Merge pull request #84 from Nunigan/master

Multiplikation #4

37 files changed, 1152 insertions, 336 deletions

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
+0.296292,256
+0.297592,256
+0.302464,256
+0.299557,256
+0.299367,256
+0.306394,256
+0.287616,256
+0.292630,256
+0.289542,256
+0.277019,256
+2.331956,512
+2.224501,512
+2.203910,512
+2.198937,512
+2.206083,512
+2.199477,512
+2.199847,512
+2.225379,512
+2.202491,512
+2.235926,512
+17.649432,1024
+17.636769,1024
+17.639024,1024
+17.625402,1024
+17.722286,1024
+17.611777,1024
+17.653120,1024
+17.748270,1024
+17.691817,1024
+17.614448,1024
+141.943689,2048
+141.580812,2048
+141.882050,2048
+141.516253,2048
+141.351237,2048
+141.641167,2048
+141.596407,2048
+141.607048,2048
+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
+0.000000,2
+0.000000,4
+0.000000,4
+0.000000,4
+0.000000,4
+0.000000,4
+0.000000,4
+0.000000,4
+0.000000,4
+0.000000,4
+0.000000,4
 0.000001,8
+0.000000,8
+0.000000,8
+0.000000,8
+0.000000,8
+0.000000,8
+0.000000,8
+0.000000,8
+0.000000,8
+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
+0.000003,16
+0.000003,16
+0.000003,16
+0.000021,32
+0.000019,32
+0.000030,32
+0.000020,32
+0.000020,32
+0.000020,32
+0.000020,32
+0.000020,32
+0.000020,32
+0.000020,32
+0.000180,64
+0.000192,64
+0.000163,64
+0.000153,64
+0.000153,64
+0.000197,64
+0.000163,64
+0.000267,64
+0.000226,64
+0.000164,64
+0.001216,128
+0.001233,128
+0.001364,128
 0.001278,128
-0.010165,256
-0.074739,512
-0.704748,1024
-6.845095,2048
-55.845038,4096
+0.001211,128
+0.001295,128
+0.001206,128
+0.001371,128
+0.001225,128
+0.001250,128
+0.009733,256
+0.009497,256
+0.009586,256
+0.009600,256
+0.009768,256
+0.009566,256
+0.009731,256
+0.009550,256
+0.009664,256
+0.009794,256
+0.077453,512
+0.076616,512
+0.088812,512
+0.075990,512
+0.076925,512
+0.076303,512
+0.075915,512
+0.075600,512
+0.075122,512
+0.075029,512
+0.769186,1024
+0.775780,1024
+0.753906,1024
+0.757834,1024
+0.772001,1024
+0.770950,1024
+0.791317,1024
+0.753319,1024
+0.747228,1024
+0.752347,1024
+7.625205,2048
+7.652278,2048
+7.640682,2048
+7.649428,2048
+7.632806,2048
+7.579347,2048
+7.612317,2048
+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
+0.000001,4
+0.000001,8
+0.000003,16
+0.000022,32
+0.000179,64
+0.001278,128
+0.010165,256
+0.074739,512
+0.704748,1024
+6.845095,2048
+55.845038,4096
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
+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
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
+0.005229,128
+0.057440,256
+0.517850,512
+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
+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
+0.000004,4
+0.000002,4
+0.000002,4
+0.000002,4
+0.000002,4
+0.000002,4
+0.000002,4
+0.000002,4
+0.000002,4
+0.000001,4
+0.000020,8
+0.000018,8
+0.000008,8
+0.000008,8
+0.000008,8
+0.000008,8
+0.000008,8
+0.000008,8
+0.000008,8
+0.000019,8
+0.000080,16
+0.000075,16
+0.000078,16
+0.000085,16
+0.000065,16
+0.000065,16
+0.000065,16
+0.000064,16
+0.000065,16
+0.000065,16
+0.000546,32
+0.000480,32
+0.000563,32
+0.000551,32
+0.000502,32
+0.000504,32
+0.000463,32
+0.000462,32
+0.000508,32
+0.000462,32
+0.003675,64
+0.003665,64
+0.003493,64
+0.003708,64
+0.003465,64
+0.003502,64
+0.003710,64
+0.003537,64
+0.003637,64
+0.003568,64
+0.025342,128
+0.025179,128
+0.026475,128
+0.025758,128
+0.025333,128
+0.024988,128
+0.025727,128
+0.025298,128
+0.025283,128
+0.025098,128
+0.181311,256
+0.178432,256
+0.177075,256
+0.177474,256
+0.177025,256
+0.177805,256
+0.177944,256
+0.178151,256
+0.177858,256
+0.178742,256
+1.283374,512
+1.246682,512
+1.245898,512
+1.251547,512
+1.250288,512
+1.250495,512
+1.257037,512
+1.255247,512
+1.255382,512
+1.259050,512
+8.784102,1024
+8.845725,1024
+8.771100,1024
+8.770184,1024
+8.955977,1024
+8.849161,1024
+8.806902,1024
+8.808937,1024
+8.848900,1024
+8.861383,1024
+61.787123,2048
+61.972599,2048
+61.822434,2048
+62.051331,2048
+61.946171,2048
+61.911404,2048
+61.872671,2048
+61.791260,2048
+61.818110,2048
+62.045588,2048
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
+0.000000,2
+0.000000,2
+0.000000,2
+0.000000,2
+0.000000,2
+0.000000,2
+0.000000,2
+0.000000,2
+0.000000,4
+0.000000,4
+0.000000,4
+0.000000,4
+0.000000,4
+0.000000,4
+0.000000,4
+0.000000,4
+0.000000,4
+0.000000,4
 0.000002,8
+0.000002,8
+0.000002,8
+0.000002,8
+0.000002,8
+0.000002,8
+0.000002,8
+0.000002,8
+0.000002,8
+0.000002,8
+0.000011,16
+0.000011,16
+0.000011,16
+0.000011,16
+0.000011,16
+0.000011,16
+0.000011,16
+0.000021,16
+0.000011,16
 0.000011,16
-0.000100,32
-0.000654,64
-0.005229,128
-0.057440,256
-0.517850,512
-4.539413,1024
-130.627663,2048
-1179.261048,4096
+0.000092,32
+0.000092,32
+0.000081,32
+0.000081,32
+0.000081,32
+0.000081,32
+0.000088,32
+0.000079,32
+0.000079,32
+0.000079,32
+0.000670,64
+0.000739,64
+0.000609,64
+0.000609,64
+0.000700,64
+0.000648,64
+0.000626,64
+0.000626,64
+0.000626,64
+0.000626,64
+0.005321,128
+0.005286,128
+0.005180,128
+0.005223,128
+0.005249,128
+0.005299,128
+0.005205,128
+0.005268,128
+0.005464,128
+0.005378,128
+0.053123,256
+0.052325,256
+0.052729,256
+0.052930,256
+0.052207,256
+0.053178,256
+0.052122,256
+0.052681,256
+0.052965,256
+0.052486,256
+0.527028,512
+0.525201,512
+0.521822,512
+0.525147,512
+0.525241,512
+0.527725,512
+0.526321,512
+0.526479,512
+0.524020,512
+0.520768,512
+4.732299,1024
+4.617253,1024
+4.647425,1024
+4.519233,1024
+4.917471,1024
+4.564929,1024
+4.870771,1024
+4.555407,1024
+4.727473,1024
+4.559349,1024
+136.409028,2048
+136.390557,2048
+136.541672,2048
+136.598491,2048
+137.720790,2048
+136.825926,2048
+136.367686,2048
+136.650627,2048
+136.642195,2048
+136.622805,2048
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)
+(128,0.0254481)
+(256,0.1781817)
+(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)
+(64,0.0044339)
+(128,0.0348443)
+(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)
+(128,0.0056842)
+(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}