-87,48,-84,-100,37,80,53},
+    {-27,88,-5,-82,-57,-27,20,10},
+    {-91,-47,54,-90,-99,-76,50,-18},
+    {69,-36,76,5,-67,-38,-95,91}
   };
 const int B2[][8] = 
   {
-    {-71,-82,-80,-78,83,-97,48,-24},
-    {15,75,15,-60,-63,-53,1,-50},
-    {-84,63,67,-2,78,93,-13,95},
-    {61,-26,-88,56,56,27,26,1},
-    {2,54,21,36,9,-41,53,53},
-    {85,-11,42,-51,-6,3,27,97},
-    {10,-2,90,-76,-75,0,8,-37},
-    {10,-64,47,-69,66,-50,89,-66}
+    {-84,22,-13,-66,-42,51,66,0},
+    {37,-65,66,-85,-10,-23,77,5},
+    {1,41,-79,0,63,-37,-10,29},
+    {72,66,-99,92,-28,65,25,-40},
+    {69,-49,65,-18,64,-97,-47,30},
+    {36,86,66,-12,-17,89,1,-37},
+    {-100,11,27,23,-75,-23,96,-9},
+    {68,90,-87,-99,-70,-28,98,-76}
   };
 const double dB2[][8] = 
   {
-    {-71,-82,-80,-78,83,-97,48,-24},
-    {15,75,15,-60,-63,-53,1,-50},
-    {-84,63,67,-2,78,93,-13,95},
-    {61,-26,-88,56,56,27,26,1},
-    {2,54,21,36,9,-41,53,53},
-    {85,-11,42,-51,-6,3,27,97},
-    {10,-2,90,-76,-75,0,8,-37},
-    {10,-64,47,-69,66,-50,89,-66}
+    {-84,22,-13,-66,-42,51,66,0},
+    {37,-65,66,-85,-10,-23,77,5},
+    {1,41,-79,0,63,-37,-10,29},
+    {72,66,-99,92,-28,65,25,-40},
+    {69,-49,65,-18,64,-97,-47,30},
+    {36,86,66,-12,-17,89,1,-37},
+    {-100,11,27,23,-75,-23,96,-9},
+    {68,90,-87,-99,-70,-28,98,-76}
   };
 const double dA2[][8] = 
   {
-    {-54,-87,87,69,52,-21,-86,55},
-    {19,-75,-61,-50,-55,-23,66,-92},
-    {-73,-67,-36,19,84,-11,24,46},
-    {-98,62,-76,57,-100,6,-23,-51},
-    {62,46,1,-64,42,-9,85,-12},
-    {35,-59,-17,-47,78,86,-50,74},
-    {-15,45,33,-59,-9,-81,49,96},
-    {-57,22,-43,7,-30,-45,-5,13}
-  };
-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;
+    {-36,-2,-58,-32,34,-89,49,-55},
+    {-68,-73,52,-3,-51,-37,-31,70},
+    {73,-90,-21,-79,-15,96,-99,12},
+    {68,-25,38,-73,-60,35,-99,72},
+    {-43,-87,48,-84,-100,37,80,53},
+    {-27,88,-5,-82,-57,-27,20,10},
+    {-91,-47,54,-90,-99,-76,50,-18},
+    {69,-36,76,5,-67,-38,-95,91}
+  };
+const int A3[][16] = 
+  {
+    {-24,65,21,19,94,70,-90,-81,53,-41,-23,-1,58,-80,-54,59},
+    {-42,76,-19,98,29,-56,92,14,45,11,82,83,48,-13,81,66},
+    {43,-57,-67,95,5,72,11,0,-47,55,-24,36,84,54,-31,-54},
+    {-39,-40,19,97,-82,-56,27,95,81,-21,-50,-74,-35,-87,-28,-26},
+    {-74,-98,79,92,-24,-48,99,94,55,-83,70,98,-24,18,-67,14},
+    {20,76,11,-23,-56,21,0,42,64,86,-74,44,93,-76,-30,97},
+    {13,20,-73,-11,-30,80,53,-8,60,21,17,-42,82,-72,-6,-80},
+    {36,-93,-64,-21,20,-85,15,24,99,81,-52,64,71,-56,52,63},
+    {32,9,-2,-85,17,62,-98,-35,75,-58,-44,-20,-47,89,-95,52},
+    {93,-43,86,68,-6,-25,90,57,60,-10,65,-97,43,46,-60,-41},
+    {43,-33,0,50,-100,26,-60,95,39,-70,-61,-81,9,-23,-99,-4},
+    {20,61,15,43,-96,93,-55,38,-29,-1,-10,26,-87,18,64,6},
+    {-98,-84,51,16,-14,86,52,59,44,-39,-2,10,82,-66,54,19},
+    {89,-49,-37,-6,-53,40,-11,46,-51,-56,86,34,11,13,-20,-49},
+    {-90,14,28,-45,-25,-56,-51,-61,28,-8,51,91,95,-10,-85,58},
+    {8,-44,88,-71,-27,11,89,37,86,-78,-44,-56,-87,0,-42,-61}
+  };
+const int B3[][16] = 
+  {
+    {62,-30,62,92,29,-93,-95,44,-33,-88,-29,9,-88,-42,-90,-70},
+    {60,37,-44,-93,-87,6,-53,2,-29,53,-49,59,6,83,-15,50},
+    {-19,85,-49,-14,84,-4,12,88,-83,-81,-24,-16,-12,-42,-63,-71},
+    {-42,-78,-58,-61,-29,67,-28,-46,64,7,6,-13,88,-42,95,-24},
+    {-90,-56,8,-30,-89,70,37,-29,24,-8,-10,-2,-25,-63,-95,-91},
+    {10,-81,42,-28,-13,-68,-72,-20,-22,5,-79,-50,-88,62,57,69},
+    {-67,24,-71,-43,11,48,33,-93,-82,-65,-4,5,-15,25,-54,-45},
+    {-49,19,-29,90,-97,-87,78,-39,-75,-85,-79,-35,54,3,-73,7},
+    {-7,39,70,-42,32,-100,56,4,-24,-57,38,-49,-50,-44,79,-42},
+    {37,-65,-55,22,-97,-42,-76,95,97,-27,38,11,0,-81,-23,35},
+    {26,-70,10,-29,47,-70,-52,29,-13,-18,5,34,18,32,87,91},
+    {-84,41,-19,96,-51,-19,81,75,81,92,2,-40,-42,-69,-10,-61},
+    {-30,98,71,-51,91,-59,58,86,86,-22,-84,7,66,-55,-52,23},
+    {-71,-44,-9,90,26,18,26,-10,-85,64,-47,3,72,81,74,-8},
+    {52,-59,-91,22,8,-63,84,9,-11,-54,-78,-71,-98,42,96,57},
+    {18,-39,34,-50,-62,-96,-2,-78,52,94,-33,2,-19,-9,-86,-75}
+  };
+const double dB3[][16] = 
+  {
+    {62,-30,62,92,29,-93,-95,44,-33,-88,-29,9,-88,-42,-90,-70},
+    {60,37,-44,-93,-87,6,-53,2,-29,53,-49,59,6,83,-15,50},
+    {-19,85,-49,-14,84,-4,12,88,-83,-81,-24,-16,-12,-42,-63,-71},
+    {-42,-78,-58,-61,-29,67,-28,-46,64,7,6,-13,88,-42,95,-24},
+    {-90,-56,8,-30,-89,70,37,-29,24,-8,-10,-2,-25,-63,-95,-91},
+    {10,-81,42,-28,-13,-68,-72,-20,-22,5,-79,-50,-88,62,57,69},
+    {-67,24,-71,-43,11,48,33,-93,-82,-65,-4,5,-15,25,-54,-45},
+    {-49,19,-29,90,-97,-87,78,-39,-75,-85,-79,-35,54,3,-73,7},
+    {-7,39,70,-42,32,-100,56,4,-24,-57,38,-49,-50,-44,79,-42},
+    {37,-65,-55,22,-97,-42,-76,95,97,-27,38,11,0,-81,-23,35},
+    {26,-70,10,-29,47,-70,-52,29,-13,-18,5,34,18,32,87,91},
+    {-84,41,-19,96,-51,-19,81,75,81,92,2,-40,-42,-69,-10,-61},
+    {-30,98,71,-51,91,-59,58,86,86,-22,-84,7,66,-55,-52,23},
+    {-71,-44,-9,90,26,18,26,-10,-85,64,-47,3,72,81,74,-8},
+    {52,-59,-91,22,8,-63,84,9,-11,-54,-78,-71,-98,42,96,57},
+    {18,-39,34,-50,-62,-96,-2,-78,52,94,-33,2,-19,-9,-86,-75}
+  };
+const double dA3[][16] = 
+  {
+    {-24,65,21,19,94,70,-90,-81,53,-41,-23,-1,58,-80,-54,59},
+    {-42,76,-19,98,29,-56,92,14,45,11,82,83,48,-13,81,66},
+    {43,-57,-67,95,5,72,11,0,-47,55,-24,36,84,54,-31,-54},
+    {-39,-40,19,97,-82,-56,27,95,81,-21,-50,-74,-35,-87,-28,-26},
+    {-74,-98,79,92,-24,-48,99,94,55,-83,70,98,-24,18,-67,14},
+    {20,76,11,-23,-56,21,0,42,64,86,-74,44,93,-76,-30,97},
+    {13,20,-73,-11,-30,80,53,-8,60,21,17,-42,82,-72,-6,-80},
+    {36,-93,-64,-21,20,-85,15,24,99,81,-52,64,71,-56,52,63},
+    {32,9,-2,-85,17,62,-98,-35,75,-58,-44,-20,-47,89,-95,52},
+    {93,-43,86,68,-6,-25,90,57,60,-10,65,-97,43,46,-60,-41},
+    {43,-33,0,50,-100,26,-60,95,39,-70,-61,-81,9,-23,-99,-4},
+    {20,61,15,43,-96,93,-55,38,-29,-1,-10,26,-87,18,64,6},
+    {-98,-84,51,16,-14,86,52,59,44,-39,-2,10,82,-66,54,19},
+    {89,-49,-37,-6,-53,40,-11,46,-51,-56,86,34,11,13,-20,-49},
+    {-90,14,28,-45,-25,-56,-51,-61,28,-8,51,91,95,-10,-85,58},
+    {8,-44,88,-71,-27,11,89,37,86,-78,-44,-56,-87,0,-42,-61}
+  };
+const int *Ap[4] = {(int*) A0,(int*) A1,(int*) A2,(int*) A3}; 
+const int *Bp[4] = {(int*) B0,(int*) B1,(int*) B2,(int*) B3}; 
+const double *dAp[4] = {(double*) dA0,(double*) dA1,(double*) dA2,(double*) dA3}; 
+const double *dBp[4] = {(double*) dB0,(double*) dB1,(double*) dB2,(double*) dB3}; 
+int n[4] = {2,4,8,16}; 
+int n_arrays = 4;
diff --git a/buch/papers/multiplikation/code/c_meas_4096.pdf b/buch/papers/multiplikation/code/c_meas_4096.pdf
index 5236afb..f637ae4 100644
--- a/buch/papers/multiplikation/code/c_meas_4096.pdf
+++ b/buch/papers/multiplikation/code/c_meas_4096.pdf
diff --git a/buch/papers/multiplikation/code/ci.txt b/buch/papers/multiplikation/code/ci.txt
new file mode 100644
index 0000000..e69de29
--- /dev/null
+++ b/buch/papers/multiplikation/code/ci.txt
diff --git a/buch/papers/multiplikation/code/helper_class.py b/buch/papers/multiplikation/code/helper_class.py
index 485fa76..3b74f67 100755
--- a/buch/papers/multiplikation/code/helper_class.py
+++ b/buch/papers/multiplikation/code/helper_class.py
@@ -101,5 +101,6 @@ 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)
+    n = np.logspace(1,11,11,base=2,dtype=(np.int))
+    # n=[8192]
+    # C = helper.write_c_matrix(n)
diff --git a/buch/papers/multiplikation/code/meas/MM.txt b/buch/papers/multiplikation/code/meas/MM.txt
index e296dd7..7bffb6e 100644
--- a/buch/papers/multiplikation/code/meas/MM.txt
+++ b/buch/papers/multiplikation/code/meas/MM.txt
@@ -1,12 +1,110 @@
-0.000001,2
+0.000000,2
+0.000000,2
+0.000000,2
+0.000000,2
+0.000000,2
+0.000000,2
+0.000000,2
+0.000000,2
+0.000000,2
+0.000000,2
 0.000001,4
+0.000000,4
+0.000000,4
+0.000000,4
+0.000000,4
+0.000000,4
+0.000000,4
+0.000000,4
+0.000001,4
+0.000001,4
+0.000002,8
+0.000002,8
+0.000002,8
+0.000002,8
+0.000002,8
+0.000002,8
+0.000002,8
+0.000002,8
+0.000001,8
 0.000001,8
-0.000010,16
-0.000081,32
-0.000654,64
-0.005556,128
-0.054253,256
-0.487317,512
-4.162845,1024
-125.909034,2048
-1111.312696,4096
+0.000011,16
+0.000011,16
+0.000011,16
+0.000011,16
+0.000011,16
+0.000021,16
+0.000011,16
+0.000011,16
+0.000011,16
+0.000011,16
+0.000090,32
+0.000093,32
+0.000083,32
+0.000082,32
+0.000090,32
+0.000080,32
+0.000080,32
+0.000080,32
+0.000089,32
+0.000126,32
+0.000771,64
+0.000651,64
+0.000651,64
+0.000651,64
+0.000731,64
+0.000673,64
+0.000745,64
+0.000672,64
+0.000671,64
+0.000707,64
+0.005642,128
+0.005579,128
+0.005768,128
+0.005745,128
+0.005518,128
+0.005877,128
+0.005513,128
+0.005850,128
+0.005769,128
+0.005581,128
+0.052188,256
+0.051988,256
+0.051888,256
+0.051518,256
+0.051709,256
+0.051543,256
+0.051707,256
+0.051845,256
+0.051495,256
+0.051834,256
+0.507020,512
+0.504111,512
+0.502049,512
+0.529743,512
+0.501028,512
+0.502097,512
+0.503490,512
+0.502079,512
+0.506688,512
+0.504163,512
+4.538722,1024
+4.291473,1024
+4.516302,1024
+4.374630,1024
+4.719557,1024
+4.438999,1024
+4.641680,1024
+4.407959,1024
+4.441451,1024
+4.677313,1024
+129.433279,2048
+129.277802,2048
+129.284817,2048
+129.086884,2048
+129.197444,2048
+129.350999,2048
+129.264250,2048
+129.295723,2048
+129.402601,2048
+129.300820,2048
diff --git a/buch/papers/multiplikation/code/meas/MM_dc.txt b/buch/papers/multiplikation/code/meas/MM_dc.txt
index f6be928..b78b925 100644
--- a/buch/papers/multiplikation/code/meas/MM_dc.txt
+++ b/buch/papers/multiplikation/code/meas/MM_dc.txt
@@ -1,12 +1,110 @@
 0.000003,2
+0.000000,2
+0.000000,2
+0.000000,2
+0.000000,2
+0.000000,2
+0.000000,2
+0.000000,2
+0.000000,2
+0.000000,2
 0.000002,4
-0.000010,8
-0.000068,16
-0.000594,32
-0.004264,64
-0.036289,128
-0.324645,256
-2.612010,512
-19.928951,1024
-159.333884,2048
-1147.106865,4096
+0.000001,4
+0.000001,4
+0.000001,4
+0.000001,4
+0.000001,4
+0.000001,4
+0.000001,4
+0.000001,4
+0.000001,4
+0.000008,8
+0.000008,8
+0.000008,8
+0.000008,8
+0.000007,8
+0.000007,8
+0.000007,8
+0.000007,8
+0.000018,8
+0.000008,8
+0.000075,16
+0.000063,16
+0.000088,16
+0.000062,16
+0.000086,16
+0.000092,16
+0.000081,16
+0.000080,16
+0.000070,16
+0.000085,16
+0.000581,32
+0.000659,32
+0.000584,32
+0.000714,32
+0.000666,32
+0.000574,32
+0.000616,32
+0.000534,32
+0.000506,32
+0.000506,32
+0.004567,64
+0.004502,64
+0.004332,64
+0.004578,64
+0.004543,64
+0.004426,64
+0.004497,64
+0.004329,64
+0.004288,64
+0.004277,64
+0.036456,128
+0.034901,128
+0.034545,128
+0.034283,128
+0.035150,128
+0.034663,128
+0.034901,128
+0.034022,128
+0.034368,128
+0.035154,128
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+0.297592,256
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+0.306394,256
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+2.331956,512
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+17.639024,1024
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+17.748270,1024
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+141.596407,2048
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+141.469723,2048
+141.515550,2048
diff --git a/buch/papers/multiplikation/code/meas/blas.txt b/buch/papers/multiplikation/code/meas/blas.txt
index 92a61b9..9414d8f 100644
--- a/buch/papers/multiplikation/code/meas/blas.txt
+++ b/buch/papers/multiplikation/code/meas/blas.txt
@@ -1,12 +1,110 @@
 0.000001,2
-0.000001,4
+0.000000,2
+0.000000,2
+0.000000,2
+0.000000,2
+0.000000,2
+0.000000,2
+0.000000,2
+0.000000,2
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+0.000000,4
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+0.000000,4
+0.000000,4
+0.000000,4
+0.000000,4
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+0.000000,4
+0.000000,4
 0.000001,8
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+0.000000,8
+0.000000,8
+0.000000,8
+0.000000,8
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+0.000000,8
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+0.000000,8
 0.000003,16
-0.000022,32
-0.000179,64
+0.000003,16
+0.000003,16
+0.000003,16
+0.000003,16
+0.000003,16
+0.000012,16
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+0.000021,32
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+0.000030,32
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+0.000226,64
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 0.001278,128
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-0.074739,512
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-6.845095,2048
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+7.676742,2048
+7.632979,2048
+7.619210,2048
diff --git a/buch/papers/multiplikation/code/meas/ci/MM.txt b/buch/papers/multiplikation/code/meas/ci/MM.txt
new file mode 100644
index 0000000..e69de29
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/ci/MM.txt
diff --git a/buch/papers/multiplikation/code/meas/ci/Wino.txt b/buch/papers/multiplikation/code/meas/ci/Wino.txt
new file mode 100644
index 0000000..e69de29
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/ci/Wino.txt
diff --git a/buch/papers/multiplikation/code/meas/ci/blas.txt b/buch/papers/multiplikation/code/meas/ci/blas.txt
new file mode 100644
index 0000000..e69de29
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/ci/blas.txt
diff --git a/buch/papers/multiplikation/code/meas/ci/dc.txt b/buch/papers/multiplikation/code/meas/ci/dc.txt
new file mode 100644
index 0000000..e69de29
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/ci/dc.txt
diff --git a/buch/papers/multiplikation/code/meas/ci/strassen.txt b/buch/papers/multiplikation/code/meas/ci/strassen.txt
new file mode 100644
index 0000000..e69de29
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/ci/strassen.txt
diff --git a/buch/papers/multiplikation/code/meas/old/8196/MM.txt b/buch/papers/multiplikation/code/meas/old/8196/MM.txt
new file mode 100644
index 0000000..0edf9f6
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/old/8196/MM.txt
@@ -0,0 +1 @@
+9376.173434,8192
diff --git a/buch/papers/multiplikation/code/meas/old/8196/MM_dc.txt b/buch/papers/multiplikation/code/meas/old/8196/MM_dc.txt
new file mode 100644
index 0000000..36f6ff0
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/old/8196/MM_dc.txt
@@ -0,0 +1 @@
+9606.402522,8192
diff --git a/buch/papers/multiplikation/code/meas/old/8196/blas.txt b/buch/papers/multiplikation/code/meas/old/8196/blas.txt
new file mode 100644
index 0000000..b5989fb
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/old/8196/blas.txt
@@ -0,0 +1 @@
+478.429957,8192
diff --git a/buch/papers/multiplikation/code/meas/old/8196/strassen.txt b/buch/papers/multiplikation/code/meas/old/8196/strassen.txt
new file mode 100644
index 0000000..ca06e97
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/old/8196/strassen.txt
@@ -0,0 +1 @@
+3014.235467,8192
diff --git a/buch/papers/multiplikation/code/meas/old/8196/winograd.txt b/buch/papers/multiplikation/code/meas/old/8196/winograd.txt
new file mode 100644
index 0000000..2a529c4
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/old/8196/winograd.txt
@@ -0,0 +1 @@
+10071.512655,8192
diff --git a/buch/papers/multiplikation/code/meas/old/MM.txt b/buch/papers/multiplikation/code/meas/old/MM.txt
new file mode 100644
index 0000000..e296dd7
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/old/MM.txt
@@ -0,0 +1,12 @@
+0.000001,2
+0.000001,4
+0.000001,8
+0.000010,16
+0.000081,32
+0.000654,64
+0.005556,128
+0.054253,256
+0.487317,512
+4.162845,1024
+125.909034,2048
+1111.312696,4096
diff --git a/buch/papers/multiplikation/code/meas/old/MM_dc.txt b/buch/papers/multiplikation/code/meas/old/MM_dc.txt
new file mode 100644
index 0000000..f6be928
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/old/MM_dc.txt
@@ -0,0 +1,12 @@
+0.000003,2
+0.000002,4
+0.000010,8
+0.000068,16
+0.000594,32
+0.004264,64
+0.036289,128
+0.324645,256
+2.612010,512
+19.928951,1024
+159.333884,2048
+1147.106865,4096
diff --git a/buch/papers/multiplikation/code/meas/old/blas.txt b/buch/papers/multiplikation/code/meas/old/blas.txt
new file mode 100644
index 0000000..92a61b9
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/old/blas.txt
@@ -0,0 +1,12 @@
+0.000001,2
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+0.000003,16
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diff --git a/buch/papers/multiplikation/code/meas/old/strassen.txt b/buch/papers/multiplikation/code/meas/old/strassen.txt
new file mode 100644
index 0000000..fdfbf2b
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/old/strassen.txt
@@ -0,0 +1,12 @@
+0.000001,2
+0.000003,4
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+0.000066,16
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diff --git a/buch/papers/multiplikation/code/meas/old/winograd.txt b/buch/papers/multiplikation/code/meas/old/winograd.txt
new file mode 100644
index 0000000..d185906
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/old/winograd.txt
@@ -0,0 +1,12 @@
+0.000001,2
+0.000001,4
+0.000002,8
+0.000011,16
+0.000100,32
+0.000654,64
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+4.539413,1024
+130.627663,2048
+1179.261048,4096
diff --git a/buch/papers/multiplikation/code/meas/strassen.txt b/buch/papers/multiplikation/code/meas/strassen.txt
index fdfbf2b..d6e040e 100644
--- a/buch/papers/multiplikation/code/meas/strassen.txt
+++ b/buch/papers/multiplikation/code/meas/strassen.txt
@@ -1,12 +1,110 @@
-0.000001,2
-0.000003,4
-0.000010,8
-0.000066,16
-0.000470,32
-0.003368,64
-0.024232,128
-0.172000,256
-1.209262,512
-8.457472,1024
-59.267256,2048
-414.648901,4096
+0.000000,2
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diff --git a/buch/papers/multiplikation/code/meas/winograd.txt b/buch/papers/multiplikation/code/meas/winograd.txt
index d185906..970a3f4 100644
--- a/buch/papers/multiplikation/code/meas/winograd.txt
+++ b/buch/papers/multiplikation/code/meas/winograd.txt
@@ -1,12 +1,110 @@
 0.000001,2
-0.000001,4
+0.000000,2
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 0.000002,8
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 0.000011,16
-0.000100,32
-0.000654,64
-0.005229,128
-0.057440,256
-0.517850,512
-4.539413,1024
-130.627663,2048
-1179.261048,4096
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diff --git a/buch/papers/multiplikation/code/meas_4096.pdf b/buch/papers/multiplikation/code/meas_4096.pdf
index e889d17..ecf2cff 100644
--- a/buch/papers/multiplikation/code/meas_4096.pdf
+++ b/buch/papers/multiplikation/code/meas_4096.pdf
diff --git a/buch/papers/multiplikation/code/meas_4096.txt b/buch/papers/multiplikation/code/meas_4096.txt
index e69de29..cae1bc6 100644
--- a/buch/papers/multiplikation/code/meas_4096.txt
+++ b/buch/papers/multiplikation/code/meas_4096.txt
@@ -0,0 +1,6 @@
+2.048000000000000000e+03 4.096000000000000000e+03
+6.154183513402938843e+03 4.681333474493026733e+04
+7.375929301261901855e+03 5.846600176072120667e+04
+3.860573610544204712e+03 2.290433094644546509e+04
+4.884613198995590210e+03 4.359707747149467468e+04
+2.157390117645263672e-01 1.491588830947875977e+00
diff --git a/buch/papers/multiplikation/images/algo_tab.pdf b/buch/papers/multiplikation/images/algo_tab.pdf
new file mode 100644
index 0000000..7f2bb4f
--- /dev/null
+++ b/buch/papers/multiplikation/images/algo_tab.pdf
diff --git a/buch/papers/multiplikation/images/algo_tab.tex b/buch/papers/multiplikation/images/algo_tab.tex
new file mode 100644
index 0000000..50ce392
--- /dev/null
+++ b/buch/papers/multiplikation/images/algo_tab.tex
@@ -0,0 +1,122 @@
+\documentclass{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{algorithm}
+\usepackage{algpseudocode}
+\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{multicol}
+\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{table}[t]
+	\begin{tabular}{ll}
+		\begin{minipage}{0.4\textwidth}
+			\begin{algorithm}[H]\footnotesize\caption{}
+				\label{multiplikation:alg:b1}
+				\setlength{\lineskip}{7pt}
+				\begin{algorithmic}
+					\Function{B1}{$a, b$}
+					\State \textbf{return} $a+b$
+					\EndFunction
+					\State
+					\State
+				\end{algorithmic}
+			\end{algorithm}
+		\end{minipage}	
+		&
+		\begin{minipage}{0.4\textwidth}
+	\begin{algorithm}[H]\footnotesize\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}
+
+		\end{minipage}
+	\end{tabular}
+\end{table}
+
+\begin{table}
+	\begin{tabular}[t]{ll} 
+		\begin{minipage}{0.4\textwidth}
+			\begin{algorithm}[H]\footnotesize\caption{}
+				\setlength{\lineskip}{7pt}
+				\begin{algorithmic}
+					\label{multiplikation:alg:linear}
+					\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
+					\State
+					\State
+				\end{algorithmic}
+			\end{algorithm}
+		\end{minipage}	
+		&
+		\begin{minipage}{0.4\textwidth}
+			\begin{algorithm}[H]\footnotesize\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}
+		\end{minipage}
+	\end{tabular}
+\end{table}
+
+dhdfh
+\end{document}
diff --git a/buch/papers/multiplikation/images/meas_c.pdf b/buch/papers/multiplikation/images/meas_c.pdf
index 3a4cfd8..faf347e 100644
--- a/buch/papers/multiplikation/images/meas_c.pdf
+++ b/buch/papers/multiplikation/images/meas_c.pdf
diff --git a/buch/papers/multiplikation/images/meas_c.tex b/buch/papers/multiplikation/images/meas_c.tex
index 818a7e6..fe2bd2f 100644
--- a/buch/papers/multiplikation/images/meas_c.tex
+++ b/buch/papers/multiplikation/images/meas_c.tex
@@ -43,8 +43,8 @@
 \begin{tikzpicture}
 \begin{axis}[
 xmode=log, ymode=log,
-xmin=60, xmax=5000,
-ymin=1e-4, ymax=2e3,
+xmin=30, xmax=10000,
+ymin=1e-5, ymax=2e4,
 grid=both,
 major grid style={black!50},
 xlabel = data Input ($n$),
@@ -57,85 +57,91 @@ width=12cm, height=8cm,
 ]
 \addlegendentry{Winograd}
 \addplot[    color=purple,
+        error bars/.cd, y dir=both, y explicit,
 ] coordinates {
-% (2,   0.000001)
-% (4,   0.000001)
-% (8,   0.000002)
-% (16,  0.000011)
-% (32,  0.000100)
-(64,  0.000654)
-(128, 0.005229)
-(256, 0.057440)
-(512, 0.517850)
-(1024,4.539413)
-(2048,130.627663)
+%(2,1e-07)
+%(4,5e-07)
+%(8,2.0000000000000003e-06)
+%(16,1.1999999999999999e-05)
+(32,8.329999999999999e-05)
+(64,0.0006479)
+(128,0.0052873)
+(256,0.052674599999999995)
+(512,0.5249752000000001)
+(1024,4.671161)
+(2048,136.6769777)
 (4096,1179.261048)
+(8192,10071.512655)
 };
 \addlegendentry{Strassen}
 \addplot [    color=black,
 ]coordinates {
-  %  (2,0.000001 )
-  %  (4,0.000003 )
-  %  (8,0.000010 )
-  % (16,0.000066 )
-  % (32,0.000470 )
-  (64,0.003368 )
- (128,0.024232 )
- (256,0.172000 )
- (512,1.209262 )
-(1024,8.457472 )
-(2048,59.267256)
+%(2,1e-07)
+%(4,2.1e-06)
+%(8,1.13e-05)
+%(16,7.07e-05)
+(32,0.0005041)
+(64,0.003596)
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+(512,1.2555)
+(1024,8.8302371)
+(2048,61.9018691)
 (4096,414.648901)
+(8192,3014.235467)
 };
 
 \addlegendentry{MM div and conq}
 \addplot[    color=green,
 ] coordinates {
-  %  (2,0.000003   )
-  %  (4,0.000002   )
-  %  (8,0.000010   )
-  % (16,0.000068   )
-  % (32,0.000594   )
-  (64,0.004264   )
- (128,0.036289   )
- (256,0.324645   )
- (512,2.612010   )
-(1024,19.928951  )
-(2048,159.333884 )
+%(2,3e-07)
+%(4,1.1e-06)
+%(8,8.6e-06)
+%(16,7.819999999999999e-05)
+(32,0.0005940000000000001)
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+(256,0.29484730000000003)
+(512,2.2228507)
+(1024,17.659234500000004)
+(2048,141.6103936)
 (4096,1147.106865)
+(8192,9606.402522)
 };
 
 \addlegendentry{MM}
 \addplot [    color=red,
 ]coordinates {
-  %  (2,0.000001   )
-  %  (4,0.000001   )
-  %  (8,0.000001   )
-  % (16,0.000010   )
-  % (32,0.000081   )
-  (64,0.000654   )
- (128,0.005556   )
- (256,0.054253   )
- (512,0.487317   )
-(1024,4.162845   )
-(2048,125.909034 )
+%(2,0.0)
+%(4,3e-07)
+%(8,1.8000000000000001e-06)
+%(16,1.1999999999999999e-05)
+(32,8.93e-05)
+(64,0.0006923)
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+(256,0.051771500000000005)
+(512,0.5062468000000001)
+(1024,4.5048086)
+(2048,129.2894619)
 (4096,1111.312696)
+(8192,9376.173434)
 };
 \addlegendentry{BLAS}
 \addplot[    color=blue,
 ] coordinates {
-  %  (2,0.000001 )
-  %  (4,0.000001 )
-  %  (8,0.000001 )
-  % (16,0.000003 )
-  % (32,0.000022 )
-  (64,0.000179 )
- (128,0.001278  )
- (256,0.010165  )
- (512,0.074739  )
-(1024,0.704748  )
-(2048,6.845095  )
+%(2,1e-07)
+%(4,0.0)
+%(8,1e-07)
+%(16,3.9e-06)
+(32,2.1000000000000002e-05)
+(64,0.00018580000000000002)
+(128,0.0012649)
+(256,0.0096489)
+(512,0.0773765)
+(1024,0.7643868)
+(2048,7.6320993999999995)
 (4096,55.845038)
+(8192,478.429957)
 };
 \end{axis}
 \end{tikzpicture}
diff --git a/buch/papers/multiplikation/images/meas_python.pdf b/buch/papers/multiplikation/images/meas_python.pdf
index cea2232..ab3b14b 100644
--- a/buch/papers/multiplikation/images/meas_python.pdf
+++ b/buch/papers/multiplikation/images/meas_python.pdf
diff --git a/buch/papers/multiplikation/images/meas_python.tex b/buch/papers/multiplikation/images/meas_python.tex
index ee4db43..d942f46 100644
--- a/buch/papers/multiplikation/images/meas_python.tex
+++ b/buch/papers/multiplikation/images/meas_python.tex
@@ -43,8 +43,8 @@
 \begin{tikzpicture}
 \begin{axis}[
 xmode=log, ymode=log,
-xmin=30, xmax=1050,
-ymin=0.01, ymax=900,
+xmin=30, xmax=4200,
+ymin=0.01, ymax=70000,
 grid=both,
 major grid style={black!50},
 xlabel = data input ($n$),
@@ -68,7 +68,8 @@ width=12cm, height=8cm,
 (256, 8.29899    )
 (512, 68.3699    )
 (1024,537.374    )
-
+(2046,4884.61)
+(4096,43597.1)
 };
 \addlegendentry{Strassen}
 \addplot [    color=black,
@@ -79,10 +80,12 @@ width=12cm, height=8cm,
   % (16,0.00475407  )
   (32,0.0485256   )
   (64,0.220414    )
- (128,1.44718    2 )
- (256,9.93866    0 )
- (512,63.961     2 )
-(1024,461.494    2 )
+ (128,1.44718    )
+ (256,9.93866    )
+ (512,63.961     )
+(1024,461.494    )
+(2046,3860.57)
+(4096,22904.3)
 };
 
 \addlegendentry{MM div and conq}
@@ -98,6 +101,8 @@ width=12cm, height=8cm,
  (256,13.27        )
  (512,105.397      )
 (1024,847.321      )
+(2046,7375.93)
+(4096,58466)
 };
 
 \addlegendentry{MM}
@@ -113,25 +118,27 @@ width=12cm, height=8cm,
  (256,  11.0062 )
  (512,   85.4768)
 (1024,750.757 )
+(2046,6154.18)
+(4096,46813.3)
 };
-% \addlegendentry{NumPy}
-% \addplot[    color=blue,
-% ] coordinates {
-%    (2,1.83582e-05 )
-%    (4,7.86781e-06)
-%    (8,1.00136e-05)
-%   (16,5.4121e-05 )
-%   (32,4.26769e-05)
-%   (64,0.000118494)
-%  (128,0.000244141 )
-%  (256,0.000695705  )
-%  (512,0.00221705   )
-% (1024,0.0188088    )
-% };
+%  \addlegendentry{NumPy}
+%  \addplot[    color=blue,
+%  ] coordinates {
+% %    (2,1.83582e-05 )
+% %    (4,7.86781e-06)
+% %    (8,1.00136e-05)
+% %   (16,5.4121e-05 )
+%    (32,4.26769e-05)
+%  (64,0.000118494)
+%   (128,0.000244141 )
+%   (256,0.000695705  )
+%   (512,0.00221705   )
+%  (1024,0.0188088    )
+% (2046,0.215739)
+% (4096,1.49159)
+%  };
+
 \end{axis}
 \end{tikzpicture}
 
 \end{document}
-
-
-
diff --git a/buch/papers/multiplikation/loesungsmethoden.tex b/buch/papers/multiplikation/loesungsmethoden.tex
index a7612e1..0760719 100755
--- a/buch/papers/multiplikation/loesungsmethoden.tex
+++ b/buch/papers/multiplikation/loesungsmethoden.tex
@@ -39,13 +39,13 @@ Die \texttt{for i} Schleife iteriert \"uber alle Zeilen der $\mathbf{A}$ Matrix,
 	\end{algorithmic}
 \end{algorithm}
 
-Die Laufzeit dieser Struktur mit drei \texttt{For} Schleifen ist $\mathcal{O}\left(n^3\right)$
+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 \cite{multiplikation:DAC} zu markant besseren Laufzeiten.
 Die Grundidee ist, dass ein Problem in mehrere, meist simplere und kleinere Teilprobleme aufgeteilt wird.
-Das bekannteste Beispiel ist wohl die \textit{Fast Fourier Transform} wobei die Laufzeit von $\mathcal{O}\left(n^2\right)$ zu $\mathcal{O}(n \log n)$ verbessert werden kann.
+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.
@@ -68,7 +68,7 @@ Das Matrizen Produkt
 \end{bmatrix},
 \end{equation}
 \begin{equation}
-\mathbf{C}_{ij} = \sum_{k=1}2n \mathbf{A}_{ik} \mathbf{B}_{kj}
+\mathbf{C}_{ij} = \sum_{k=1}^{2n} \mathbf{A}_{ik} \mathbf{B}_{kj}
 \label{multiplikation:eq:MM_block}
 \end{equation}
 ist identisch zu der Gleichung \eqref{multiplikation:eq:MM}, f\"ur die Multiplikation der Untermatrize $\mathbf{A}_{ik}$ und $\mathbf{B}_{kj}$ wird die Matrizenmultiplikation verwendet.
@@ -109,7 +109,7 @@ Die Laufzeit dieser rekursiven Funktion kann mit dem \textit{Master Theorem} \ci
 Ohne auf dieses vertieft einzugehen, bestimmt die Anzahl rekursiver Aufrufe $\mathcal{T} $ 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) =	8 \cdot \mathcal{T}\left (\frac{n}{2}\right ) + n^2  = \mathcal{O}(n^{\log_2 8}) = \mathcal{O}\left (n^{3} \right )
+	\mathcal{T}(n) =	8 \cdot \mathcal{T} \left(\frac{n}{2}\right ) + n^2  = \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.
@@ -202,7 +202,7 @@ Die Funktion wird sieben mal rekursiv aufgerufen.
 Dies f\"uhrt nach dem \textit{Master Theorem} zu einer Laufzeit von
 \begin{equation} \label{multiplikation:eq:laufzeitstrassen}
 \mathcal{T}(n) =
-7 \cdot \mathcal{T}(\frac{n}{2}) + n^2  = \mathcal{O}\left(n^{\log_2 7}\right ) = \mathcal{O}\left(n^{2.8074} \right )
+7 \cdot \mathcal{T}\left(\frac{n}{2}\right) + n^2  = \mathcal{O}(n^{\log_2 7} ) = \mathcal{O}(n^{2.8074}  )
 \end{equation}
 und ist somit schneller als die Standardmethode.
 Man beachte, dass die Anzahl von Additionen und Subtraktionen gr\"osser und die Anzahl der Multiplikationen kleiner wurde.
@@ -265,9 +265,9 @@ N=2n, \quad T = n^2 \\
 \end{equation}
 sein, damit man etwas einspart.
 Die Implementation kann Algorithmus \ref{multiplikation:alg:winograd} entnommen werden.
-Falls $m=n=p$ werden $\frac{n^3}/{2}$ Multiplikationen benötigt.
+Falls $m=n=p$ werden $\frac{n^3}{2}$ Multiplikationen benötigt.
 Im Abschnitt \ref{muliplikation:sec:bigo} wurde bereits erläutert: falls $n \rightarrow \infty$ können Konstanten vernachlässigt werden und
- somit entsteht für diesen Algorithmus wieder die Ursprüngliche Laufzeit von $\mathcal{O}\left(n^3 \right)$.
+ somit entsteht für diesen Algorithmus wieder die Ursprüngliche Laufzeit von $\mathcal{O}(n^3 )$.
 \begin{algorithm}\footnotesize\caption{Winograds Matrizenmultiplikation}
 	\setlength{\lineskip}{7pt}
 	\label{multiplikation:alg:winograd}
@@ -336,33 +336,33 @@ Die meisten Numerischen Bibliotheken von High-Level Skriptsprachen wie \texttt{M
 	\item Level 2
 	\begin{itemize}
 		\item Operationen der Art: $\mathbf{y} \leftarrow \alpha \mathbf{A}\mathbf{x}+\beta  \mathbf{y}$
-		\item Dieses Level hat $\mathcal{O}\left(n^2\right)$ Charakteristik
+		\item Dieses Level hat $\mathcal{O}(n^2)$ Charakteristik
 		\end{itemize}
 		\item Level 3
 		\begin{itemize}
 			\item Operationen der Art: $\mathbf{C} \leftarrow \alpha \mathbf{A}\mathbf{B}+\beta\mathbf{C}$
-			\item Dieses Level hat $\mathcal{O}\left(n^3\right)$ Charakteristik
+			\item Dieses Level hat $\mathcal{O}(n^3)$ Charakteristik
 			\end{itemize}
 \end{itemize}
 
 Die \textit{BLAS} sind auf die modernen Computer Prozessoren optimiert und k\"onnen dank einer ausgeklügelter Verwendung der Speicherarchitektur zu erheblichen Leistungsoptimierungen f\"uhren.
 
 
-\subsubsection{General Matrix Multiplication (GEMM)}
-
-Die \textit{Double-GEMM} \cite{multiplikation:DGEMM} ist definiert als:
-
-\textit{DGEMM  performs one of the matrix-matrix operations}
-$$
- C := \alpha \cdot op( A )\cdot op( B ) + \beta \cdot C,
- $$
- \textit{where  op( X ) is one of}
-$$
-op( X ) = X  \quad \text{ or } \quad  op( X ) = X^T,
-$$
- \textit{alpha and beta are scalars, and A, B and C are matrices, with op( A )
- an m by k matrix,  op( B )  a  k by n matrix and  C an m by n matrix.
- }
+%\subsubsection{General Matrix Multiplication (GEMM)}
+%
+%Die \textit{Double-GEMM} \cite{multiplikation:DGEMM} ist definiert als:
+%
+%\textit{DGEMM  performs one of the matrix-matrix operations}
+%$$
+% C := \alpha \cdot op( A )\cdot op( B ) + \beta \cdot C,
+% $$
+% \textit{where  op( X ) is one of}
+%$$
+%op( X ) = X  \quad \text{ or } \quad  op( X ) = X^T,
+%$$
+% \textit{alpha and beta are scalars, and A, B and C are matrices, with op( A )
+% an m by k matrix,  op( B )  a  k by n matrix and  C an m by n matrix.
+% }
 
 %Die Implementation von $\alpha\mathbf{A}\mathbf{B} + \beta \mathbf{C} = \mathbf{C}$, wobei  $\alpha = 1.0$ und $\beta = 0.0$ in der \texttt{C}-Version von \textit{BLAS}, ist als
 %\begin{lstlisting}[style=multiplikationC]
@@ -379,7 +379,7 @@ $$
 Folgende Algorithmen wurden jeweils in \texttt{C} und \texttt{Python} implementiert.
 \begin{itemize}
 	\item Standard Matrizenmultiplikation
-	\item \textit{Devide and Conquer} Matrizenmultiplikation
+	\item \textit{Divide and Conquer} Matrizenmultiplikation
 	\item Strassens Matrizenmultiplikation
 	\item Winograds Matrizenmultiplikation
 	\item \texttt{BLAS} Matrizenmultiplikation in \texttt{C}
@@ -389,6 +389,14 @@ Folgende Algorithmen wurden jeweils in \texttt{C} und \texttt{Python} implementi
 Der Code kann im zum Buch gehörigem \textit{GitHub} \footnote{\url{https://github.com/AndreasFMueller/SeminarMatrizen.git}} Repository gefunden werden.
 Anzumerken ist, dass die Matrizenmultiplikation von \texttt{NumPy} als einzige Implementation Multiprocessing und Multithreading verwendet, dies f\"uhrt zu den tiefen Messzeiten.
 In Abbildung \ref{multiplikation:fig:python} und Abbildung \ref{multiplikation:fig:c_meas_4096} sind de Messresultate grafisch dargestellt. Die selben Messresultate sind tabellarisch in Tabelle \ref{multiplikation:tab:messung_Python} und Tabelle \ref{multiplikation:tab:messung_C} ersichtlich.
+
+In der Messung mit der Programmiersprache \texttt{C}, kann ein typischer Cache-Effekt beobachtet wer-
+den.
+Bei den Algorithmen von Winograd und der Standardmethode hat bei einer Matrizengrösse von $n = 2048$ wohl eine Zeile der Matrize nicht an einer Cache Speicherstelle platzt.
+Diese beiden Algorithmen sind die Einzigen, welche \texttt{for}-Schleifen über die ganze Breite der Matrizen verwenden.
+Dies führt dazu, dass ganze Zeilen zwischengespeichert werden müssen.
+Bei den anderen Algorithmen ist dies nicht der Fall.
+
 Die Hardwareinformationen des verwendeten Computers sind in der Tabelle \ref{multiplikation:tab:pc_config} aufgelistet.
 
 
@@ -400,14 +408,15 @@ Die Hardwareinformationen des verwendeten Computers sind in der Tabelle \ref{mul
 							 \textbf{n} & \textbf{MM (\textit{s})} &  \textbf{MM DC (\textit{s})} & \textbf{Strassen (\textit{s})}  & \textbf{Winograd (\textit{s})} & \textbf{BLAS (\textit{s})} \\
 							 \hline
 							 \multicolumn{6}{c}{} \\
-							 \textbf{32}   & 0.000081 &0.000594 &  0.00047& 0.00010 & 0.000022  \\
-							 \textbf{64}   & 0.00065  & 0.0042&  0.0033& 0.00065& 0.00017 \\
-							 \textbf{128}  & 0.0055   & 0.036&  0.024&  0.0052 & 0.0012 \\
-							 \textbf{256}  & 0.054    & 0.32 & 0.17 &  0.057& 0.010 \\
-							 \textbf{512}  & 0.48     & 2.61 & 1.20 & 0.51 &  0.074\\
-							 \textbf{1024} & 4.16     & 19.92& 8.45  & 4.53 & 0.704 \\
-							 \textbf{2048} & 125.90   & 159.33& 59.26 & 130.62 &  6.84 \\
-							 \textbf{4096} & 1111.31  & 1147.10& 414.64 & 1179.26  &  55.84\\
+							 \textbf{32}   & 0.000089 & 0.000594 & 0.0005 & 0.00008 & 0.000021  \\
+							 \textbf{64}   & 0.00069  & 0.0044   & 0.0036  & 0.00064 & 0.00018   \\
+							 \textbf{128}  & 0.0057   & 0.035    & 0.025   & 0.0052  & 0.0012    \\
+							 \textbf{256}  & 0.052    & 0.29     & 0.178    & 0.053   & 0.0096     \\
+							 \textbf{512}  & 0.51     & 2.22     & 1.25    & 0.55    & 0.077     \\
+							 \textbf{1024} & 4.50     & 17.65    & 8.83    & 4.67    & 0.764     \\
+							 \textbf{2048} & 129.28   & 141.61   & 61.901   & 136.67  & 7.63      \\
+							 \textbf{4096} & 1111.31  & 1147.10  & 414.64  & 1179.26 & 55.84     \\
+							 \textbf{8192} & 9376.17  & 9606.40  & 3014.23  & 10071.51& 478.42     \\
 							 \multicolumn{6}{c}{} \\
 							 \hline
 							 \hline
@@ -427,13 +436,14 @@ Die Hardwareinformationen des verwendeten Computers sind in der Tabelle \ref{mul
 	 							 \textbf{n} & \textbf{MM (\textit{s})} &  \textbf{MM DC (\textit{s})} & \textbf{Strassen (\textit{s})}  & \textbf{Winograd (\textit{s})} & \textbf{\texttt{NumPy}(\textit{s})} \\
 	 							 \hline
 	 							 \multicolumn{6}{c}{} \\
-	 							 \textbf{32}   & 0.0240 &0.0271 &  0.04852& 0.01871 & 4.26e-05  \\
+	 							 \textbf{32}   & 0.0240 &0.0271 &  0.04852& 0.01871 & 0.0000426  \\
 	 							 \textbf{64}   & 0.186  & 0.265&  0.2204& 0.1530& 0.000118 \\
 	 							 \textbf{128}  & 1.563   & 1.777&  1.447&  1.1947 & 0.000244 \\
 	 							 \textbf{256}  & 11.006    & 13.27 & 9.938 &  8.298& 0.000695 \\
 	 							 \textbf{512}  & 85.476    & 105.397 & 63.961 & 68.36 &  0.00221\\
 	 							 \textbf{1024} & 750.757     & 847.321& 461.494  & 537.374 & 0.0188 \\
-								 \textbf{4096} & -     & - & -  & - & 1.633 \\
+								 \textbf{2048} & 6154.18     & 7375.93& 3860.57  & 4884.61 & 0.215 \\
+								 \textbf{4096} & 46813.3     & 58466 & 22904.3  & 43597.1 & 1.49 \\
 	 							 \multicolumn{6}{c}{} \\
 	 							 \hline
 	 							 \hline
diff --git a/buch/papers/multiplikation/problemstellung.tex b/buch/papers/multiplikation/problemstellung.tex
index e53b0de..c8ba274 100755
--- a/buch/papers/multiplikation/problemstellung.tex
+++ b/buch/papers/multiplikation/problemstellung.tex
@@ -14,87 +14,102 @@ Gezielt wird auf Algorithmen eingegangen, welche das Problem schneller als der S
 Die Big $\mathcal{O}$ Notation beschreibt die Laufzeitkomplexit\"at eines Algorithmus in Abhängigkeit zur Inputgrösse \cite{multiplikation:bigo}.
 $f(x) \in \mathcal{O}(g(x))$ besagt, dass die Funktion $f$ nicht wesentlich schneller w\"achst als $g$ wenn $x \rightarrow \infty$.
 % Es gibt eine Konstante $K$ derart, dass $f(x) \le K g(x)$ für $x\to\infty$
-Als Beispiel: benötigt eine Funktion $g$ $\mathcal{O}\left(n^2 \right)$ Multiplikationen, so wächst $f$ mit $\mathcal{O}\left(n+ n^2 \right)$ nicht wesentlich schneller falls $x\to\infty$.
+Als Beispiel: benötigt eine Funktion $g$ $\mathcal{O} (n^2  )$ Multiplikationen, so wächst $f$ mit $\mathcal{O} (n+ n^2  )$ nicht wesentlich schneller falls $x\to\infty$.
 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 $f \in \mathcal{O}\left (n^2 \right ) \rightarrow f$ w\"achst quadratisch
+	\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 $f \in \mathcal{O}\left (e^n \right ) \rightarrow f$ w\"achst exponentiell
+	\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.
 Bei einer logarithmischen Darstellung werden Polynome der Form $f(x) = x^k$ als Gerade und Exponentialfunktionen der Form $f(x) = a^x$ als nach oben gekr\"ummte Kurven dargestellt.
-Sch\"on zu erkennen ist, dass Logarithmische Kurven beschr\"ankt sind.
+
 
 
 \subsubsection{Beispiel Algorithmen}
 
 Es folgen einige Beispiele von Algorithmen welche zu einer bestimmten Zeitkomplexit\"atsklasse zugeteilt werden k\"onnen.
 
-\begin{minipage}{0.4\textwidth}
-	\begin{algorithm}[H]\footnotesize\caption{}
-		\label{multiplikation:alg:b1}
-		\setlength{\lineskip}{7pt}
-		\begin{algorithmic}
-			\Function{B1}{$a, b$}
-			\State \textbf{return} $a+b$
-			\EndFunction
-		\end{algorithmic}
-	\end{algorithm}
-
-	\begin{algorithm}[H]\footnotesize\caption{}
-		\setlength{\lineskip}{7pt}
-		\begin{algorithmic}
-			\label{multiplikation:alg:linear}
-			\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}
-\end{minipage}
-\hspace{2cm}
-\begin{minipage}{0.4\textwidth}
-
-	\begin{algorithm}[H]\footnotesize\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}
-
-
-	\begin{algorithm}[H]\footnotesize\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}
-
-\end{minipage}
+
+\begin{table}[t]
+	\begin{tabular}{ll}
+		\begin{minipage}{0.48\textwidth}
+			\begin{algorithm}[H]\footnotesize\caption{}
+				\label{multiplikation:alg:b1}
+				\setlength{\lineskip}{7pt}
+				\begin{algorithmic}
+					\Function{B1}{$a, b$}
+					\State \textbf{return} $a+b$
+					\EndFunction
+					\State
+					\State
+				\end{algorithmic}
+			\end{algorithm}
+		\end{minipage}	
+		&
+		\begin{minipage}{0.48\textwidth}
+			\begin{algorithm}[H]\footnotesize\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}
+			
+		\end{minipage}
+	\end{tabular}
+\end{table}
+
+\begin{table}
+	\begin{tabular}[t]{ll} 
+		\begin{minipage}{0.48\textwidth}
+			\begin{algorithm}[H]\footnotesize\caption{}
+				\setlength{\lineskip}{7pt}
+				\begin{algorithmic}
+					\label{multiplikation:alg:linear}
+					\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
+					\State
+					\State
+				\end{algorithmic}
+			\end{algorithm}
+		\end{minipage}	
+		&
+		\begin{minipage}{0.48\textwidth}
+			\begin{algorithm}[H]\footnotesize\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}
+		\end{minipage}
+	\end{tabular}
+\end{table}
 
 \paragraph{Beschr\"ankter Algorithmus}
 
@@ -111,7 +126,7 @@ Die \texttt{for}-Schleife wird $n$-mal durchlaufen und f\"uhrt deshalb zu $\math
 \paragraph{Quadratischer Algorithmus}
 
 Der Algorithmus \ref{multiplikation:alg:q1} hat ein quadratisches Verhalten.
-Die beiden \texttt{for}-Schleifen werden jeweils $n$-mal durchlaufen und f\"uhrt deshalb zu $\mathcal{O}\left(n^2\right)$.
+Die beiden \texttt{for}-Schleifen werden jeweils $n$-mal durchlaufen und f\"uhrt deshalb zu $\mathcal{O} (n^2 )$.
 
 
 \begin{figure}