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-rwxr-xr-x[-rw-r--r--]buch/papers/multiplikation/Makefile0
-rwxr-xr-x[-rw-r--r--]buch/papers/multiplikation/Makefile.inc7
-rwxr-xr-xbuch/papers/multiplikation/code/Figure_1.pngbin0 -> 144173 bytes
-rwxr-xr-xbuch/papers/multiplikation/code/MMbin0 -> 26848 bytes
-rwxr-xr-xbuch/papers/multiplikation/code/MM.c465
-rw-r--r--buch/papers/multiplikation/code/MM.py311
-rw-r--r--buch/papers/multiplikation/code/__pycache__/MM.cpython-38.pycbin0 -> 4160 bytes
-rw-r--r--buch/papers/multiplikation/code/c_matrix.h101
-rw-r--r--buch/papers/multiplikation/code/c_meas_1024.pdfbin0 -> 16748 bytes
-rw-r--r--buch/papers/multiplikation/code/c_meas_128.pdfbin0 -> 16454 bytes
-rw-r--r--buch/papers/multiplikation/code/c_meas_16.pdfbin0 -> 16376 bytes
-rw-r--r--buch/papers/multiplikation/code/c_meas_2048.pdfbin0 -> 16281 bytes
-rw-r--r--buch/papers/multiplikation/code/c_meas_256.pdfbin0 -> 15286 bytes
-rw-r--r--buch/papers/multiplikation/code/c_meas_32.pdfbin0 -> 15163 bytes
-rw-r--r--buch/papers/multiplikation/code/c_meas_4096.pdfbin0 -> 15865 bytes
-rw-r--r--buch/papers/multiplikation/code/c_meas_512.pdfbin0 -> 15472 bytes
-rw-r--r--buch/papers/multiplikation/code/c_meas_64.pdfbin0 -> 16358 bytes
-rw-r--r--buch/papers/multiplikation/code/c_meas_8.pdfbin0 -> 16766 bytes
-rwxr-xr-xbuch/papers/multiplikation/code/helper_class.py105
-rw-r--r--buch/papers/multiplikation/code/meas/MM.txt12
-rw-r--r--buch/papers/multiplikation/code/meas/MM_dc.txt12
-rw-r--r--buch/papers/multiplikation/code/meas/blas.txt12
-rw-r--r--buch/papers/multiplikation/code/meas/strassen.txt12
-rw-r--r--buch/papers/multiplikation/code/meas/test/4096/MM.txt12
-rw-r--r--buch/papers/multiplikation/code/meas/test/4096/strassen.txt12
-rw-r--r--buch/papers/multiplikation/code/meas/test/MM.txt14900
-rw-r--r--buch/papers/multiplikation/code/meas/test/blas.txt14900
-rw-r--r--buch/papers/multiplikation/code/meas/test/winograd.txt14900
-rw-r--r--buch/papers/multiplikation/code/meas/winograd.txt11
-rw-r--r--buch/papers/multiplikation/code/meas_1024.pdfbin0 -> 17660 bytes
-rw-r--r--buch/papers/multiplikation/code/meas_1024.txt6
-rw-r--r--buch/papers/multiplikation/code/meas_128.pdfbin0 -> 17961 bytes
-rw-r--r--buch/papers/multiplikation/code/meas_128.txt6
-rw-r--r--buch/papers/multiplikation/code/meas_16.pdfbin0 -> 17766 bytes
-rw-r--r--buch/papers/multiplikation/code/meas_16.txt6
-rw-r--r--buch/papers/multiplikation/code/meas_256.pdfbin0 -> 18067 bytes
-rw-r--r--buch/papers/multiplikation/code/meas_256.txt6
-rw-r--r--buch/papers/multiplikation/code/meas_32.pdfbin0 -> 17078 bytes
-rw-r--r--buch/papers/multiplikation/code/meas_32.txt6
-rw-r--r--buch/papers/multiplikation/code/meas_512.pdfbin0 -> 18028 bytes
-rw-r--r--buch/papers/multiplikation/code/meas_512.txt6
-rw-r--r--buch/papers/multiplikation/code/meas_64.pdfbin0 -> 17678 bytes
-rw-r--r--buch/papers/multiplikation/code/meas_64.txt6
-rw-r--r--buch/papers/multiplikation/code/meas_8.pdfbin0 -> 18400 bytes
-rw-r--r--buch/papers/multiplikation/code/meas_8.txt6
-rw-r--r--buch/papers/multiplikation/code/test.tex92
-rwxr-xr-xbuch/papers/multiplikation/einlteung.tex52
-rw-r--r--buch/papers/multiplikation/images/bigo.pdfbin0 -> 24288 bytes
-rw-r--r--buch/papers/multiplikation/images/bigo.tex107
-rw-r--r--buch/papers/multiplikation/images/mm_visualisation.pdfbin0 -> 21665 bytes
-rw-r--r--buch/papers/multiplikation/images/mm_visualisation.tex45
-rw-r--r--buch/papers/multiplikation/images/strassen.pdfbin0 -> 15850 bytes
-rw-r--r--buch/papers/multiplikation/images/strassen.tex140
-rwxr-xr-xbuch/papers/multiplikation/loesungsmethoden.tex309
-rwxr-xr-x[-rw-r--r--]buch/papers/multiplikation/main.tex34
-rwxr-xr-x[-rw-r--r--]buch/papers/multiplikation/packages.tex0
-rwxr-xr-xbuch/papers/multiplikation/papers/Strassen_GPU.pdfbin0 -> 254508 bytes
-rwxr-xr-xbuch/papers/multiplikation/papers/Strassen_original_1969.pdfbin0 -> 151265 bytes
-rwxr-xr-xbuch/papers/multiplikation/papers/assay_fast_MM.pdfbin0 -> 484352 bytes
-rwxr-xr-xbuch/papers/multiplikation/papers/strassen_video.txt1
-rwxr-xr-xbuch/papers/multiplikation/papers/winograd_original.pdfbin0 -> 533604 bytes
-rw-r--r--buch/papers/multiplikation/presentation/common.tex79
-rw-r--r--buch/papers/multiplikation/presentation/presentation.nav59
-rw-r--r--buch/papers/multiplikation/presentation/presentation.pdfbin0 -> 717544 bytes
-rw-r--r--buch/papers/multiplikation/presentation/presentation.snm0
-rw-r--r--buch/papers/multiplikation/presentation/presentation.tex12
-rw-r--r--buch/papers/multiplikation/presentation/slides/algo.tex111
-rw-r--r--buch/papers/multiplikation/presentation/slides/bigO.tex251
-rw-r--r--buch/papers/multiplikation/presentation/slides/blas.tex18
-rw-r--r--buch/papers/multiplikation/presentation/slides/conclusuion.tex0
-rw-r--r--buch/papers/multiplikation/presentation/slides/logo.pdfbin0 -> 8987 bytes
-rw-r--r--buch/papers/multiplikation/presentation/slides/meas.tex42
-rw-r--r--buch/papers/multiplikation/presentation/slides/nn.tex97
-rw-r--r--buch/papers/multiplikation/presentation/slides/parcomp.tex66
-rw-r--r--buch/papers/multiplikation/presentation/slides/slides.tex15
-rw-r--r--buch/papers/multiplikation/presentation/slides/strassen.tex429
-rw-r--r--buch/papers/multiplikation/presentation/tikz/algo.pdfbin0 -> 33396 bytes
-rw-r--r--buch/papers/multiplikation/presentation/tikz/algo.tex52
-rwxr-xr-xbuch/papers/multiplikation/problemstellung.tex104
-rwxr-xr-x[-rw-r--r--]buch/papers/multiplikation/references.bib30
-rw-r--r--buch/papers/multiplikation/teil0.tex22
-rw-r--r--buch/papers/multiplikation/teil1.tex55
-rw-r--r--buch/papers/multiplikation/teil2.tex40
-rw-r--r--buch/papers/multiplikation/teil3.tex40
-rw-r--r--buch/papers/multiplikation/tikz_formulas/algo.fdb_latexmk254
-rw-r--r--buch/papers/multiplikation/tikz_formulas/algo.fls438
-rw-r--r--buch/papers/multiplikation/tikz_formulas/algo.pdfbin0 -> 33785 bytes
-rwxr-xr-xbuch/papers/multiplikation/tikz_formulas/algo.tex131
-rw-r--r--buch/papers/multiplikation/tikz_formulas/algo_graph.fdb_latexmk245
-rw-r--r--buch/papers/multiplikation/tikz_formulas/algo_graph.fls485
-rwxr-xr-xbuch/papers/multiplikation/tikz_formulas/algo_graph.pdfbin0 -> 15850 bytes
-rwxr-xr-xbuch/papers/multiplikation/tikz_formulas/algo_graph.tex140
92 files changed, 49628 insertions, 187 deletions
diff --git a/buch/papers/multiplikation/Makefile b/buch/papers/multiplikation/Makefile
index 8f04c2c..8f04c2c 100644..100755
--- a/buch/papers/multiplikation/Makefile
+++ b/buch/papers/multiplikation/Makefile
diff --git a/buch/papers/multiplikation/Makefile.inc b/buch/papers/multiplikation/Makefile.inc
index b78d67e..074020f 100644..100755
--- a/buch/papers/multiplikation/Makefile.inc
+++ b/buch/papers/multiplikation/Makefile.inc
@@ -7,8 +7,7 @@ dependencies-multiplikation = \
papers/multiplikation/packages.tex \
papers/multiplikation/main.tex \
papers/multiplikation/references.bib \
- papers/multiplikation/teil0.tex \
- papers/multiplikation/teil1.tex \
- papers/multiplikation/teil2.tex \
- papers/multiplikation/teil3.tex
+ papers/multiplikation/einlteung.tex \
+ papers/multiplikation/loesungsmethoden.tex \
+ papers/multiplikation/problemstellung.tex
diff --git a/buch/papers/multiplikation/code/Figure_1.png b/buch/papers/multiplikation/code/Figure_1.png
new file mode 100755
index 0000000..9def15a
--- /dev/null
+++ b/buch/papers/multiplikation/code/Figure_1.png
Binary files differ
diff --git a/buch/papers/multiplikation/code/MM b/buch/papers/multiplikation/code/MM
new file mode 100755
index 0000000..f07985f
--- /dev/null
+++ b/buch/papers/multiplikation/code/MM
Binary files differ
diff --git a/buch/papers/multiplikation/code/MM.c b/buch/papers/multiplikation/code/MM.c
new file mode 100755
index 0000000..04c4dab
--- /dev/null
+++ b/buch/papers/multiplikation/code/MM.c
@@ -0,0 +1,465 @@
+#include <stdio.h>
+#include <stdint.h>
+#include <stdlib.h>
+#include <time.h>
+#include <omp.h>
+#include "c_matrix.h"
+#include <gsl/gsl_cblas.h>
+#include <string.h>
+
+void MM(int *A, int *B, int *C, int n);
+void openMP_MM(int *A, int *B, int *C, int n);
+void winograd(int *A, int *B, int *C, int n);
+int winograd_inner(int *a, int *b, int n);
+void run_algo(void (*algo)(), char alog_name[], int print);
+void run_algo_cblas(int print);
+void MM_dc(int *A, int *B, int *C, int n);
+void strassen(int *A, int *B, int *C, int n);
+void printMatrix(int *C, int n);
+void printMatrix_double(double *C, int n);
+void split(int *in, int *out, int n, int col, int row);
+void join(int *in, int *out, int n, int col, int row);
+void add(int *A, int *B, int *C, int n);
+void sub(int *A, int *B, int *C, int n);
+void multiply(int *A, int *B, int *C, int n);
+
+int main() {
+ // omp_set_dynamic(0);
+ // omp_set_num_threads(4);
+// run_algo(openMP_MM, "openMP_MM",0);
+ run_algo(MM_dc, "MM_dc",0);
+ run_algo(strassen, "strassen",0);
+
+ run_algo(MM, "MM", 0);
+ // run_algo(winograd, "winograd", 0);
+ run_algo_cblas(0);
+
+ return 0;
+}
+
+void MM(int *A, int *B, int *C, int n) {
+ for (int i = 0; i < n; ++i) {
+ for (int j = 0; j < n; ++j) {
+ int sum = 0;
+ for (int k = 0; k < n; ++k) {
+ sum += (*((A + i * n) + k)) * (*((B + k * n) + j));
+ }
+ *((C + i * n) + j) = sum;
+ }
+ }
+}
+
+int winograd_inner(int *a, int *b, int n){
+ int ab = 0;
+ if(n%2==0)
+ {
+ int xi = 0;
+ int etha = 0;
+ for(int i = 0; i<n/2;++i)
+ {
+ xi += a[2*i]*a[2*i+1];
+ etha += b[2*i]*b[2*i+1];
+ ab += (a[2*i]+b[2*i+1])*(a[2*i+1]+b[2*i]);
+ }
+ ab = ab-etha-xi;
+ }
+ return ab;
+ }
+
+ void winograd(int *A, int *B, int *C, int n) {
+
+ int xi_array[n];
+ int etha_array[n];
+ int xi = 0;
+ int etha = 0;
+ int ab = 0;
+
+ for (int i = 0; i < n; ++i) {
+ xi = 0;
+ etha = 0;
+ for(int k = 0;k<n/2;++k)
+ {
+ xi += (*((A + i * n) + 2*k))*(*((A + i * n) + (2*k+1)));
+ etha += (*((B + 2*k * n) + i))*(*((B + (2*k+1) * n) + i));
+ }
+ xi_array[i] = xi;
+ etha_array[i] = etha;
+ }
+
+ for (int i = 0; i < n; ++i) {
+ for (int j = 0; j < n; ++j) {
+ ab = 0;
+ for(int k = 0;k<n/2;++k)
+ {
+ ab += ((*((A + i * n) + 2*k))+(*((B + (2*k+1) * n) + j)))*((*((A + i * n) + (2*k+1)))+(*((B + 2*k * n) + j)));
+ }
+ *((C + i * n) + j) = ab-etha_array[j]-xi_array[i];
+ }
+ }
+
+
+
+
+ // for (int i = 0; i < n; ++i) {
+ // int *a = (int*) malloc(n * sizeof(int));
+ // for(int k = 0; k<n; ++k)
+ // {
+ // a[k] = (*((A + i * n) + k));
+ // }
+ //
+ // for (int j = 0; j < n; ++j) {
+ // int *b = (int*) malloc(n * sizeof(int));
+ // for(int k = 0; k<n; ++k)
+ // {
+ // b[k] =(*((B + k * n) + j));
+ // }
+ // *((C + i * n) + j) = winograd_inner(a,b,n);
+ // }
+ // }
+ }
+
+
+void openMP_MM(int *A, int *B, int *C, int n) {
+
+ #pragma omp parallel for
+ for (int i = 0; i < n; ++i) {
+ for (int j = 0; j < n; ++j) {
+ int sum = 0;
+ for (int k = 0; k < n; ++k) {
+ sum += (*((A + i * n) + k)) * (*((B + k * n) + j));
+ }
+ *((C + i * n) + j) = sum;
+ }
+ }
+}
+
+void MM_dc(int *A, int *B, int *C, int n) {
+ if (n <= 2) {
+ MM((int*) A, (int*) B, (int*) C, n);
+ } else {
+ int *A11 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *A12 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *A21 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *A22 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *B11 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *B12 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *B21 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *B22 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+
+ split((int*) A, (int*) A11, n / 2, 0, 0);
+ split((int*) A, (int*) A12, n / 2, 0, n / 2);
+ split((int*) A, (int*) A21, n / 2, n / 2, 0);
+ split((int*) A, (int*) A22, n / 2, n / 2, n / 2);
+ split((int*) B, (int*) B11, n / 2, 0, 0);
+ split((int*) B, (int*) B12, n / 2, 0, n / 2);
+ split((int*) B, (int*) B21, n / 2, n / 2, 0);
+ split((int*) B, (int*) B22, n / 2, n / 2, n / 2);
+
+ int *tmp1 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *tmp2 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *tmp3 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *tmp4 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *tmp5 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *tmp6 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *tmp7 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *tmp8 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+
+ MM_dc((int*) A11, (int*) B11, (int*) tmp1, n / 2);
+ MM_dc((int*) A12, (int*) B21, (int*) tmp2, n / 2);
+ MM_dc((int*) A11, (int*) B12, (int*) tmp3, n / 2);
+ MM_dc((int*) A12, (int*) B22, (int*) tmp4, n / 2);
+ MM_dc((int*) A21, (int*) B11, (int*) tmp5, n / 2);
+ MM_dc((int*) A22, (int*) B21, (int*) tmp6, n / 2);
+ MM_dc((int*) A21, (int*) B12, (int*) tmp7, n / 2);
+ MM_dc((int*) A22, (int*) B22, (int*) tmp8, n / 2);
+
+ free(A11);
+ free(A12);
+ free(A21);
+ free(A22);
+ free(B11);
+ free(B12);
+ free(B21);
+ free(B22);
+
+ int *C11 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *C12 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *C21 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *C22 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+
+ add((int*) tmp1, (int*) tmp2, (int*) C11, n / 2);
+ add((int*) tmp3, (int*) tmp4, (int*) C12, n / 2);
+ add((int*) tmp5, (int*) tmp6, (int*) C21, n / 2);
+ add((int*) tmp7, (int*) tmp8, (int*) C22, n / 2);
+
+ free(tmp1);
+ free(tmp2);
+ free(tmp3);
+ free(tmp4);
+ free(tmp5);
+ free(tmp6);
+ free(tmp7);
+ free(tmp8);
+
+ join((int*) C11, (int*) C, n / 2, 0, 0);
+ join((int*) C12, (int*) C, n / 2, 0, n / 2);
+ join((int*) C21, (int*) C, n / 2, n / 2, 0);
+ join((int*) C22, (int*) C, n / 2, n / 2, n / 2);
+
+ free(C11);
+ free(C12);
+ free(C21);
+ free(C22);
+
+ }
+}
+
+void strassen(int *A, int *B, int *C, int n) {
+ if (n <= 2) {
+
+ int P, Q, R, S, T, U, V;
+ P = ((*((A + 0 * n) + 0)) + (*((A + 1 * n) + 1)))
+ * ((*((B + 0 * n) + 0)) + (*((B + 1 * n) + 1)));
+ Q = ((*((A + 1 * n) + 0)) + (*((A + 1 * n) + 1)))
+ * ((*((B + 0 * n) + 0)));
+ R = ((*((A + 0 * n) + 0)))
+ * ((*((B + 0 * n) + 1)) - (*((B + 1 * n) + 1)));
+ S = ((*((A + 1 * n) + 1)))
+ * ((*((B + 1 * n) + 0)) - (*((B + 0 * n) + 0)));
+ T = ((*((A + 0 * n) + 0)) + (*((A + 0 * n) + 1)))
+ * ((*((B + 1 * n) + 1)));
+ U = ((*((A + 1 * n) + 0)) - (*((A + 0 * n) + 0)))
+ * ((*((B + 0 * n) + 0)) + (*((B + 0 * n) + 1)));
+ V = ((*((A + 0 * n) + 1)) - (*((A + 1 * n) + 1)))
+ * ((*((B + 1 * n) + 0)) + (*((B + 1 * n) + 1)));
+ (*((C + 0 * n) + 0)) = P + S - T + V;
+ (*((C + 0 * n) + 1)) = R + T;
+ (*((C + 1 * n) + 0)) = Q + S;
+ (*((C + 1 * n) + 1)) = P + R - Q + U;
+
+ } else {
+ int *A11 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *A12 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *A21 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *A22 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *B11 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *B12 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *B21 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *B22 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+
+ split((int*) A, (int*) A11, n / 2, 0, 0);
+ split((int*) A, (int*) A12, n / 2, 0, n / 2);
+ split((int*) A, (int*) A21, n / 2, n / 2, 0);
+ split((int*) A, (int*) A22, n / 2, n / 2, n / 2);
+ split((int*) B, (int*) B11, n / 2, 0, 0);
+ split((int*) B, (int*) B12, n / 2, 0, n / 2);
+ split((int*) B, (int*) B21, n / 2, n / 2, 0);
+ split((int*) B, (int*) B22, n / 2, n / 2, n / 2);
+
+ int *P = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *Q = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *R = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *S = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *T = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *U = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *V = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+
+ int *addA = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *addB = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+
+ add((int*) A11, (int*) A22, (int*) addA, n / 2);
+ add((int*) B11, (int*) B22, (int*) addB, n / 2);
+ strassen((int*) addA, (int*) addB, (int*) P, n / 2);
+
+ add((int*) A21, (int*) A22, (int*) addA, n / 2);
+ strassen((int*) addA, (int*) B11, (int*) Q, n / 2);
+
+ sub((int*) B12, (int*) B22, (int*) addB, n / 2);
+ strassen((int*) A11, (int*) addB, (int*) R, n / 2);
+
+ sub((int*) B21, (int*) B11, (int*) addB, n / 2);
+ strassen((int*) A22, (int*) addB, (int*) S, n / 2);
+
+ add((int*) A11, (int*) A12, (int*) addA, n / 2);
+ strassen((int*) addA, (int*) B22, (int*) T, n / 2);
+
+ sub((int*) A21, (int*) A11, (int*) addA, n / 2);
+ add((int*) B11, (int*) B12, (int*) addB, n / 2);
+ strassen((int*) addA, (int*) addB, (int*) U, n / 2);
+
+ sub((int*) A12, (int*) A22, (int*) addA, n / 2);
+ add((int*) B21, (int*) B22, (int*) addB, n / 2);
+ strassen((int*) addA, (int*) addB, (int*) V, n / 2);
+
+ free(A11);
+ free(A12);
+ free(A21);
+ free(A22);
+ free(B11);
+ free(B12);
+ free(B21);
+ free(B22);
+ free(addA);
+ free(addB);
+
+ int *C11 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *C12 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *C21 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *C22 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+
+ int *resAdd1 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+ int *resAdd2 = (int*) malloc(n / 2 * n / 2 * sizeof(int));
+
+ add((int*) R, (int*) T, (int*) C12, n / 2);
+ add((int*) Q, (int*) S, (int*) C21, n / 2);
+
+ add((int*) P, (int*) S, (int*) resAdd1, n / 2);
+ add((int*) resAdd1, (int*) V, (int*) resAdd2, n / 2);
+ sub((int*) resAdd2, (int*) T, (int*) C11, n / 2);
+
+ add((int*) P, (int*) R, (int*) resAdd1, n / 2);
+ add((int*) resAdd1, (int*) U, (int*) resAdd2, n / 2);
+ sub((int*) resAdd2, (int*) Q, (int*) C22, n / 2);
+
+ free(P);
+ free(Q);
+ free(R);
+ free(S);
+ free(T);
+ free(U);
+ free(V);
+ free(resAdd1);
+ free(resAdd2);
+
+ join((int*) C11, (int*) C, n / 2, 0, 0);
+ join((int*) C12, (int*) C, n / 2, 0, n / 2);
+ join((int*) C21, (int*) C, n / 2, n / 2, 0);
+ join((int*) C22, (int*) C, n / 2, n / 2, n / 2);
+
+ free(C11);
+ free(C12);
+ free(C21);
+ free(C22);
+ }
+}
+
+void add(int *A, int *B, int *C, int n) {
+ for (int i = 0; i < n; i++) {
+ for (int j = 0; j < n; j++) {
+ *((C + i * n) + j) = *((A + i * n) + j) + *((B + i * n) + j);
+ }
+ }
+}
+
+void sub(int *A, int *B, int *C, int n) {
+ for (int i = 0; i < n; i++) {
+ for (int j = 0; j < n; j++) {
+ *((C + i * n) + j) = *((A + i * n) + j) - *((B + i * n) + j);
+ }
+ }
+}
+
+void multiply(int *A, int *B, int *C, int n) {
+ int mul;
+
+ for (int i = 0; i < n; ++i) {
+ for (int j = 0; j < n; ++j) {
+ mul = (*((A + i * n) + j)) * (*((B + i * n) + j));
+ *((C + i * n) + j) = mul;
+ }
+ }
+}
+
+void split(int *in, int *out, int n, int col, int row) {
+ for (int i1 = 0, i2 = col; i1 < n; i1++, i2++)
+ for (int j1 = 0, j2 = row; j1 < n; j1++, j2++) {
+ *((out + i1 * n) + j1) = *((in + i2 * n * 2) + j2);
+
+ }
+}
+
+void join(int *in, int *out, int n, int col, int row) {
+ for (int i1 = 0, i2 = col; i1 < n; i1++, i2++)
+ for (int j1 = 0, j2 = row; j1 < n; j1++, j2++)
+ *((out + i2 * n * 2) + j2) = *((in + i1 * n) + j1);
+}
+
+void printMatrix(int *C, int n) {
+ for (int i = 0; i < n; ++i) {
+ for (int j = 0; j < n; ++j) {
+ printf("%d ", *((C + i * n) + j));
+ }
+ printf("\n");
+ }
+}
+
+void printMatrix_double(double *C, int n) {
+ for (int i = 0; i < n; ++i) {
+ for (int j = 0; j < n; ++j) {
+ printf("%.0f ", *((C + i * n) + j));
+ }
+ printf("\n");
+ }
+}
+
+void run_algo(void (*algo)(), char alog_name[], int print)
+{
+ FILE *fptr;
+
+ char fileName[40] = "meas/";
+ strcat(fileName, alog_name);
+ strcat(fileName, ".txt");
+ fptr = fopen(fileName, "w");
+
+
+ for(int i=0; i<n_arrays; ++i)
+ {
+ for(int j = 0; j<1; ++j)
+ {
+ int *C = (int*) malloc(n[i] * n[i] * sizeof(int));
+ double dtime = omp_get_wtime();
+ algo(Ap[i], Bp[i], (int*) C, n[i]);
+ dtime = omp_get_wtime() - dtime;
+ // printf("The %s program took %f seconds to execute \n", alog_name, dtime);
+ fprintf(fptr, "%f,%d\n", dtime, n[i]);
+
+ if(print==1)
+ {
+ printMatrix((int*)C, n[i]);
+ }
+ free(C);
+ }
+ }
+ fclose(fptr);
+
+}
+
+void run_algo_cblas(int print)
+{
+
+ FILE *fptr;
+
+ fptr = fopen("meas/blas.txt", "w");
+ for(int i=0; i<n_arrays; ++i)
+ {
+ for(int j = 0; j<1; ++j)
+ {
+ double *dC = (double*) malloc(n[i] * n[i] * sizeof(double));
+ double dtime = omp_get_wtime();
+ cblas_dgemm(CblasRowMajor, CblasNoTrans, CblasNoTrans, n[i], n[i], n[i], 1.0, dAp[i], n[i],
+ dBp[i], n[i], 0.0, dC, n[i]);
+ dtime = omp_get_wtime() - dtime;
+ // printf("The cblas program took %f seconds to execute \n", dtime);
+ fprintf(fptr, "%f,%d\n",dtime, n[i]);
+
+ if(print==1)
+ {
+ printMatrix_double( (double*)dC, n[i]);
+ }
+
+ free(dC);
+ }
+ }
+ fclose(fptr);
+
+}
diff --git a/buch/papers/multiplikation/code/MM.py b/buch/papers/multiplikation/code/MM.py
new file mode 100644
index 0000000..626b82d
--- /dev/null
+++ b/buch/papers/multiplikation/code/MM.py
@@ -0,0 +1,311 @@
+#!/usr/bin/env python3
+# -*- coding: utf-8 -*-
+"""
+Created on Fri Mar 19 07:31:29 2021
+
+@author: nunigan
+"""
+import numpy as np
+import time
+import matplotlib.pyplot as plt
+from scipy.optimize import curve_fit
+import tikzplotlib
+def MM(A, B):
+ n = np.shape(A)[0]
+ C = np.zeros((n, n))
+ for i in range(n):
+ for j in range(n):
+ C[i, j] = 0
+ for k in range(n):
+ C[i, j] += A[i, k]*B[k, j]
+ return C
+
+
+def MM_dc(A, B):
+ n = np.shape(A)[0]
+ if(n <= 2):
+ C = np.zeros((n, n))
+ C[0, 0] = A[0, 0]*B[0, 0]+A[0, 1]*B[1, 0]
+ C[0, 1] = A[0, 0]*B[0, 1]+A[0, 1]*B[1, 1]
+ C[1, 0] = A[1, 0]*B[0, 0]+A[1, 1]*B[1, 0]
+ C[1, 1] = A[1, 0]*B[0, 1]+A[1, 1]*B[1, 1]
+ return C
+ else:
+ A11, A12, A21, A22 = A[:n//2, :n//2], A[:n//2, n//2:], A[n//2:, :n//2], A[n//2:, n//2:]
+ B11, B12, B21, B22 = B[:n//2, :n//2], B[:n//2, n//2:], B[n//2:, :n//2], B[n//2:, n//2:]
+ C11 = MM_dc(A11, B11) + MM_dc(A12, B21)
+ C12 = MM_dc(A11, B12) + MM_dc(A12, B22)
+ C21 = MM_dc(A21, B11) + MM_dc(A22, B21)
+ C22 = MM_dc(A21, B12) + MM_dc(A22, B22)
+ C = np.vstack((np.hstack((C11, C12)), np.hstack((C21, C22))))
+ return C
+
+
+def strassen(A, B):
+ n = np.shape(A)[0]
+ if(n <= 2):
+ C = np.zeros((n, n))
+ P = (A[0, 0]+A[1, 1])*(B[0, 0]+B[1, 1])
+ Q = (A[1, 0]+A[1, 1])*B[0, 0]
+ R = A[0, 0]*(B[0, 1]-B[1, 1])
+ S = A[1, 1]*(B[1, 0]-B[0, 0])
+ T = (A[0, 0]+A[0, 1])*B[1, 1]
+ U = (A[1, 0]-A[0, 0])*(B[0, 0]+B[0, 1])
+ V = (A[0, 1]-A[1, 1])*(B[1, 0]+B[1, 1])
+ C[0, 0] = P+S-T+V
+ C[0, 1] = R+T
+ C[1, 0] = Q+S
+ C[1, 1] = P+R-Q+U
+ return C
+ else:
+ m = n//2
+ A11, A12, A21, A22 = A[:m, :m], A[:m, m:], A[m:, :m], A[m:, m:]
+ B11, B12, B21, B22 = B[:m, :m], B[:m, m:], B[m:, :m], B[m:, m:]
+ P = strassen((A11+A22),(B11+B22))
+ Q = strassen((A21+A22),B11)
+ R = strassen(A11,(B12-B22))
+ S = strassen(A22,(B21-B11))
+ T = strassen((A11+A12),B22)
+ U = strassen((A21-A11),(B11+B12))
+ V = strassen((A12-A22),(B21+B22))
+
+ C11 = P+S-T+V
+ C12 = R+T
+ C21 = Q+S
+ C22 = P+R-Q+U
+
+ C = np.vstack((np.hstack((C11, C12)), np.hstack((C21, C22))))
+ return C
+
+def winograd_inner(a, b):
+ n = np.shape(a)[0]
+ if n%2 == 0:
+ xi = np.sum(a[::2]*a[1::2])
+ etha = np.sum(b[::2]*b[1::2])
+ # print("xi = {}, etha = {}".format(xi, etha))
+ ab = np.sum((a[::2]+b[1::2])*(a[1::2]+b[::2]))-xi-etha
+ else:
+ xi = np.sum(a[0:-1:2]*a[1::2])
+ etha = np.sum(b[0:-1:2]*b[1::2])
+ ab = np.sum((a[0:-1:2]+b[1::2])*(a[1::2]+b[0:-1:2]))-xi-etha+a[-1]*b[-1]
+ return ab
+
+def winograd(A, B):
+ m,n = np.shape(A)
+ n2,p = np.shape(B)
+ C = np.zeros((m,p))
+ for i in range(np.shape(A)[0]):
+ for j in range(np.shape(B)[1]):
+ C[i,j] = winograd_inner(A[i,:], B[:,j])
+ return C
+
+def winograd2(A, B):
+ m,n = np.shape(A)
+ n2,p = np.shape(B)
+ C = np.zeros((m,p))
+ xi = np.zeros((m))
+ eta = np.zeros((p))
+ ab = 0
+ for i in range(m):
+ for j in range(n//2):
+ xi[i] += A[i,2*j]*A[i,2*j+1]
+
+ for i in range(p):
+ for j in range(n//2):
+ eta[i] += B[2*j,i]*B[2*j+1,i]
+
+ if n%2==0:
+ for i in range(m):
+ for j in range(p):
+ ab = 0
+ for k in range(n//2):
+ ab += (A[i,2*k]+B[2*k+1,j])*(A[i,2*k+1]+B[2*k,j])
+ C[i,j] = ab-eta[j]-xi[i]
+ else:
+ for i in range(m):
+ for j in range(p):
+ ab = 0
+ for k in range(n//2):
+ ab += (A[i,2*k]+B[2*k+1,j])*(A[i,2*k+1]+B[2*k,j])
+ C[i,j] = ab-eta[j]-xi[i]+A[i,-1]*B[-1,j]
+
+ return C
+
+def test_perfomance(n):
+ t_mm = []
+ t_mm_dc = []
+ t_mm_strassen = []
+ t_wino = []
+ t_np = []
+
+ for i in n:
+ A = np.random.randn(i, i)
+ B = np.random.randn(i, i)
+ # A = np.random.randint(-100, 100,(i, i))
+ # B = np.random.randint(-100, 100,(i, i))
+
+ start = time.time()
+ C3 = strassen(A, B)
+ t_mm_strassen.append(time.time() - start)
+
+ start = time.time()
+ C1 = MM(A, B)
+ t_mm.append(time.time() - start)
+
+ start = time.time()
+ C2 = MM_dc(A, B)
+ t_mm_dc.append(time.time() - start)
+
+ start = time.time()
+ C4 = winograd2(A, B)
+ t_wino.append(time.time() - start)
+
+ start = time.time()
+ C = A@B
+ t_np.append(time.time() - start)
+
+ plt.figure(figsize=(13,8))
+ plt.rcParams['font.family'] = 'STIXGeneral'
+ plt.rc('axes', labelsize=23)
+ plt.rc('xtick', labelsize=23)
+ plt.rc('ytick', labelsize=23)
+ plt.plot(n, t_mm, label='Standard', lw=5)
+ plt.plot(n, t_mm_dc, label='Divide and conquer', lw=5)
+ plt.plot(n, t_mm_strassen, label='Strassen', lw=5)
+ plt.plot(n, t_wino, label='Winograd', lw=5)
+ plt.plot(n, t_np, label='NumPy A@B', lw=5)
+ plt.legend()
+ plt.xlabel("n")
+ plt.ylabel("time (s)")
+ plt.grid(True)
+ plt.tight_layout()
+ # plt.yscale('log')
+ plt.legend(fontsize=19)
+ plt.savefig('meas_' + str(max(n))+ '.pdf')
+ arr = np.array([n, t_mm, t_mm_dc, t_mm_strassen, t_wino, t_np])
+ np.savetxt('meas_' + str(max(n))+ '.txt',arr)
+ return arr
+
+
+def plot(num):
+ arr = np.loadtxt('meas_{}.txt'.format(num))
+ n, t_mm, t_mm_dc, t_mm_strassen, t_wino, t_np = arr
+ plt.figure(figsize=(13,8))
+ plt.rcParams['font.family'] = 'STIXGeneral'
+ plt.rc('axes', labelsize=23)
+ plt.rc('xtick', labelsize=23)
+ plt.rc('ytick', labelsize=23)
+ plt.plot(n, t_mm, label='3 For Loops', lw=5)
+ plt.plot(n, t_mm_dc, label='Divide and Conquer', lw=5)
+ plt.plot(n, t_mm_strassen, label='Strassen', lw=5)
+ # plt.plot(n, t_wino, label='Winograd', lw=5)
+ plt.plot(n, t_np, label='NumPy A@B', lw=5)
+ plt.legend()
+ plt.xlabel("n")
+ plt.ylabel("time (s)")
+ plt.grid(True)
+ plt.tight_layout()
+ # plt.yscale('log')
+ plt.legend(fontsize=19)
+ plt.savefig('meas_' + str(num)+ '.pdf')
+ return arr
+
+def plot_c_res(ave, num):
+ MM = np.loadtxt("meas/MM.txt", delimiter=',')
+ # winograd = np.loadtxt("meas/winograd.txt", delimiter=',')
+ blas = np.loadtxt("meas/blas.txt", delimiter=',')
+ MM_dc = np.loadtxt("meas/MM_dc.txt", delimiter=',')
+ strassen = np.loadtxt("meas/strassen.txt", delimiter=',')
+
+ MM_t = MM[:,0]
+ MM_n = MM[:,1]
+ MM_t = np.mean(MM_t.reshape(-1,ave),axis=1)
+ MM_n = np.mean(MM_n.reshape(-1,ave),axis=1)
+
+ MM_dc_t = MM_dc[:,0]
+ MM_dc_n = MM_dc[:,1]
+ MM_dc_t = np.mean(MM_dc_t.reshape(-1,ave),axis=1)
+ MM_dc_n = np.mean(MM_dc_n.reshape(-1,ave),axis=1)
+
+ strassen_t = strassen[:,0]
+ strassen_n = strassen[:,1]
+ strassen_t = np.mean(strassen_t.reshape(-1,ave),axis=1)
+ strassen_n = np.mean(strassen_n.reshape(-1,ave),axis=1)
+
+ # winograd_t = winograd[:,0]
+ # winograd_n = winograd[:,1]
+ # winograd_t = np.mean(winograd_t.reshape(-1,ave),axis=1)
+ # winograd_n = np.mean(winograd_n.reshape(-1,ave),axis=1)
+
+ blas_t = blas[:,0]
+ blas_n = blas[:,1]
+ blas_t = np.mean(blas_t.reshape(-1,ave),axis=1)
+ blas_n = np.mean(blas_n.reshape(-1,ave),axis=1)
+
+ def func(x, a,b):
+ return b*x**a
+
+ # popt, pcov = curve_fit(func, blas_n, blas_t)
+ # popt1, pcov2 = curve_fit(func, blas_n, winograd_t)
+ # popt2, pcov2 = curve_fit(func, blas_n, MM_t)
+
+ plt.figure(figsize=(13,8))
+ plt.rcParams['font.family'] = 'STIXGeneral'
+ plt.rc('axes', labelsize=23)
+ plt.rc('xtick', labelsize=23)
+ plt.rc('ytick', labelsize=23)
+ plt.plot(MM_n, MM_t, label='3 For Loops', lw=5)
+ # plt.plot(winograd_n, winograd_t, label='Winograd MM', lw=5)
+ plt.plot(blas_n, blas_t, label='Blas', lw=5)
+ plt.plot(strassen_n, strassen_t, label='Strassen', lw=5)
+ plt.plot(MM_dc_n, MM_dc_t, label='Divide and Conquer', lw=5)
+ plt.xlabel("n")
+ plt.ylabel("time (s)")
+ plt.grid(True)
+ plt.tight_layout()
+ plt.legend(fontsize=19)
+ plt.savefig('c_meas_' + str(num)+ '.pdf')
+
+ # plt.plot(blas_n, func(blas_n, *popt), 'r-', label='fit blas: a=%5.5f, b=%5.10f' % tuple(popt))
+ # plt.plot(blas_n, func(blas_n, *popt1), 'r-', label='fit winograd: a=%5.5f, b=%5.10f' % tuple(popt1))
+ # plt.plot(blas_n, func(blas_n, *popt2), 'r-', label='fit MM: a=%5.5f, b=%5.10f' % tuple(popt2))
+
+ plt.legend()
+
+
+# test%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+if __name__ == '__main__':
+ plot_c_res(1, 4096)
+
+
+ # plot(8)
+ # n = np.logspace(1,10,10,base=2,dtype=(np.int))
+ # n = np.arange(1,50,2)
+ A = np.random.randint(-10, 10, (5,3))
+ B = np.random.randint(-10, 10, (3,5))
+
+ C = winograd2(A, B)
+ C_test = A@B
+ print(C)
+ print(C_test)
+ # print(np.equal(C, C_test))
+
+ # t_np = test_perfomance(n)
+ # C = strassen(A, B)
+ # C_test = A@B
+
+
+ # plot_c_res()
+ # def func(x, a):
+ # return x**a
+
+ # popt, pcov = curve_fit(func, n, t_np, bounds=(2, 3))
+
+
+ # plt.figure()
+ # plt.plot(n, t_np, 'b-', label='data')
+ # plt.plot(n, func(n, *popt), 'r-', label='fit: a=%5.3f' % tuple(popt))
+ # plt.xlabel('x')
+ # plt.ylabel('y')
+ # plt.legend()
+ \ No newline at end of file
diff --git a/buch/papers/multiplikation/code/__pycache__/MM.cpython-38.pyc b/buch/papers/multiplikation/code/__pycache__/MM.cpython-38.pyc
new file mode 100644
index 0000000..7768772
--- /dev/null
+++ b/buch/papers/multiplikation/code/__pycache__/MM.cpython-38.pyc
Binary files differ
diff --git a/buch/papers/multiplikation/code/c_matrix.h b/buch/papers/multiplikation/code/c_matrix.h
new file mode 100644
index 0000000..13df55d
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_matrix.h
@@ -0,0 +1,101 @@
+/* Seminar Matrizen, autogenerated File, Michael Schmid, 30/05/2021, 22:00:57 */
+
+#include <stdint.h>
+const int A0[][2] =
+ {
+ {-15,68},
+ {49,86}
+ };
+const int B0[][2] =
+ {
+ {33,73},
+ {38,-76}
+ };
+const double dB0[][2] =
+ {
+ {33,73},
+ {38,-76}
+ };
+const double dA0[][2] =
+ {
+ {-15,68},
+ {49,86}
+ };
+const int A1[][4] =
+ {
+ {75,-38,-32,-65},
+ {37,74,-31,29},
+ {15,-62,-20,-20},
+ {-31,-35,-89,47}
+ };
+const int B1[][4] =
+ {
+ {71,90,78,-98},
+ {4,63,12,-47},
+ {11,-44,75,-69},
+ {95,-15,64,23}
+ };
+const double dB1[][4] =
+ {
+ {71,90,78,-98},
+ {4,63,12,-47},
+ {11,-44,75,-69},
+ {95,-15,64,23}
+ };
+const double dA1[][4] =
+ {
+ {75,-38,-32,-65},
+ {37,74,-31,29},
+ {15,-62,-20,-20},
+ {-31,-35,-89,47}
+ };
+const int A2[][8] =
+ {
+ {80,42,3,-16,6,55,87,16},
+ {-99,-14,21,-1,-94,-56,91,10},
+ {-47,-55,-59,62,12,-53,87,-65},
+ {-60,94,-67,23,-62,33,-63,-72},
+ {12,-75,16,21,22,-37,1,16},
+ {-100,-99,82,-66,2,64,-13,44},
+ {59,-100,-90,8,36,-24,18,88},
+ {73,-58,75,-100,-19,-29,85,-19}
+ };
+const int B2[][8] =
+ {
+ {-61,88,69,49,-53,47,73,45},
+ {16,14,-88,-11,-67,-73,-20,43},
+ {-60,-63,26,32,-29,18,-44,-69},
+ {1,21,21,38,7,-100,-61,-76},
+ {-90,95,-99,88,49,-80,27,-36},
+ {24,-12,-47,-7,29,15,52,37},
+ {-98,-76,29,76,-41,-75,97,79},
+ {62,-90,-35,-14,-30,-42,-95,52}
+ };
+const double dB2[][8] =
+ {
+ {-61,88,69,49,-53,47,73,45},
+ {16,14,-88,-11,-67,-73,-20,43},
+ {-60,-63,26,32,-29,18,-44,-69},
+ {1,21,21,38,7,-100,-61,-76},
+ {-90,95,-99,88,49,-80,27,-36},
+ {24,-12,-47,-7,29,15,52,37},
+ {-98,-76,29,76,-41,-75,97,79},
+ {62,-90,-35,-14,-30,-42,-95,52}
+ };
+const double dA2[][8] =
+ {
+ {80,42,3,-16,6,55,87,16},
+ {-99,-14,21,-1,-94,-56,91,10},
+ {-47,-55,-59,62,12,-53,87,-65},
+ {-60,94,-67,23,-62,33,-63,-72},
+ {12,-75,16,21,22,-37,1,16},
+ {-100,-99,82,-66,2,64,-13,44},
+ {59,-100,-90,8,36,-24,18,88},
+ {73,-58,75,-100,-19,-29,85,-19}
+ };
+const int *Ap[3] = {(int*) A0,(int*) A1,(int*) A2};
+const int *Bp[3] = {(int*) B0,(int*) B1,(int*) B2};
+const double *dAp[3] = {(double*) dA0,(double*) dA1,(double*) dA2};
+const double *dBp[3] = {(double*) dB0,(double*) dB1,(double*) dB2};
+int n[3] = {2,4,8};
+int n_arrays = 3;
diff --git a/buch/papers/multiplikation/code/c_meas_1024.pdf b/buch/papers/multiplikation/code/c_meas_1024.pdf
new file mode 100644
index 0000000..95b68b5
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+++ b/buch/papers/multiplikation/code/c_meas_1024.pdf
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diff --git a/buch/papers/multiplikation/code/c_meas_128.pdf b/buch/papers/multiplikation/code/c_meas_128.pdf
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index 0000000..56b9200
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_128.pdf
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diff --git a/buch/papers/multiplikation/code/c_meas_16.pdf b/buch/papers/multiplikation/code/c_meas_16.pdf
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index 0000000..2edc82d
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_16.pdf
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diff --git a/buch/papers/multiplikation/code/c_meas_2048.pdf b/buch/papers/multiplikation/code/c_meas_2048.pdf
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index 0000000..caba698
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_2048.pdf
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diff --git a/buch/papers/multiplikation/code/c_meas_256.pdf b/buch/papers/multiplikation/code/c_meas_256.pdf
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index 0000000..383ae86
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_256.pdf
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diff --git a/buch/papers/multiplikation/code/c_meas_32.pdf b/buch/papers/multiplikation/code/c_meas_32.pdf
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index 0000000..180fd22
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_32.pdf
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diff --git a/buch/papers/multiplikation/code/c_meas_4096.pdf b/buch/papers/multiplikation/code/c_meas_4096.pdf
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index 0000000..547d794
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_4096.pdf
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diff --git a/buch/papers/multiplikation/code/c_meas_512.pdf b/buch/papers/multiplikation/code/c_meas_512.pdf
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index 0000000..5e8894e
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_512.pdf
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diff --git a/buch/papers/multiplikation/code/c_meas_64.pdf b/buch/papers/multiplikation/code/c_meas_64.pdf
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index 0000000..8ff905c
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_64.pdf
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diff --git a/buch/papers/multiplikation/code/c_meas_8.pdf b/buch/papers/multiplikation/code/c_meas_8.pdf
new file mode 100644
index 0000000..9682aca
--- /dev/null
+++ b/buch/papers/multiplikation/code/c_meas_8.pdf
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diff --git a/buch/papers/multiplikation/code/helper_class.py b/buch/papers/multiplikation/code/helper_class.py
new file mode 100755
index 0000000..485fa76
--- /dev/null
+++ b/buch/papers/multiplikation/code/helper_class.py
@@ -0,0 +1,105 @@
+#!/usr/bin/env python3
+# -*- coding: utf-8 -*-
+"""
+Created on Fri Mar 12 09:02:48 2021
+
+@author: nunigan
+"""
+
+from datetime import datetime
+import numpy as np
+
+class Helper():
+ def __init__(self):
+ pass
+
+ def write_c_matrix(self, n_array):
+
+ with open('c_matrix.h', 'w') as file:
+ file.writelines('/* Seminar Matrizen, autogenerated File, Michael Schmid, {} */ \n \n'.format(datetime.now().strftime("%d/%m/%Y, %H:%M:%S")))
+
+ file.writelines('#include <stdint.h> \n')
+
+
+
+ for k, n in enumerate(n_array):
+ A = np.random.randint(-100,100,(n,n))
+ B = np.random.randint(-100,100,(n,n))
+ file.writelines('const int A{}[][{}] = \n'.format(k, n))
+ file.writelines(' {\n')
+ for i in range(n):
+ file.writelines(' {')
+ for j in range(n):
+ if j == n-1:
+ file.writelines('{}'.format(A[i,j]))
+ else:
+ file.writelines('{},'.format(A[i,j]))
+ if i == n-1:
+ file.writelines('}\n')
+ else:
+ file.writelines('},\n')
+
+ file.writelines(' };\n')
+
+ file.writelines('const int B{}[][{}] = \n'.format(k,n))
+ file.writelines(' {\n')
+ for i in range(n):
+ file.writelines(' {')
+ for j in range(n):
+ if j == n-1:
+ file.writelines('{}'.format(B[i,j]))
+ else:
+ file.writelines('{},'.format(B[i,j]))
+ if i == n-1:
+ file.writelines('}\n')
+ else:
+ file.writelines('},\n')
+
+ file.writelines(' };\n')
+
+ file.writelines('const double dB{}[][{}] = \n'.format(k,n))
+ file.writelines(' {\n')
+ for i in range(n):
+ file.writelines(' {')
+ for j in range(n):
+ if j == n-1:
+ file.writelines('{}'.format(B[i,j]))
+ else:
+ file.writelines('{},'.format(B[i,j]))
+ if i == n-1:
+ file.writelines('}\n')
+ else:
+ file.writelines('},\n')
+
+ file.writelines(' };\n')
+
+ file.writelines('const double dA{}[][{}] = \n'.format(k,n))
+ file.writelines(' {\n')
+ for i in range(n):
+ file.writelines(' {')
+ for j in range(n):
+ if j == n-1:
+ file.writelines('{}'.format(A[i,j]))
+ else:
+ file.writelines('{},'.format(A[i,j]))
+ if i == n-1:
+ file.writelines('}\n')
+ else:
+ file.writelines('},\n')
+
+ file.writelines(' };\n')
+
+ file.writelines('const int *Ap[{}] = {{{}}}; \n'.format(len(n_array),",".join(['(int*) A'+str(element) for element in np.arange(len(n_array))])))
+ file.writelines('const int *Bp[{}] = {{{}}}; \n'.format(len(n_array),",".join(['(int*) B'+str(element) for element in np.arange(len(n_array))])))
+ file.writelines('const double *dAp[{}] = {{{}}}; \n'.format(len(n_array),",".join(['(double*) dA'+str(element) for element in np.arange(len(n_array))])))
+ file.writelines('const double *dBp[{}] = {{{}}}; \n'.format(len(n_array),",".join(['(double*) dB'+str(element) for element in np.arange(len(n_array))])))
+ file.writelines('int n[{}] = {{{}}}; \n'.format(len(n_array),",".join([str(element) for element in n_array])))
+ file.writelines('int n_arrays = {};\n'.format(len(n_array)))
+
+# test%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+if __name__ == '__main__':
+
+ helper = Helper()
+ # n = np.arange(2,10)
+ n = np.logspace(1,3,3,base=2,dtype=(np.int))
+ C = helper.write_c_matrix(n)
diff --git a/buch/papers/multiplikation/code/meas/MM.txt b/buch/papers/multiplikation/code/meas/MM.txt
new file mode 100644
index 0000000..1a0cd5d
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/MM.txt
@@ -0,0 +1,12 @@
+0.000000,2
+0.000000,4
+0.000002,8
+0.000011,16
+0.000080,32
+0.000653,64
+0.005397,128
+0.045147,256
+0.487710,512
+3.964180,1024
+128.863544,2048
+996.370209,4096
diff --git a/buch/papers/multiplikation/code/meas/MM_dc.txt b/buch/papers/multiplikation/code/meas/MM_dc.txt
new file mode 100644
index 0000000..0d5580a
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/MM_dc.txt
@@ -0,0 +1,12 @@
+0.000006,2
+0.000007,4
+0.000035,8
+0.000228,16
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+0.007204,64
+0.034338,128
+0.267511,256
+2.131212,512
+17.177403,1024
+146.112874,2048
+1156.777565,4096
diff --git a/buch/papers/multiplikation/code/meas/blas.txt b/buch/papers/multiplikation/code/meas/blas.txt
new file mode 100644
index 0000000..6b7cd0b
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/blas.txt
@@ -0,0 +1,12 @@
+0.000001,2
+0.000000,4
+0.000001,8
+0.000003,16
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+0.000164,64
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+0.735149,1024
+6.895747,2048
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diff --git a/buch/papers/multiplikation/code/meas/strassen.txt b/buch/papers/multiplikation/code/meas/strassen.txt
new file mode 100644
index 0000000..89cf41a
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/strassen.txt
@@ -0,0 +1,12 @@
+0.000000,2
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+1.248713,512
+9.007700,1024
+61.079879,2048
+424.493037,4096
diff --git a/buch/papers/multiplikation/code/meas/test/4096/MM.txt b/buch/papers/multiplikation/code/meas/test/4096/MM.txt
new file mode 100644
index 0000000..25e40e1
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/test/4096/MM.txt
@@ -0,0 +1,12 @@
+0.000000,2
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+4.006544,1024
+124.427496,2048
+993.405615,4096
diff --git a/buch/papers/multiplikation/code/meas/test/4096/strassen.txt b/buch/papers/multiplikation/code/meas/test/4096/strassen.txt
new file mode 100644
index 0000000..eb2a496
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/test/4096/strassen.txt
@@ -0,0 +1,12 @@
+0.000007,2
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+1.198323,512
+8.421896,1024
+58.803644,2048
+415.115401,4096
diff --git a/buch/papers/multiplikation/code/meas/test/MM.txt b/buch/papers/multiplikation/code/meas/test/MM.txt
new file mode 100644
index 0000000..e0754ab
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/test/MM.txt
@@ -0,0 +1,14900 @@
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diff --git a/buch/papers/multiplikation/code/meas/test/blas.txt b/buch/papers/multiplikation/code/meas/test/blas.txt
new file mode 100644
index 0000000..7b0a9d1
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/test/blas.txt
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diff --git a/buch/papers/multiplikation/code/meas/test/winograd.txt b/buch/papers/multiplikation/code/meas/test/winograd.txt
new file mode 100644
index 0000000..d01fefd
--- /dev/null
+++ b/buch/papers/multiplikation/code/meas/test/winograd.txt
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new file mode 100644
index 0000000..3a4d88b
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diff --git a/buch/papers/multiplikation/code/test.tex b/buch/papers/multiplikation/code/test.tex
new file mode 100644
index 0000000..40ea239
--- /dev/null
+++ b/buch/papers/multiplikation/code/test.tex
@@ -0,0 +1,92 @@
+% This file was created by tikzplotlib v0.9.8.
+\begin{tikzpicture}
+
+\definecolor{color0}{rgb}{0.886274509803922,0.290196078431373,0.2}
+\definecolor{color1}{rgb}{0.203921568627451,0.541176470588235,0.741176470588235}
+\definecolor{color2}{rgb}{0.596078431372549,0.556862745098039,0.835294117647059}
+\definecolor{color3}{rgb}{0.984313725490196,0.756862745098039,0.368627450980392}
+
+\begin{axis}[
+axis background/.style={fill=white!89.8039215686275!black},
+axis line style={white},
+legend cell align={left},
+legend style={
+ fill opacity=0.8,
+ draw opacity=1,
+ text opacity=1,
+ at={(0.03,0.97)},
+ anchor=north west,
+ draw=white!80!black,
+ fill=white!89.8039215686275!black
+},
+tick align=outside,
+tick pos=left,
+x grid style={white},
+xlabel={n},
+xmajorgrids,
+xmin=-4.3, xmax=134.3,
+xtick style={color=white!33.3333333333333!black},
+y grid style={white},
+ylabel={time (s)},
+ymajorgrids,
+ymin=-0.0834965705871582, ymax=1.75356960296631,
+ytick style={color=white!33.3333333333333!black}
+]
+\addplot [line width=2pt, color0]
+table {%
+2 1.57356262207031e-05
+4 5.96046447753906e-05
+8 0.000428915023803711
+16 0.00276041030883789
+32 0.0217020511627197
+64 0.160412073135376
+128 1.3419406414032
+};
+\addlegendentry{Standard MM}
+\addplot [line width=2pt, color1]
+table {%
+2 6.43730163574219e-06
+4 6.69956207275391e-05
+8 0.00048065185546875
+16 0.00336766242980957
+32 0.0257236957550049
+64 0.231612205505371
+128 1.67006659507751
+};
+\addlegendentry{Divide and conquer MM}
+\addplot [line width=2pt, color2]
+table {%
+2 2.90870666503906e-05
+4 0.000133275985717773
+8 0.000703096389770508
+16 0.00453472137451172
+32 0.0282893180847168
+64 0.181003332138062
+128 1.40816903114319
+};
+\addlegendentry{Strassen MM}
+\addplot [line width=2pt, white!46.6666666666667!black]
+table {%
+2 2.19345092773438e-05
+4 9.01222229003906e-05
+8 0.000406503677368164
+16 0.00258469581604004
+32 0.0171687602996826
+64 0.126588344573975
+128 1.02698183059692
+};
+\addlegendentry{Winograd MM}
+\addplot [line width=2pt, color3]
+table {%
+2 1.45435333251953e-05
+4 1.1444091796875e-05
+8 7.39097595214844e-06
+16 1.28746032714844e-05
+32 2.83718109130859e-05
+64 0.000111103057861328
+128 0.000159025192260742
+};
+\addlegendentry{np MM}
+\end{axis}
+
+\end{tikzpicture}
diff --git a/buch/papers/multiplikation/einlteung.tex b/buch/papers/multiplikation/einlteung.tex
new file mode 100755
index 0000000..bc4bfcf
--- /dev/null
+++ b/buch/papers/multiplikation/einlteung.tex
@@ -0,0 +1,52 @@
+%
+% einleitung.tex -- Beispiel-File für die Einleitung
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\section{Einleitung \label{multiplikation:section:einleitung}}
+\rhead{Einleitung}
+
+Die Multiplikation zweier Matrizen ist eine wichtige Operation die in verschiedensten Teilen der Mathematik Anwendung findet.
+Die Beschreibung der Multiplikation aus der Definition 2.10 (\textcolor{blue} {Kein Hyperlink zu einer Definition?)}:
+
+Eine $m\times n$-Matrix $\mathbf{A}\in M_{m\times n}(\Bbbk)$ und eine
+$n\times p$-Matrix $\mathbf{B}\in M_{n\times l}(\Bbbk)$ haben als Produkt
+eine $n\times l$-Matrix $\mathbf{C}=\mathbf{AB}\in M_{n\times l}(\Bbbk)$ mit den
+Koeffizienten
+\begin{equation}
+c_{ij} = \sum_{k=1}^n a_{ik} b_{kj}.
+\label{multiplikation:eq:MM}
+\end{equation}
+Grafisch kann die Matrizenmultiplikation $AB=C$ wie in \ref{multiplikation:fig:mm_viz} visualisiert werden.
+\begin{figure}
+ \center
+ \includegraphics[]{papers/multiplikation/images/mm_visualisation}
+ \caption{Matrizen Multiplikation}
+ \label{multiplikation:fig:mm_viz}
+\end{figure}
+Im Fall einer Matrizengr\"osse von $2\times 2$
+\begin{equation}
+ \begin{bmatrix}
+A_{11} & A_{12}\\
+A_{21} & A_{22}
+\end{bmatrix}
+\begin{bmatrix}
+B_{11} & B_{12}\\
+B_{21} & B_{22}
+\end{bmatrix}
+=
+\begin{bmatrix}
+C_{11} & C_{12}\\
+C_{21} & C_{22}
+\end{bmatrix}
+\end{equation}
+kann die Gleichung der einzelnen Terme
+\begin{equation} \label{multiplikation:eq:MM_exp}
+\begin{split}
+C_{11} &= A_{11} \cdot B_{11} + A_{12} \cdot B_{21}\\
+C_{12} &= A_{11} \cdot B_{12} + A_{12} \cdot B_{22}\\
+C_{21} &= A_{21} \cdot B_{11} + A_{22} \cdot B_{21}\\
+C_{22} &= A_{21} \cdot B_{12} + A_{22} \cdot B_{22}
+\end{split}
+\end{equation}
+explizit geschrieben werden.
diff --git a/buch/papers/multiplikation/images/bigo.pdf b/buch/papers/multiplikation/images/bigo.pdf
new file mode 100644
index 0000000..dfa2ba4
--- /dev/null
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diff --git a/buch/papers/multiplikation/images/bigo.tex b/buch/papers/multiplikation/images/bigo.tex
new file mode 100644
index 0000000..e3293e4
--- /dev/null
+++ b/buch/papers/multiplikation/images/bigo.tex
@@ -0,0 +1,107 @@
+\documentclass[border=10pt,varwidth]{standalone}
+\usepackage[left=25mm,right=25mm,top=25mm,bottom=25mm]{geometry}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{times}
+\usepackage{geometry}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{mathrsfs}
+\usepackage{amsfonts}
+\usepackage{amsthm}
+\usepackage{lipsum}
+\usepackage{amscd}
+\usepackage{graphicx}
+\usepackage{fancyhdr}
+\usepackage{textcomp}
+\usepackage{pgfplots}
+\usepackage{txfonts}
+\usepackage[all]{xy}
+\usepackage{paralist}
+\usepackage[colorlinks=true]{hyperref}
+\usepackage{array}
+\usepackage{tikz}
+\usepackage{slashed}
+\usepackage{pdfpages}
+\usepackage{cite}
+\usepackage{url}
+\usepackage{amsmath,amsfonts,amssymb}
+\usepackage{tikz}
+\usetikzlibrary{arrows,matrix,positioning}
+\usetikzlibrary{overlay-beamer-styles}
+\usetikzlibrary{matrix.skeleton}
+\usetikzlibrary{automata,positioning}
+\usetikzlibrary{decorations.text}
+\usepackage{listings}
+\usepackage{multirow}
+\usepackage{color}
+
+\begin{document}
+
+\begin{tikzpicture}
+\begin{axis}[
+ axis lines = left,
+ xlabel = $n$ (Data Input),
+ ylabel = {$t$ (time)},
+ legend pos=north east,
+ very thick,
+ ymax = 500,
+ yticklabels=\empty,
+ xticklabels=\empty,
+ scale only axis=true,
+ width=12cm, height=6cm,
+ ]
+\addplot [
+ domain= 1:20,
+ samples=100,
+ color=red,
+]
+{1};
+\addlegendentry{$\mathcal{O}(1)$}
+\addplot [
+ domain= 1:20,
+ samples=100,
+ color=green,
+]
+{x};
+\addlegendentry{$\mathcal{O}(n)$}
+\addplot [
+ domain= 1:20,
+ samples=100,
+ color=blue,
+]
+{x^2};
+\addlegendentry{$\mathcal{O}(n^2)$}
+\addplot [
+ domain= 1:10,
+ samples=100,
+ color=purple,
+]
+{x^3};
+\addlegendentry{$\mathcal{O}(n^3)$}
+\addplot [
+ domain= 1:10,
+ samples=100,
+ color=black,
+]
+{exp(x)};
+\addlegendentry{$\mathcal{O}(e^n)$}
+\addplot [
+ domain= 1:20,
+ samples=100,
+ color=orange,
+]
+{log2(x)};
+\addlegendentry{$\mathcal{O}(\log n)$}
+
+\addplot [
+ domain= 1:20,
+ samples=100,
+ color=gray,
+]
+{x*log2(x)};
+\addlegendentry{$\mathcal{O}(n \log n)$}
+\end{axis}
+\end{tikzpicture}
+
+\end{document}
diff --git a/buch/papers/multiplikation/images/mm_visualisation.pdf b/buch/papers/multiplikation/images/mm_visualisation.pdf
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diff --git a/buch/papers/multiplikation/images/mm_visualisation.tex b/buch/papers/multiplikation/images/mm_visualisation.tex
new file mode 100644
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--- /dev/null
+++ b/buch/papers/multiplikation/images/mm_visualisation.tex
@@ -0,0 +1,45 @@
+
+ \begin{tikzpicture}[ampersand replacement=\&]
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+ \vdots \& \& \vdots \& \& \vdots \\
+ A_{m,1} \& \cdots \& A_{m,k} \& \cdots \& A_{m,n} \\
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+ \vdots \& \& \vdots \& \& \vdots \\
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+ };
+
+
+ \node[opacity=0.5, rounded corners=2pt, inner sep=-1pt, fill=green, fit=(A-3-1)(A-3-5)] {};
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+
+ \end{tikzpicture}
+
+\end{document}
+
diff --git a/buch/papers/multiplikation/images/strassen.pdf b/buch/papers/multiplikation/images/strassen.pdf
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diff --git a/buch/papers/multiplikation/images/strassen.tex b/buch/papers/multiplikation/images/strassen.tex
new file mode 100644
index 0000000..797772b
--- /dev/null
+++ b/buch/papers/multiplikation/images/strassen.tex
@@ -0,0 +1,140 @@
+\documentclass[border=10pt]{standalone}
+\usepackage[left=25mm,right=25mm,top=25mm,bottom=25mm]{geometry}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{times}
+\usepackage{geometry}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{mathrsfs}
+\usepackage{amsfonts}
+\usepackage{amsthm}
+\usepackage{lipsum}
+\usepackage{amscd}
+\usepackage{graphicx}
+\usepackage{fancyhdr}
+\usepackage{textcomp}
+\usepackage{pgfplots}
+\usepackage{txfonts}
+\usepackage[all]{xy}
+\usepackage{paralist}
+\usepackage[colorlinks=true]{hyperref}
+\usepackage{array}
+\usepackage{tikz}
+\usepackage{slashed}
+\usepackage{pdfpages}
+\usepackage{cite}
+\usepackage{url}
+\usepackage{amsmath,amsfonts,amssymb}
+\usepackage{tikz}
+\usetikzlibrary{arrows,matrix,positioning}
+\usetikzlibrary{overlay-beamer-styles}
+\usetikzlibrary{matrix.skeleton}
+\usetikzlibrary{automata,positioning}
+\usetikzlibrary{decorations.text}
+\usepackage{listings}
+\usepackage{multirow}
+\usepackage{color}
+
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+
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+ \node at (10,-2) {II};
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+ \node at (20,-2) {IV};
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+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M42-1-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M42-1-3)] {};
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+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M46-3-1)] {};
+\end{tikzpicture}
+
+\end{document}
diff --git a/buch/papers/multiplikation/loesungsmethoden.tex b/buch/papers/multiplikation/loesungsmethoden.tex
new file mode 100755
index 0000000..83be814
--- /dev/null
+++ b/buch/papers/multiplikation/loesungsmethoden.tex
@@ -0,0 +1,309 @@
+%
+% teil2.tex -- Beispiel-File für teil2
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+
+\section{L\"osungsmethoden}
+\rhead{L\"osungsmethoden}
+
+In diesem Abschnitt werden mehrere Algorithmen zur Berechnung der Matrizenmultiplikation vorgestellt, auch werden Libraries zur automatisierten Verwendung von vordefinierten Algorithmen gezeigt.
+
+\subsection{Standard Algorithmus}
+
+Der Standard Methode kann im Algorithmus \ref{multiplikation:alg:smm} entnommen werden.
+Hierf\"ur wurde die Gleichung \eqref{multiplikation:eq:MM} direkt implementiert.
+Die \texttt{For i} Schleife iteriert \"uber alle Zeilen der $\mathbf{A}$ Matrix, die \texttt{For j} Schleife iteriert \"uber alle Spalten der $\mathbf{B}$ Matrix und die \texttt{For k} Schleife iteriert \"uber alle Eintr\"age dieser Zeilen bzw. Spalten.
+
+\begin{algorithm}\caption{Matrix Multiplication}
+ \label{multiplikation:alg:smm}
+ \setlength{\lineskip}{7pt}
+ \begin{algorithmic}[1]
+ \Function{MM}{$\textbf{A}, \textbf{B}$}
+ \State $sum \gets 0$
+ \State $n \gets columns(\textbf{A}) == rows(\textbf{B})$
+ \State $m \gets rows(\textbf{A})$
+ \State $p \gets columns(\textbf{B})$
+ \State $\textbf{C} \gets zeros(m,p)$
+ \For{$i = 0,1,2 \dots,m-1$}
+ \For{$j = 0,1,2 \dots,p-1$}
+ \State $sum \gets 0$
+ \For{$k = 0,1,2 \dots,n-1$}
+ \State $sum \gets sum + \textbf{A}[i][k] \cdot \textbf{B}[k][j]$
+ \EndFor
+ \State $\textbf{C}[i][j] \gets sum $
+ \EndFor
+ \EndFor
+ \State \textbf{return} $\textbf{C}$
+ \EndFunction
+ \end{algorithmic}
+\end{algorithm}
+
+Die Laufzeit dieser Struktur mit drei \texttt{For} Schleifen ist $\mathcal{O}(n^3)$
+
+\subsubsection{Divide and Conquer Methode}
+
+F\"ur gewisse Algorithmen f\"uhren \textit{Divide and Conquer} Ans\"atze zu markant besseren Laufzeiten.
+Das bekannteste Beispiel ist wohl die \textit{Fast Fourier Transform} wobei die Laufzeit von $\mathcal{O}(n^2)$ zu $\mathcal{O}(n \log n)$ verbessert werden kann.
+
+Die Matrizenmultiplikation kann ebenfalls mit solch einem Ansatz berechnet werden.
+Zur vereinfachten Veranschaulichung kann die Situation, mit $\mathbf{A}$ und $\mathbf{B}$ der gr\"osse $2^n \times 2^n$ verwendet werden.
+Die Matrizen $\mathbf{A}$ und $\mathbf{B}$ werden in jeweils vier Blockmatrizen der gr\"osse $2^{n-1} \times 2^{n-1}$
+\begin{equation}
+\mathbf{A}\mathbf{B}=
+\begin{bmatrix}
+\mathbf{A}_{11} & \mathbf{A}_{12}\\
+\mathbf{A}_{21} & \mathbf{A}_{22}
+\end{bmatrix}
+\begin{bmatrix}
+\mathbf{B}_{11} & \mathbf{B}_{12}\\
+\mathbf{B}_{21} & \mathbf{B}_{22}
+\end{bmatrix}
+=
+\begin{bmatrix}
+\mathbf{C}_{11} & \mathbf{C}_{12}\\
+\mathbf{C}_{21} & \mathbf{C}_{22}
+\end{bmatrix}
+\end{equation}
+aufgeteilt.
+Die Berechnung
+\begin{equation}
+\mathbf{C}_{ij} = \sum_{k=1}^n \mathbf{A}_{ik} \mathbf{B}_{kj}
+\label{multiplikation:eq:MM_block}
+\end{equation}
+ist identisch zu der Gleichung \eqref{multiplikation:eq:MM}, wobei hier f\"ur die Multiplikation die Matrizenmultiplikation verwendet wird.
+
+Der Algorithmus \ref{multiplikation:alg:devide_mm} zeigt den \textit{Divide and Conquer} Ansatz,
+Der Grundstruktur dieser Methode besteht aus dem rekursiven Aufruf der Funktion mit den erzeugten Blockmatrizen.
+Der rekursive Aufruf wird bis zu der Gr\"osse der Matrizen von $N = 2 \times 2$ durchgef\"uhrt.
+\begin{algorithm}\caption{Divide and Conquer Matrix Multiplication}
+ \setlength{\lineskip}{7pt}
+ \label{multiplikation:alg:devide_mm}
+ \begin{algorithmic}
+ \Function{MM}{$\textbf{A}, \textbf{B}, n$}
+ \If{$n = 2$}
+ \State $ \mathbf{C} \gets zeros(n, n)$
+ \State $C[0, 0] \gets A[0][0]\cdot B[0][0]+A[0][1]\cdot B[1][0]$
+ \State $C[0, 1] \gets A[0][0]\cdot B[0][1]+A[0][1]\cdot B[1][1]$
+ \State $C[1, 0] \gets A[1][0]\cdot B[0][0]+A[1][1]\cdot B[1][0]$
+ \State $C[1, 1] \gets A[1][0]\cdot B[0][1]+A[1][1]\cdot B[1][1]$
+ \Else
+ \State $ m \gets n/2$
+ \State $\mathbf{A11}, \mathbf{A12}, \mathbf{A21}, \mathbf{A22} \gets \mathbf{A}[:m][:m], \mathbf{A}[:m][m:], \mathbf{A}[m:][:m], \mathbf{A}[m:][m:]$
+ \State $\mathbf{B11}, \mathbf{B12}, \mathbf{B21}, \mathbf{B22} \gets \mathbf{B}[:m][:m], \mathbf{B}[:m][m:], \mathbf{B}[m:][:m], \mathbf{B}[m:][m:]$
+
+ \State $\mathbf{C11} \gets \text{MM}(\mathbf{A11}, \mathbf{B11},n) + \text{MM}(\mathbf{A12}, \mathbf{B21},n)$
+ \State $\mathbf{C12} \gets \text{MM}(\mathbf{A11},\mathbf{B12},n) + \text{MM}(\mathbf{A12}, \mathbf{B22},n)$
+ \State $\mathbf{C21} \gets \text{MM}(\mathbf{A21}, \mathbf{B11},n) + \text{MM}(\mathbf{A22}, \mathbf{B21},n)$
+ \State $\mathbf{C22} \gets \text{MM}(\mathbf{A21}, \mathbf{B12},n) + \text{MM}(\mathbf{A22}, \mathbf{B22},n)$
+ \State $ C \gets vstack(hstack(C11, C12), hstack(C21, C22))$
+
+ \EndIf
+ \State \textbf{return} $\textbf{C}$
+
+ \EndFunction
+ \end{algorithmic}
+\end{algorithm}
+
+Die Laufzeit dieser rekursiven Funktion kann mit dem \textit{Master Theorem} berechnet werden.
+Ohne auf diesen vertieft einzugehen, bestimmt die Anzahl rekursiver Aufrufe der Funktion die Laufzeit.
+In diesem Fall wird die Funktion pro Durchlauf acht mal rekursiv aufgerufen, dies f\"uhrt
+\begin{equation} \label{multiplikation:eq:laufzeitdac}
+ \mathcal{T}(n) =
+ \begin{cases}
+ 1 & \text{if } n \leq 2\\
+ 8 \cdot \mathcal{T}(\frac{n}{2}) + n^2 & \text{if } n > 2
+ \end{cases} = \mathcal{O}(n^{\log_2 8}) = \mathcal{O}(n^{3})
+\end{equation}
+zu einer kubischen Laufzeit.
+Die Addition zweier Matrizen $\mathbf{A} + \mathbf{B} = \mathbf{C}$ hat eine Laufzeit von $\mathcal{O}(n^{2})$ und kann neben dem dominierendem Anteil von $\mathcal{O}(n^{3})$ ignoriert werden.
+In diesem Fall hat der \textit{Divide and Conquer} Ansatz zu keiner Verbesserung gef\"uhrt.
+
+
+\subsection{Strassen's Algorithmus}
+
+Strassen's Algorithmus \cite{multiplikation:strassen_1969} beschreibt die Matrizenmultiplikation mit einer Vielzahl von Additionen, Subtraktionen und Multiplikationen.
+Die Grundlegenden Terme
+\begin{equation} \label{multiplikation:eq:strassen}
+\begin{split}
+\text{\textbf{P}} &= (\mathbf{A}_{11} + \mathbf{A}_{22}) \cdot (\mathbf{B}_{11} + \mathbf{B}_{22}) \\
+\text{\textbf{Q}} &= (\mathbf{A}_{21} + \mathbf{A}_{22}) \cdot \mathbf{B}_{11} \\
+\text{\textbf{R}} &= \mathbf{A}_{11} \cdot (\mathbf{B}_{12}-\mathbf{B}_{22}) \\
+\text{\textbf{S}} &= \mathbf{A}_{22} \cdot (-\mathbf{B}_{11}+\mathbf{B}_{21}) \\
+\text{\textbf{T}} &= (\mathbf{A}_{11} + \mathbf{A}_{12}) \cdot \mathbf{B}_{22} \\
+\text{\textbf{U}} &= (-\mathbf{A}_{11} + \mathbf{A}_{21}) \cdot (\mathbf{B}_{11} + \mathbf{B}_{12}) \\
+\text{\textbf{V}} &= (\mathbf{A}_{12} - \mathbf{A}_{22}) \cdot (\mathbf{B}_{21} + \mathbf{B}_{22})
+\end{split}
+\end{equation}
+aus $\mathbf{A}$ und $\mathbf{B}$, werden f\"ur die Berechnung der Matrix $\mathbf{C}$
+\begin{equation} \label{multiplikation:eq:strassen2}
+\begin{split}
+\mathbf{C}_{11} &= \text{\textbf{P}} + \text{\textbf{S}} - \text{\textbf{T}} + \text{\textbf{V}} \\
+\mathbf{C}_{21} &= \text{\textbf{R}} + \text{\textbf{T}} \\
+\mathbf{C}_{12} &= \text{\textbf{Q}} + \text{\textbf{S}}\\
+\mathbf{C}_{22} &= \text{\textbf{P}} + \text{\textbf{R}} - \text{\textbf{Q}} + \text{\textbf{U}}
+\end{split}
+\end{equation}
+gebraucht.
+\begin{algorithm}\caption{Strassen Matrix Multiplication}
+ \label{multiplikation:alg:strassen}
+ \setlength{\lineskip}{7pt}
+ \begin{algorithmic}
+ \Function{strassen}{$\textbf{A}, \textbf{B}, n$}
+ \If{$n = 2$}
+ \State $ \mathbf{C} \gets zeros((n, n))$
+ \State $P \gets (A[0][0]+A[1][1])\cdot( B[0][0]+B[1][1])$
+ \State $Q \gets (A[1][0]+A[1][1])\cdot B[0][0]$
+ \State $R \gets A[0][0]\cdot (B[0][1]-B[1][1])$
+ \State $S \gets A[1][1]\cdot (B[1][0]-B[0][0])$
+ \State $T \gets (A[0][0]+A[0][1])\cdot B[1][1]$
+ \State $U \gets (A[1][0]-A[0][0])\cdot (B[0][0]+B[0][1])$
+ \State $V \gets (A[0][1]-A[1][1])\cdot (B[1][0]+B[1][1])$
+ \State $C[0][0] \gets P+S-T+V$
+ \State $C[0][1] \gets R+T$
+ \State $C[1][0] \gets Q+S$
+ \State $C[1][1] \gets P+R-Q+U$
+ \Else
+ \State $ m \gets n/2$
+ \State $\mathbf{A11}, \mathbf{A12}, \mathbf{A21}, \mathbf{A22} \gets \mathbf{A}[:m][:m], \mathbf{A}[:m][m:], \mathbf{A}[m:][:m], \mathbf{A}[m:][m:]$
+ \State $\mathbf{B11}, \mathbf{B12}, \mathbf{B21}, \mathbf{B22} \gets \mathbf{B}[:m][:m], \mathbf{B}[:m][m:], \mathbf{B}[m:][:m], \mathbf{B}[m:][m:]$
+
+ \State $ \mathbf{P} \gets \text{strassen}((\mathbf{A11}+ \mathbf{A22}),(\mathbf{B11}+\mathbf{B22}), m)$
+ \State $ \mathbf{Q} \gets \text{strassen}((\mathbf{A21}+ \mathbf{A22}), \mathbf{B11},m)$
+ \State $ \mathbf{R} \gets \text{strassen}( \mathbf{A11},(\mathbf{B12}- \mathbf{B22}),m)$
+ \State $ \mathbf{S} \gets \text{strassen}( \mathbf{A22},(\mathbf{B21}- \mathbf{B11}),m)$
+ \State $ \mathbf{T} \gets \text{strassen}((\mathbf{A11}+ \mathbf{A12}), \mathbf{B22},m)$
+ \State $ \mathbf{U} \gets \text{strassen}((\mathbf{A21}- \mathbf{A11}),(\mathbf{B11}+\mathbf{B12}),m)$
+ \State $ \mathbf{V} \gets \text{strassen}((\mathbf{A12}- \mathbf{A22}),(\mathbf{B21}+\mathbf{B22}),m)$
+
+
+
+ \State $\mathbf{C11} \gets \mathbf{P+S-T+V}$
+ \State $\mathbf{C12} \gets \mathbf{R+T}$
+ \State $\mathbf{C21} \gets \mathbf{Q+S}$
+ \State $\mathbf{C22} \gets \mathbf{P+R-Q+U}$
+ \State $ C \gets vstack(hstack(C11, C12), hstack(C21, C22))$
+
+ \EndIf
+ \State \textbf{return} $\textbf{C}$
+
+ \EndFunction
+ \end{algorithmic}
+\end{algorithm}
+Strassens's Methode wird in der Abbildung \ref{multiplikation:fig:strassen} grafisch dargestellt.
+\begin{figure}
+ \center
+ \includegraphics[width=\linewidth]{papers/multiplikation/images/strassen.pdf}
+ \caption{Strassen's Algorithmus}
+ \label{multiplikation:fig:strassen}
+\end{figure}
+
+Die Funktion wird sieben mal rekursiv aufgerufen.
+Dies f\"uhrt zu einer Laufzeit von
+\begin{equation} \label{multiplikation:eq:laufzeitstrassen}
+\mathcal{T}(n) =
+\begin{cases}
+1 & \text{if } n \leq 2\\
+7 \cdot \mathcal{T}(\frac{n}{2}) + n^2 & \text{if } n > 2
+\end{cases} = \mathcal{O}(n^{\log_2 7}) = \mathcal{O}(n^{2.8074})
+\end{equation}
+und ist somit schneller als die Standard Methode.
+
+\subsection{Winograd's Algorithmus}
+
+Ein weiterer Ansatz lieferte Shmuel Winograd im Jahre 1968 \cite{multiplikation:winograd_1968}.
+Er zeigte einen neuen Algorithmus f\"ur das
+\begin{equation}
+ \langle x,y \rangle = \sum_{i=1}^{n}x_i y_i
+\end{equation}
+Skalarprodukt.
+F\"ur jeden Vektor berechne
+\begin{equation}
+ \xi = \sum_{j=1}^{ \lfloor n/2 \rfloor} x_{2j-1} \cdot x_{2j}
+\end{equation}
+und
+\begin{equation}
+ \eta = \sum_{j=1}^{ \lfloor n/2 \rfloor} y_{2j-1} \cdot y_{2j}.
+\end{equation}
+Das Skalarprodukt ist nun geben mit
+\begin{equation}
+ \langle x,y \rangle =
+ \begin{cases}
+ \displaystyle \quad \sum_{j=1}^{ \lfloor n/2 \rfloor} (x_{2j-1} + y_{2j})(x_{2j}+y_{2j-1})-\xi - \eta & \text{if $n$ is even}\\
+ \displaystyle \quad \sum_{j=1}^{ \lfloor n/2 \rfloor} (x_{2j-1} + y_{2j})(x_{2j}+y_{2j-1})-\xi - \eta + x_n y_n & \text{if $n$ is odd}.
+ \end{cases}
+\end{equation}
+
+Angenommen man hat $N$ Vektoren mit welchen man $T$ Skalarprodukte berechnen m\"ochte.
+Daf\"ur werden $N\lfloor n/2 \rfloor + T\lfloor (n+1)/2 \rfloor $ Multiplikationen ben\"otigt.
+Eine Matrizenmultiplikation mit $\mathbf{A}$ einer $m \times n$ und $\mathbf{B}$ einer $n \times p$ Matrix, entspricht $N=m+p$ Vektoren mit welchen man $T=mp$ Skalarprodukte berechnet.
+Dies f\"uhrt zu
+\begin{equation}
+ (m+p) \left \lfloor \frac{n}{2} \right \rfloor + mp \left \lfloor \frac{n+1}{2} \right \rfloor = \frac{mn}{2} + \frac{pn}{2} + \frac{mpn}{2} + \frac{mp}{2}
+\end{equation}
+Multiplikationen.
+Wenn $m,p,n$ gross werden, dominiert der Term $\frac{mpn}{2}$ und es werden $\frac{mpn}{2}$ Multiplikationen ben\"otigt.
+Was im Vergleich zu den $mpn$ Multiplikation der Standard Methode nur die H\"alfte ist.
+Die Implementation kann im Algorithmus \ref{multiplikation:alg:winograd} entnommen werden.
+
+\begin{algorithm}\caption{Winograd Matrix Multiplication}
+ \setlength{\lineskip}{7pt}
+ \label{multiplikation:alg:winograd}
+ \begin{algorithmic}
+ \Function{Winograd}{$\textbf{A}, \textbf{B}, n$}
+ \State $ m \gets rows(\mathbf{A})$
+ \State $ n \gets columns(\mathbf{A}) == rows(\mathbf{B})$
+ \State $ p \gets columns(\mathbf{B})$
+ \State $ \mathbf{\xi} \gets zeros(m)$
+ \State $ \mathbf{\eta} \gets zeros(p)$
+
+
+ \For{$i = 0,1,2 \dots,m-1$}
+ \For{$j = 0,1,2 \dots,\lfloor n/2 \rfloor-1$}
+ \State $\xi[i] \gets \xi[i]+A[i,2 j]A[i,2 j+1]$
+ \EndFor
+ \EndFor
+
+ \For{$i = 0,1,2 \dots,p-1$}
+ \For{$j = 0,1,2 \dots,\lfloor n/2 \rfloor-1$}
+ \State $\eta[i] \gets \eta[i]+B[2 j,i]B[2 j+1,i]$
+ \EndFor
+ \EndFor
+
+ \If{$n \% 2 == 0$}
+ \For{$i = 0,1,2 \dots,m-1$}
+ \For{$j = 0,1,2 \dots,p-1$}
+ \State $ab \gets 0$
+ \For{$k = 0,1,2 \dots,\lfloor n/2 \rfloor-1$}
+ \State $ab \gets ab + (A[i,2k]+B[2k+1,j])(A[i,2k+1]+B[2k,j])$
+ \EndFor
+ \State $C[i,j] \gets ab-\eta[j]-\xi[i]$
+ \EndFor
+ \EndFor
+ \Else
+ \For{$i = 0,1,2 \dots,n-1$}
+ \For{$j = 0,1,2 \dots,n-1$}
+ \State $ab \gets 0$
+ \For{$k = 0,1,2 \dots,\lfloor n/2 \rfloor-1$}
+ \State $ab \gets ab + (A[i,2k]+B[2k+1,j])(A[i,2k+1]+B[2k,j])$
+ \EndFor
+ \State $C[i,j] \gets ab-\eta[j]-\xi[i]+A[i,-1]B[-1,j]$
+ \EndFor
+ \EndFor
+ \EndIf
+ \State \textbf{return} $\textbf{C}$
+
+ \EndFunction
+ \end{algorithmic}
+\end{algorithm}
+
+\subsection{Weitere Algorithmen}
+
+\textcolor{red}{TODO: BLAS}
+
+\section{Implementation}
+\rhead{Implementation}
+\textcolor{red}{TODO: messresultate}
+
+\section{Fazit}
+\rhead{Fazit}
diff --git a/buch/papers/multiplikation/main.tex b/buch/papers/multiplikation/main.tex
index 42f2768..8d0a8df 100644..100755
--- a/buch/papers/multiplikation/main.tex
+++ b/buch/papers/multiplikation/main.tex
@@ -1,36 +1,18 @@
+% !TEX root = ../../buch.tex
%
% main.tex -- Paper zum Thema <multiplikation>
%
-% (c) 2020 Hochschule Rapperswil
+% (c) 2021 Hochschule Rapperswil
%
-\chapter{Thema\label{chapter:multiplikation}}
-\lhead{Thema}
+\chapter{Schnelle Matrizen Multiplikation\label{chapter:multiplikation}}
+\lhead{FMM}
\begin{refsection}
-\chapterauthor{Hans Muster}
+\chapterauthor{Michael Schmid}
-Ein paar Hinweise für die korrekte Formatierung des Textes
-\begin{itemize}
-\item
-Absätze werden gebildet, indem man eine Leerzeile einfügt.
-Die Verwendung von \verb+\\+ ist nur in Tabellen und Arrays gestattet.
-\item
-Die explizite Platzierung von Bildern ist nicht erlaubt, entsprechende
-Optionen werden gelöscht.
-Verwenden Sie Labels und Verweise, um auf Bilder hinzuweisen.
-\item
-Beginnen Sie jeden Satz auf einer neuen Zeile.
-Damit ermöglichen Sie dem Versionsverwaltungssysteme, Änderungen
-in verschiedenen Sätzen von verschiedenen Autoren ohne Konflikt
-anzuwenden.
-\item
-Bilden Sie auch für Formeln kurze Zeilen, einerseits der besseren
-Ãœbersicht wegen, aber auch um GIT die Arbeit zu erleichtern.
-\end{itemize}
-\input{papers/multiplikation/teil0.tex}
-\input{papers/multiplikation/teil1.tex}
-\input{papers/multiplikation/teil2.tex}
-\input{papers/multiplikation/teil3.tex}
+\input{papers/multiplikation/einlteung.tex}
+\input{papers/multiplikation/problemstellung.tex}
+\input{papers/multiplikation/loesungsmethoden.tex}
\printbibliography[heading=subbibliography]
\end{refsection}
diff --git a/buch/papers/multiplikation/packages.tex b/buch/papers/multiplikation/packages.tex
index e4173c0..e4173c0 100644..100755
--- a/buch/papers/multiplikation/packages.tex
+++ b/buch/papers/multiplikation/packages.tex
diff --git a/buch/papers/multiplikation/papers/Strassen_GPU.pdf b/buch/papers/multiplikation/papers/Strassen_GPU.pdf
new file mode 100755
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--- /dev/null
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diff --git a/buch/papers/multiplikation/papers/Strassen_original_1969.pdf b/buch/papers/multiplikation/papers/Strassen_original_1969.pdf
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index 0000000..b647fc0
--- /dev/null
+++ b/buch/papers/multiplikation/papers/Strassen_original_1969.pdf
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diff --git a/buch/papers/multiplikation/papers/assay_fast_MM.pdf b/buch/papers/multiplikation/papers/assay_fast_MM.pdf
new file mode 100755
index 0000000..3cd6b63
--- /dev/null
+++ b/buch/papers/multiplikation/papers/assay_fast_MM.pdf
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diff --git a/buch/papers/multiplikation/papers/strassen_video.txt b/buch/papers/multiplikation/papers/strassen_video.txt
new file mode 100755
index 0000000..f84122c
--- /dev/null
+++ b/buch/papers/multiplikation/papers/strassen_video.txt
@@ -0,0 +1 @@
+https://www.youtube.com/watch?v=0oJyNmEbS4w
diff --git a/buch/papers/multiplikation/papers/winograd_original.pdf b/buch/papers/multiplikation/papers/winograd_original.pdf
new file mode 100755
index 0000000..a7aba36
--- /dev/null
+++ b/buch/papers/multiplikation/papers/winograd_original.pdf
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diff --git a/buch/papers/multiplikation/presentation/common.tex b/buch/papers/multiplikation/presentation/common.tex
new file mode 100644
index 0000000..200d244
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/common.tex
@@ -0,0 +1,79 @@
+%
+% common.tex -- gemeinsame Definitionen
+%
+% (c) 2021 Michael Schmid, OST Campus Rapperswil
+%
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{epic}
+\usepackage{color}
+\usepackage{array}
+\usepackage{algorithm}
+\usepackage{ifthen}
+\usepackage{adjustbox}
+\usepackage[noend]{algpseudocode}
+\usepackage{neuralnetwork}
+\usepackage{amsmath}
+\usepackage{lmodern}
+\usepackage{tikz}
+\usetikzlibrary{decorations.text}
+\usetikzlibrary{arrows,matrix,positioning}
+\usetikzlibrary{overlay-beamer-styles}
+\usetikzlibrary{matrix.skeleton}
+\usepackage{pgfplots}
+\usepackage{listings}
+\usepackage{svg}
+
+\definecolor{codegreen}{rgb}{0,0.6,0}
+\definecolor{codegray}{rgb}{0.5,0.5,0.5}
+\definecolor{codepurple}{rgb}{0.58,0,0.82}
+\definecolor{backcolour}{rgb}{0.95,0.95,0.92}
+\definecolor{ost}{rgb}{164,0,136}
+
+\lstdefinestyle{mystyle}{
+ backgroundcolor=\color{backcolour},
+ commentstyle=\color{codegreen},
+ keywordstyle=\color{magenta},
+ numberstyle=\tiny\color{codegray},
+ stringstyle=\color{codepurple},
+ basicstyle=\footnotesize,
+ breakatwhitespace=false,
+ breaklines=true,
+ captionpos=b,
+ keepspaces=true,
+ numbers=left,
+ numbersep=2pt,
+ showspaces=false,
+ showstringspaces=false,
+ showtabs=false,
+ tabsize=2
+}
+
+\usetikzlibrary{fit}
+\tikzset{%
+ highlight/.style={rectangle,rounded corners,fill=red!15,draw,fill opacity=0.5,inner sep=0pt}
+}
+\newcommand{\tikzmark}[2]{\tikz[overlay,remember picture,baseline=(#1.base)] \node (#1) {#2};}
+%
+\newcommand{\Highlight}[1][submatrix]{%
+ \tikz[overlay,remember picture]{
+ \node[highlight,fit=(left.north west) (right.south east)] (#1) {};}
+}
+
+
+\lstset{style=mystyle}
+\lstdefinestyle{mystyle}{
+ morekeywords={cwt,contourf,datetick}
+}
+
+
+\usetikzlibrary{shapes.geometric}
+\mode<beamer>{%
+\usetheme[]{Frankfurt}}
+\beamertemplatenavigationsymbolsempty
+\title[]{Fast Matrix Multiplication}
+\author[]{Michael Schmid}
+\usecolortheme[named=ost]{structure}
+
+\date[]{31.05.2021}
+\newboolean{presentation}
diff --git a/buch/papers/multiplikation/presentation/presentation.nav b/buch/papers/multiplikation/presentation/presentation.nav
new file mode 100644
index 0000000..2a01568
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/presentation.nav
@@ -0,0 +1,59 @@
+\headcommand {\slideentry {0}{0}{1}{1/1}{}{0}}
+\headcommand {\beamer@framepages {1}{1}}
+\headcommand {\beamer@sectionpages {1}{1}}
+\headcommand {\beamer@subsectionpages {1}{1}}
+\headcommand {\sectionentry {1}{Big $\mathcal {O}$}{2}{Big $\mathcal {O}$}{0}}
+\headcommand {\slideentry {1}{0}{1}{2/4}{}{0}}
+\headcommand {\beamer@framepages {2}{4}}
+\headcommand {\slideentry {1}{0}{2}{5/6}{}{0}}
+\headcommand {\beamer@framepages {5}{6}}
+\headcommand {\slideentry {1}{0}{3}{7/8}{}{0}}
+\headcommand {\beamer@framepages {7}{8}}
+\headcommand {\slideentry {1}{0}{4}{9/10}{}{0}}
+\headcommand {\beamer@framepages {9}{10}}
+\headcommand {\slideentry {1}{0}{5}{11/12}{}{0}}
+\headcommand {\beamer@framepages {11}{12}}
+\headcommand {\slideentry {1}{0}{6}{13/13}{}{0}}
+\headcommand {\beamer@framepages {13}{13}}
+\headcommand {\slideentry {1}{0}{7}{14/14}{}{0}}
+\headcommand {\beamer@framepages {14}{14}}
+\headcommand {\beamer@sectionpages {2}{14}}
+\headcommand {\beamer@subsectionpages {2}{14}}
+\headcommand {\sectionentry {2}{Strassen's Algorithm}{15}{Strassen's Algorithm}{0}}
+\headcommand {\slideentry {2}{0}{1}{15/15}{}{0}}
+\headcommand {\beamer@framepages {15}{15}}
+\headcommand {\slideentry {2}{0}{2}{16/18}{}{0}}
+\headcommand {\beamer@framepages {16}{18}}
+\headcommand {\slideentry {2}{0}{3}{19/19}{}{0}}
+\headcommand {\beamer@framepages {19}{19}}
+\headcommand {\slideentry {2}{0}{4}{20/20}{}{0}}
+\headcommand {\beamer@framepages {20}{20}}
+\headcommand {\slideentry {2}{0}{5}{21/23}{}{0}}
+\headcommand {\beamer@framepages {21}{23}}
+\headcommand {\slideentry {2}{0}{6}{24/24}{}{0}}
+\headcommand {\beamer@framepages {24}{24}}
+\headcommand {\slideentry {2}{0}{7}{25/25}{}{0}}
+\headcommand {\beamer@framepages {25}{25}}
+\headcommand {\slideentry {2}{0}{8}{26/26}{}{0}}
+\headcommand {\beamer@framepages {26}{26}}
+\headcommand {\slideentry {2}{0}{9}{27/29}{}{0}}
+\headcommand {\beamer@framepages {27}{29}}
+\headcommand {\slideentry {2}{0}{10}{30/32}{}{0}}
+\headcommand {\beamer@framepages {30}{32}}
+\headcommand {\beamer@sectionpages {15}{32}}
+\headcommand {\beamer@subsectionpages {15}{32}}
+\headcommand {\sectionentry {3}{Measurements}{33}{Measurements}{0}}
+\headcommand {\slideentry {3}{0}{1}{33/40}{}{0}}
+\headcommand {\beamer@framepages {33}{40}}
+\headcommand {\slideentry {3}{0}{2}{41/49}{}{0}}
+\headcommand {\beamer@framepages {41}{49}}
+\headcommand {\beamer@sectionpages {33}{49}}
+\headcommand {\beamer@subsectionpages {33}{49}}
+\headcommand {\sectionentry {4}{How To Matrix Multiply}{50}{How To Matrix Multiply}{0}}
+\headcommand {\slideentry {4}{0}{1}{50/50}{}{0}}
+\headcommand {\beamer@framepages {50}{50}}
+\headcommand {\beamer@partpages {1}{50}}
+\headcommand {\beamer@subsectionpages {50}{50}}
+\headcommand {\beamer@sectionpages {50}{50}}
+\headcommand {\beamer@documentpages {50}}
+\headcommand {\gdef \inserttotalframenumber {21}}
diff --git a/buch/papers/multiplikation/presentation/presentation.pdf b/buch/papers/multiplikation/presentation/presentation.pdf
new file mode 100644
index 0000000..842e68c
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/presentation.pdf
Binary files differ
diff --git a/buch/papers/multiplikation/presentation/presentation.snm b/buch/papers/multiplikation/presentation/presentation.snm
new file mode 100644
index 0000000..e69de29
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/presentation.snm
diff --git a/buch/papers/multiplikation/presentation/presentation.tex b/buch/papers/multiplikation/presentation/presentation.tex
new file mode 100644
index 0000000..2a4af45
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/presentation.tex
@@ -0,0 +1,12 @@
+%
+% MathSem-yyy-xxx.tex -- Präsentation
+%
+% (c) 2021 Michael Schmid, OST campus Rapperswil
+%
+
+\documentclass[aspectratio=169]{beamer}
+\input{common.tex}
+%\setboolean{presentation}{true}
+\begin{document}
+\input{slides/slides.tex}
+\end{document}
diff --git a/buch/papers/multiplikation/presentation/slides/algo.tex b/buch/papers/multiplikation/presentation/slides/algo.tex
new file mode 100644
index 0000000..0c3d130
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/slides/algo.tex
@@ -0,0 +1,111 @@
+\begin{frame}
+ \frametitle{Algorithm}
+ \begin{columns}
+ \begin{column}{0.6\textwidth}
+ \begin{algorithm}[H]\caption{Square Matrix Multiplication}
+ \setlength{\lineskip}{7pt}
+ \begin{algorithmic}[1]
+ \Function{MM}{$\textbf{A}, \textbf{B}, \textbf{C}$}
+ \State $sum \gets 0$
+ \State $n \gets columns(\textbf{A}) == rows(\textbf{B})$
+ \State $m \gets rows(\textbf{A})$
+ \State $p \gets columns(\textbf{B})$
+
+ \For{$i = 0,1,2 \dots,m-1$}
+ \For{$j = 0,1,2 \dots,p-1$}
+ \State $sum \gets 0$
+ \For{$k = 0,1,2 \dots,n-1$}
+ \State $sum \gets sum + \textbf{A}[i][k] \cdot \textbf{B}[k][j]$
+ \EndFor
+ \State $\textbf{C}[i][j] \gets sum $
+ \EndFor
+ \EndFor
+ \State \textbf{return} $\textbf{C}$
+ \EndFunction
+ \end{algorithmic}
+ \end{algorithm}
+\end{column}
+\begin{column}{0.4\textwidth}
+ \scalebox{0.6}{\parbox{\linewidth}{
+
+ \begin{tikzpicture}[ampersand replacement=\&,remember picture,overlay]
+
+ \matrix (A)[matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}] at (2,-2.8)
+ {
+ A_{1,1} \& \cdots \& A_{1,k} \& \cdots \& A_{1,n} \\
+ \vdots \& \& \vdots \& \& \vdots \\
+ A_{i,1} \& \cdots \& A_{i,k} \& \cdots \& A_{i,n} \\
+ \vdots \& \& \vdots \& \& \vdots \\
+ A_{m,1} \& \cdots \& A_{m,k} \& \cdots \& A_{m,n} \\
+ };
+
+ \matrix (B)[matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}] at (7.5,1.2)
+ {
+ B_{1,1} \& \cdots \& B_{1,j} \& \cdots \& B_{1,p} \\
+ \vdots \& \& \vdots \& \& \vdots \\
+ B_{k,1} \& \cdots \& B_{k,j} \& \cdots \& B_{k,p} \\
+ \vdots \& \& \vdots \& \& \vdots \\
+ B_{n,1} \& \cdots \& B_{n,j} \& \cdots \& B_{n,p} \\
+ };
+
+ \matrix (C)[matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}] at (7.5,-2.8)
+ {
+ C_{1,1} \& \cdots \& C_{1,j} \& \cdots \& C_{1,p} \\
+ \vdots \& \& \vdots \& \& \vdots \\
+ C_{i,1} \& \cdots \& C_{i,j} \& \cdots \& C_{i,p} \\
+ \vdots \& \& \vdots \& \& \vdots \\
+ C_{m,1} \& \cdots \& C_{m,j} \& \cdots \& C_{m,p} \\
+ };
+
+
+ \begin{scope}[on background layer]
+ \node[opacity=0.5, rounded corners=2pt, inner sep=-1pt, fill=green, fit=(A-3-1)(A-3-5)] {};
+ \node[opacity=0.5, rounded corners=2pt, inner sep=-1pt, fill=blue, fit=(B-1-3)(B-5-3)] {};
+ \node[opacity=0.5, rounded corners=2pt, inner sep=-1pt, fill=red, fit=(C-3-3)] {};
+
+ \end{scope}
+
+
+
+
+ \end{tikzpicture}
+ }}
+ \end{column}
+\end{columns}
+\end{frame}
+
+
+\begin{frame}
+ \frametitle{Algorithm}
+
+\begin{columns}
+ \begin{column}{0.6\textwidth}
+\begin{algorithm}[H]\caption{Square Matrix Multiplication}
+ \setlength{\lineskip}{7pt}
+ \begin{algorithmic}[1]
+ \Function{MM}{$\textbf{A}, \textbf{B}, \textbf{C}$}
+ \State $sum \gets 0$
+ \State $n \gets columns(\textbf{A}) == rows(\textbf{B})$
+ \State $m \gets rows(\textbf{A})$
+ \State $p \gets columns(\textbf{B})$
+
+ \For{$i = 0,1,2 \dots,m-1$}
+ \For{$j = 0,1,2 \dots,p-1$}
+ \State $sum \gets 0$
+ \For{$k = 0,1,2 \dots,n-1$}
+ \State $sum \gets sum + \textbf{A}[i][k] \cdot \textbf{B}[k][j]$
+ \EndFor
+ \State $\textbf{C}[i][j] \gets sum $
+ \EndFor
+ \EndFor
+ \State \textbf{return} $\textbf{C}$
+ \EndFunction
+ \end{algorithmic}
+\end{algorithm}
+\end{column}
+\begin{column}{0.4\textwidth}
+\Huge$\mathcal{O}(n^3)$
+\end{column}
+\end{columns}
+
+\end{frame}
diff --git a/buch/papers/multiplikation/presentation/slides/bigO.tex b/buch/papers/multiplikation/presentation/slides/bigO.tex
new file mode 100644
index 0000000..d425da8
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/slides/bigO.tex
@@ -0,0 +1,251 @@
+
+\begin{frame}
+ \frametitle{Big $\mathcal{O}$ notation}
+\begin{itemize}
+ \item <1-> Time complexity of an algorithm
+ \item <2-> How many multiplications in a function
+ \item <3-> Drop Constants
+\end{itemize}
+\end{frame}
+
+
+\begin{frame}
+ \frametitle{Big $\mathcal{O}$ notation}
+ \onslide<1->{
+
+ \begin{algorithm}[H]\caption{Foo 1}
+ \setlength{\lineskip}{7pt}
+ \begin{algorithmic}[1]
+ \Function{foo}{$a, b$}
+ \State \textbf{return} $a+b$
+ \EndFunction
+ \end{algorithmic}
+ \end{algorithm}
+}
+\onslide<2->{
+$\mathcal{O}(1)$
+ }
+\end{frame}
+
+\begin{frame}
+ \frametitle{Big $\mathcal{O}$ notation}
+ \onslide<1->{
+
+ \begin{algorithm}[H]\caption{Foo 2}
+ \setlength{\lineskip}{7pt}
+ \begin{algorithmic}[1]
+ \Function{foo}{$a, b$}
+ \State $ x \gets a+b $
+ \State $ y \gets a \cdot b $
+ \State \textbf{return} $x+y$
+ \EndFunction
+ \end{algorithmic}
+ \end{algorithm}
+}
+\onslide<2->{
+$\mathcal{O}(1) + \mathcal{O}(1) = 2\mathcal{O}(1) = \mathcal{O}(1) $
+ }
+\end{frame}
+
+\begin{frame}
+ \frametitle{Big $\mathcal{O}$ notation}
+ \onslide<1->{
+
+ \begin{algorithm}[H]\caption{Foo 3}
+ \setlength{\lineskip}{7pt}
+ \begin{algorithmic}[1]
+ \Function{foo}{$\mathbf{A}, \mathbf{B}$,n}
+ \State $ sum \gets 0$
+ \For{$i = 0,1,2 \dots,n$}
+ \State $ sum \gets sum + A[i] \cdot B[i] $
+ \EndFor
+
+ \State \textbf{return} $sum$
+
+ \EndFunction
+ \end{algorithmic}
+ \end{algorithm}
+}
+\onslide<2->{
+$\mathcal{O}(n)$
+ }
+\end{frame}
+
+\begin{frame}
+ \frametitle{Big $\mathcal{O}$ notation}
+ \onslide<1->{
+
+ \begin{algorithm}[H]\caption{Foo 4}
+ \setlength{\lineskip}{7pt}
+ \begin{algorithmic}[1]
+ \Function{foo}{$\mathbf{A}, \mathbf{B}$,n}
+ \State $ sum \gets 0$
+ \For{$i = 0,1,2 \dots,n$}
+ \For{$j = 0,1,2 \dots,n$}
+ \State $ sum \gets sum + A[i] \cdot B[j] $
+ \EndFor
+ \EndFor
+ \State \textbf{return} $sum$
+ \EndFunction
+ \end{algorithmic}
+ \end{algorithm}
+}
+\onslide<2->{
+$\mathcal{O}(n^2)$
+ }
+\end{frame}
+
+% \begin{frame}
+% \frametitle{Big $\mathcal{O}$ notation}
+% \onslide<1->{
+%
+% \begin{algorithm}[H]\caption{Fibonacci}
+% \setlength{\lineskip}{7pt}
+% \begin{algorithmic}[1]
+% \Function{fib}{$n$}
+% \If{$n <= 1$}
+% \State \textbf{return} $1$
+% \Else
+% \State \textbf{return} fib($n-1$) + fib($n-2$)
+% \EndIf
+%
+% \EndFunction
+% \end{algorithmic}
+% \end{algorithm}
+% }
+% \onslide<2->{
+% \[
+% \langle x,y \rangle =
+% \begin{cases}
+% \displaystyle $\mathcal{O}(1)$ & \text{if $n \leq 2$}\\
+% \displaystyle $ 2 \mathcal{T}(\frac{n}{2})$ & \text{if $n > 2$}
+% \end{cases}
+% \] }
+% \end{frame}
+
+
+\begin{frame}
+ \frametitle{Big $\mathcal{O}$ notation}
+\begin{tikzpicture}
+\begin{axis}[
+ axis lines = left,
+ xlabel = $n$ (Data Input),
+ ylabel = {$t$ (time)},
+ legend pos=north east,
+ very thick,
+ ymax = 20,
+ yticklabels=\empty,
+ xticklabels=\empty,
+ scale only axis=true,
+ width=12cm, height=6cm,
+ ]
+%Below the red parabola is defined
+\addplot [
+ domain= 1:6,
+ samples=100,
+ color=red,
+]
+{1};
+\addlegendentry{$\mathcal{O}(1)$}
+%Here the blue parabloa is defined
+\addplot [
+ domain= 1:6,
+ samples=100,
+ color=green,
+]
+{x};
+\addlegendentry{$\mathcal{O}(n)$}
+\addplot [
+ domain= 1:6,
+ samples=100,
+ color=blue,
+]
+{x^2};
+\addlegendentry{$\mathcal{O}(n^2)$}
+\addplot [
+ domain= 1:6,
+ samples=100,
+ color=purple,
+]
+{x^3};
+\addlegendentry{$\mathcal{O}(n^3)$}
+\addplot [
+ domain= 1:3,
+ samples=100,
+ color=black,
+]
+{exp(x)};
+\addlegendentry{$\mathcal{O}(e^n)$}
+\addplot [
+ domain= 1:6,
+ samples=100,
+ color=orange,
+]
+{log2(x)};
+\addlegendentry{$\mathcal{O}(\log n)$}
+\end{axis}
+\end{tikzpicture}
+
+\end{frame}
+
+\begin{frame}
+ \frametitle{Big $\mathcal{O}$ notation}
+\begin{tikzpicture}
+\begin{axis}[
+ axis lines = left,
+ xlabel = $n$ (Data Input),
+ ylabel = {$t$ (time)},
+ legend pos=north east,
+ very thick,
+ ymax = 500,
+ yticklabels=\empty,
+ xticklabels=\empty,
+ scale only axis=true,
+ width=12cm, height=6cm,
+ ]
+\addplot [
+ domain= 1:20,
+ samples=100,
+ color=red,
+]
+{1};
+\addlegendentry{$\mathcal{O}(1)$}
+\addplot [
+ domain= 1:20,
+ samples=100,
+ color=green,
+]
+{x};
+\addlegendentry{$\mathcal{O}(n)$}
+\addplot [
+ domain= 1:20,
+ samples=100,
+ color=blue,
+]
+{x^2};
+\addlegendentry{$\mathcal{O}(n^2)$}
+\addplot [
+ domain= 1:10,
+ samples=100,
+ color=purple,
+]
+{x^3};
+\addlegendentry{$\mathcal{O}(n^3)$}
+\addplot [
+ domain= 1:10,
+ samples=100,
+ color=black,
+]
+{exp(x)};
+\addlegendentry{$\mathcal{O}(e^n)$}
+\addplot [
+ domain= 1:20,
+ samples=100,
+ color=orange,
+]
+{log2(x)};
+\addlegendentry{$\mathcal{O}(\log n)$}
+\end{axis}
+\end{tikzpicture}
+
+\end{frame}
diff --git a/buch/papers/multiplikation/presentation/slides/blas.tex b/buch/papers/multiplikation/presentation/slides/blas.tex
new file mode 100644
index 0000000..ed498a3
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/slides/blas.tex
@@ -0,0 +1,18 @@
+\begin{frame}
+\frametitle{BLAS, LAPACK}
+\begin{itemize}
+ \item Basic Linear Algebra Subprograms
+ \begin{itemize}
+ \item $\mathbf{y} = \alpha \mathbf{x}+\mathbf{y}$
+ \item $\mathbf{y} = \alpha \mathbf{A}\mathbf{x}+ \beta \mathbf{y}$
+ \item $\mathbf{C} = \alpha \mathbf{A}\mathbf{B}+ \beta \mathbf{C}$
+
+ \end{itemize}
+ \item Linear Algebra Package
+ \begin{itemize}
+ \item QR decomposition
+ \item Singular value decomposition
+ \item Eigenvalues
+ \end{itemize}
+\end{itemize}
+\end{frame}
diff --git a/buch/papers/multiplikation/presentation/slides/conclusuion.tex b/buch/papers/multiplikation/presentation/slides/conclusuion.tex
new file mode 100644
index 0000000..e69de29
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/slides/conclusuion.tex
diff --git a/buch/papers/multiplikation/presentation/slides/logo.pdf b/buch/papers/multiplikation/presentation/slides/logo.pdf
new file mode 100644
index 0000000..d78ca88
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/slides/logo.pdf
Binary files differ
diff --git a/buch/papers/multiplikation/presentation/slides/meas.tex b/buch/papers/multiplikation/presentation/slides/meas.tex
new file mode 100644
index 0000000..489c010
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/slides/meas.tex
@@ -0,0 +1,42 @@
+\begin{frame}
+ \frametitle{Measurements Python}
+ \only<1>{
+ \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/meas_8.pdf}}
+ \only<2>{
+ \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/meas_16.pdf}}
+ \only<3>{
+ \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/meas_32.pdf}}
+ \only<4>{
+ \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/meas_64.pdf}}
+ \only<5>{
+ \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/meas_128.pdf}}
+ \only<6>{
+ \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/meas_256.pdf}}
+ \only<7>{
+ \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/meas_512.pdf}}
+ \only<8>{
+ \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/meas_1024.pdf}}
+\end{frame}
+
+
+\begin{frame}
+ \frametitle{Measurements C}
+ \only<1>{
+ \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/c_meas_8.pdf}}
+ \only<2>{
+ \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/c_meas_16.pdf}}
+ \only<3>{
+ \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/c_meas_32.pdf}}
+ \only<4>{
+ \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/c_meas_64.pdf}}
+ \only<5>{
+ \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/c_meas_128.pdf}}
+ \only<6>{
+ \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/c_meas_256.pdf}}
+ \only<7>{
+ \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/c_meas_512.pdf}}
+ \only<8>{
+ \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/c_meas_1024.pdf}}
+ \only<9>{
+ \includegraphics[width=\textwidth,height=0.9\textheight,keepaspectratio]{../code/c_meas_2048.pdf}}
+\end{frame}
diff --git a/buch/papers/multiplikation/presentation/slides/nn.tex b/buch/papers/multiplikation/presentation/slides/nn.tex
new file mode 100644
index 0000000..e74e970
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/slides/nn.tex
@@ -0,0 +1,97 @@
+
+\begin{frame}
+ \frametitle{Neural Network}
+ \centering
+\newcommand{\inputnum}{4}
+
+% Hidden layer neurons'number
+\newcommand{\hiddennumA}{5}
+\newcommand{\hiddennumB}{6}
+
+% Output layer neurons'number
+\newcommand{\outputnum}{4}
+
+\begin{tikzpicture}
+
+
+% Input Layer
+\foreach \i in {1,...,\inputnum}
+{
+ \node[circle,
+ minimum size = 6mm,
+ fill=blue!30] (Input-\i) at (0,-\i) {};
+}
+
+% Hidden Layer1
+\foreach \i in {1,...,\hiddennumA}
+{
+ \node[circle,
+ minimum size = 6mm,
+ fill=red!50,
+ yshift=(\hiddennumA-\inputnum)*5 mm
+ ] (Hidden1-\i) at (2.5,-\i) {};
+}
+
+% Hidden Layer2
+\foreach \i in {1,...,\hiddennumB}
+{
+ \node[circle,
+ minimum size = 6mm,
+ fill=red!50,
+ yshift=(\hiddennumB-\inputnum)*5 mm
+ ] (Hidden2-\i) at (5,-\i) {};
+}
+
+% Output Layer
+\foreach \i in {1,...,\outputnum}
+{
+ \node[circle,
+ minimum size = 6mm,
+ fill=green!50,
+ yshift=(\outputnum-\inputnum)*5 mm
+ ] (Output-\i) at (7.5,-\i) {};
+}
+
+% Connect neurons In-Hidden
+\foreach \i in {1,...,\inputnum}
+{
+ \foreach \j in {1,...,\hiddennumA}
+ {
+ \draw[->, shorten >=1pt] (Input-\i) -- (Hidden1-\j);
+ }
+}
+
+% Connect neurons In-Hidden
+\foreach \i in {1,...,\hiddennumA}
+{
+ \foreach \j in {1,...,\hiddennumB}
+ {
+ \draw[->, shorten >=1pt] (Hidden1-\i) -- (Hidden2-\j);
+ }
+}
+
+% Connect neurons Hidden-Out
+\foreach \i in {1,...,\hiddennumB}
+{
+ \foreach \j in {1,...,\outputnum}
+ {
+ \draw[->, shorten >=1pt] (Hidden2-\i) -- (Output-\j);
+ }
+}
+
+% Inputs
+\foreach \i in {1,...,\inputnum}
+{
+ \draw[<-, shorten <=1pt] (Input-\i) -- ++(-1,0)
+ node[left]{\LARGE{$x_{\i}$}};
+}
+
+% Outputs
+\foreach \i in {1,...,\outputnum}
+{
+ \draw[->, shorten <=1pt] (Output-\i) -- ++(1,0)
+ node[right]{\LARGE{$y_{\i}$}};
+}
+
+\end{tikzpicture}
+\end{frame}
diff --git a/buch/papers/multiplikation/presentation/slides/parcomp.tex b/buch/papers/multiplikation/presentation/slides/parcomp.tex
new file mode 100644
index 0000000..1ba39ee
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/slides/parcomp.tex
@@ -0,0 +1,66 @@
+% !TEX root = presentation.tex
+
+\begin{frame}
+ \frametitle{Vector-Matrix Multiplication}
+\center{
+ \begin{tikzpicture}[ampersand replacement=\&]
+
+ \matrix (A)[matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}]
+ {
+ A_{1,1} \& A_{1,2} \& A_{1,3} \& A_{1,4} \\
+ };
+
+ \matrix (B)[matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}] at (5,-0.95)
+ {
+ B_{1,1} \& B_{1,2} \& B_{1,3} \& B_{1,4} \& B_{1,5} \\
+ B_{2,1} \& B_{2,2} \& B_{2,3} \& B_{2,4} \& B_{2,5} \\
+ B_{3,1} \& B_{3,2} \& B_{3,3} \& B_{3,4} \& B_{3,5} \\
+ B_{4,1} \& B_{4,2} \& B_{4,3} \& B_{4,4} \& B_{4,5} \\
+ };
+
+ \matrix (C)[matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}] at (5,-3)
+ {
+ C_{1,1} \& C_{1,2} \& C_{1,3} \& C_{1,4} \& C_{1,5}\\
+ };
+
+ \foreach \i in {1,...,4}
+ {
+ \pgfmathtruncatemacro{\ii}{\i+1}
+ \onslide<\ii>{
+
+ \foreach \j in {1,...,5}
+ {
+ \draw[thick] (A-1-\i.south) to [out=-90,in=135]node[visible on=<\i->, anchor=north]{} (B-\i-\j.center);
+
+ }
+ }
+ }
+
+
+ \end{tikzpicture}
+}
+\end{frame}
+
+
+\begin{frame}
+ \frametitle{DSP Architecture}
+\scalebox{2}{
+ \begin{tikzpicture}
+ \node (mul) at (0,0) [circle,draw=black,inner sep=0pt,minimum size=0.5cm] {X};
+ \node (mac) at (2,0) [circle,draw=black,inner sep=0pt,minimum size=0.5cm] {\textbf{+}};
+
+ \node at (-2,0.3) {$A[n]$};
+ \node at (0.4,2) {$B[n]$};
+ \node at (4,0.3) {$C[n]$};
+
+ \draw[thick, ->] (-2,0) --++ (mul);
+ \draw[thick, ->] (0,2) --++ (mul);
+ \draw[thick, ->] (mul) -- (mac);
+ \draw[thick] (mac) --++ (1,0) node (i) {};
+ \draw[thick, ->] (i.center) --++ (0,1) --++ (-1,0) -- (mac);
+ \draw[thick, ->] (i.center) --++ (1,0);
+
+
+ \end{tikzpicture}
+ }
+\end{frame}
diff --git a/buch/papers/multiplikation/presentation/slides/slides.tex b/buch/papers/multiplikation/presentation/slides/slides.tex
new file mode 100644
index 0000000..64edb86
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/slides/slides.tex
@@ -0,0 +1,15 @@
+% !TEX root = presentation.tex
+\begin{frame}
+\titlepage
+\end{frame}
+%
+\section{Big $\mathcal{O}$}
+\input{slides/BigO.tex}
+\section{Strassen's Algorithm}
+\input{slides/strassen.tex}
+% \input{slides/nn.tex}
+\section{Measurements}
+\input{slides/meas.tex}
+% \input{slides/parcomp.tex}
+\section{How To Matrix Multiply}
+\input{slides/blas.tex}
diff --git a/buch/papers/multiplikation/presentation/slides/strassen.tex b/buch/papers/multiplikation/presentation/slides/strassen.tex
new file mode 100644
index 0000000..c3398d5
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/slides/strassen.tex
@@ -0,0 +1,429 @@
+\begin{frame}
+ \frametitle{Strassen's Algorithm}
+ \includegraphics[page=1,width=\textwidth,height=0.8\textheight,keepaspectratio]{../papers/Strassen_original_1969.pdf}
+ \includegraphics[page=2,width=\textwidth,height=0.8\textheight,keepaspectratio]{../papers/Strassen_original_1969.pdf} \includegraphics[page=3,width=\textwidth,height=0.8\textheight,keepaspectratio]{../papers/Strassen_original_1969.pdf}
+ \end{frame}
+
+\begin{frame}
+ \frametitle{Strassen's Algorithm}
+ \centering
+ \large
+\onslide<1->{
+ $
+ \mathbf{A B = C}
+ $
+}
+
+\onslide<2->{
+
+
+\medskip
+ $
+ \begin{bmatrix}
+ A_{11} & A_{12}\\
+ A_{21} & A_{22}
+ \end{bmatrix}
+ \begin{bmatrix}
+ B_{11} & B_{12}\\
+ B_{21} & B_{22}
+ \end{bmatrix}
+ =
+ \begin{bmatrix}
+ C_{11} & C_{12}\\
+ C_{21} & C_{22}
+ \end{bmatrix}
+ $
+ }
+
+
+ \onslide<3->{
+
+\medskip
+$
+C_{11} = A_{11} \cdot B_{11} + A_{12} \cdot B_{21}
+$
+
+$
+C_{12} = A_{11} \cdot B_{12} + A_{12} \cdot B_{22}
+$
+
+$
+C_{21} = A_{21} \cdot B_{11} + A_{22} \cdot B_{21}
+$
+
+$
+C_{22} = A_{21} \cdot B_{12} + A_{22} \cdot B_{22}
+$
+}
+\end{frame}
+
+\input{slides/algo.tex}
+
+
+
+\begin{frame}
+ \frametitle{Strassen's Algorithm}
+ \begin{columns}
+ \begin{column}{0.5\textwidth}
+ \onslide<1->{
+ \large
+ \begin{math}
+ \begin{aligned}
+ \text{I} &= (A_{11} + A_{22}) \cdot (B_{11} + B_{22}) \\
+ \text{II} &= (A_{21} + A_{22}) \cdot B_{11} \\
+ \text{III} &= A_{11} \cdot (B_{12}-B_{22}) \\
+ \text{IV} &= A_{22} \cdot (-B_{11}+B_{21}) \\
+ \text{V} &= (A_{11} + A_{12}) \cdot B_{22} \\
+ \text{VI} &= (-A_{11} + A_{21}) \cdot (B_{11} + B_{12}) \\
+ \text{VII} &= (A_{12} - A_{22}) \cdot (B_{21} + B_{22}) \\
+ \end{aligned}
+ \end{math}
+ }
+ \end{column}
+
+ \begin{column}{0.5\textwidth}
+ \onslide<2->{
+ \large
+ \begin{math}
+ \begin{aligned}
+ C_{11} &= \text{I} + \text{IV} - \text{V} + \text{VII} \\
+ C_{21} &= \text{II} + \text{IV} \\
+ C_{12} &= \text{III} + \text{V}\\
+ C_{22} &= \text{I} + \text{III} - \text{II} + \text{VI} \\
+ \end{aligned}
+ \end{math}
+ }
+ \end{column}
+\end{columns}
+
+\onslide<3->{
+
+\bigskip
+\centering
+\tiny
+\begin{math}
+\begin{aligned}
+ C_{11} &= (A_{11} + A_{22}) \cdot (B_{11} + B_{22}) + A_{22} \cdot (-B_{11}+B_{21}) - (A_{11} + A_{12}) \cdot B_{22} + (A_{12} - A_{22}) \cdot (B_{21} + B_{22}) \\
+ C_{11} &= A_{11}B_{11} + A_{11}B_{22} + A_{22}B_{11} + A_{22}B_{22} -A_{22}B_{11}+A_{22}B_{21} - A_{11}B_{22} - A_{12}B_{22}+ A_{12}B_{21} + A_{12}B_{22} - A_{22}B_{21} - A_{22}B_{22} \\
+ C_{11} &= A_{11}B_{11} + A_{12}B_{21}
+\end{aligned}
+\end{math}
+}
+
+\end{frame}
+
+
+\begin{frame}
+\begin{adjustbox}{width=\textwidth}
+\begin{tikzpicture}[ampersand replacement=\&]
+
+ \foreach \i in {1,...,4}
+ {
+ \small{
+ \matrix (X\i)[matrix of math nodes,nodes in empty cells,
+ nodes = {draw, minimum size=10mm,
+ anchor=center,
+ inner sep=0pt, outer sep=0pt},
+ column sep=-\pgflinewidth,
+ row sep=-\pgflinewidth,
+ ] at (0,-\i*5)
+ {
+ A_{11}B_{11} \& A_{12}B_{11} \& A_{21}B_{11} \& A_{22}B_{11} \\
+ A_{11}B_{21} \& A_{12}B_{21} \& A_{21}B_{21} \& A_{22}B_{21} \\
+ A_{11}B_{11} \& A_{12}B_{12} \& A_{21}B_{12} \& A_{22}B_{12} \\
+ A_{11}B_{22} \& A_{12}B_{22} \& A_{21}B_{22} \& A_{22}B_{22} \\
+ };}
+
+ \foreach \j in {1,...,7}
+ {
+ \matrix(M\i\j)[matrix of math nodes,nodes in empty cells,
+ nodes = {draw, minimum size=10mm,
+ anchor=center,
+ inner sep=0pt, outer sep=0pt},
+ column sep=-\pgflinewidth,
+ row sep=-\pgflinewidth,
+ ] at (\j*5,-\i*5)
+ {
+ \& \& \& \\
+ \& \& \& \\
+ \& \& \& \\
+ \& \& \& \\
+ };
+ }
+ }
+
+\huge{
+ \node at (-3,-20) {$C_{22}=$};
+ \node at (-3,-15) {$C_{21}=$} ;
+ \node at (-3,-10) {$C_{12}=$} ;
+ \node at (-3,-5) {$C_{11}=$} ;
+
+ \node at (5,-2) {I};
+ \node at (10,-2) {II};
+ \node at (15,-2) {III};
+ \node at (20,-2) {IV};
+ \node at (25,-2) {V};
+ \node at (30,-2) {VI};
+ \node at (35,-2) {VII};
+ }
+
+
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X1-1-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X1-2-2)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X2-3-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X2-4-2)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X3-1-3)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X3-2-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X4-3-3)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(X4-4-4)] {};
+
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M11-4-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M11-1-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M11-4-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M11-1-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M14-1-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M14-2-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M15-4-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M15-4-2)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M17-2-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M17-4-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M17-2-2)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M17-4-2)] {};
+
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M23-3-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M23-4-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M25-4-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M25-4-2)] {};
+
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M32-1-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M32-1-3)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M34-1-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M34-2-4)] {};
+
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M41-4-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M41-1-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M41-4-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M41-1-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M42-1-4)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M42-1-3)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M43-3-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M43-4-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M46-1-3)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M46-1-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M46-3-3)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M46-3-1)] {};
+\end{tikzpicture}
+\end{adjustbox}
+\end{frame}
+
+
+\begin{frame}
+ \frametitle{Strassen's Algorithm}
+ \begin{columns}
+ \begin{column}{0.5\textwidth}
+ \large
+ \begin{math}
+ \begin{aligned}
+ \text{I} &= (A_{11} + A_{22}) \cdot (B_{11} + B_{22}) \\
+ \text{II} &= (A_{21} + A_{22}) \cdot B_{11} \\
+ \text{III} &= A_{11} \cdot (B_{12}-B_{22}) \\
+ \text{IV} &= A_{22} \cdot (-B_{11}+B_{21}) \\
+ \text{V} &= (A_{11} + A_{12}) \cdot B_{22} \\
+ \text{VI} &= (-A_{11} + A_{21}) \cdot (B_{11} + B_{12}) \\
+ \text{VII} &= (A_{12} - A_{22}) \cdot (B_{21} + B_{22}) \\
+ \end{aligned}
+ \end{math}
+
+ \end{column}
+
+ \begin{column}{0.5\textwidth}
+ \large
+ \begin{math}
+ \begin{aligned}
+ C_{11} &= \text{I} + \text{IV} - \text{V} + \text{VII} \\
+ C_{21} &= \text{II} + \text{IV} \\
+ C_{12} &= \text{III} + \text{V}\\
+ C_{22} &= \text{I} + \text{III} - \text{II} + \text{VI} \\
+ \end{aligned}
+ \end{math}
+
+ \end{column}
+\end{columns}
+\end{frame}
+
+
+
+\begin{frame}
+ \frametitle{Strassen's Algorithm}
+
+\begin{columns}
+ \begin{column}{0.5\textwidth}
+\large
+\begin{math}
+\begin{aligned}
+\text{\textbf{I}} &= (\mathbf{A_{11}} + \mathbf{A_{22}}) \cdot (\mathbf{B_{11}} + \mathbf{B_{22}}) \\
+\text{\textbf{II}} &= (\mathbf{A_{21}} + \mathbf{A_{22}}) \cdot \mathbf{B_{11}} \\
+\text{\textbf{III}} &= \mathbf{A_{11}} \cdot (\mathbf{B_{12}}-\mathbf{B_{22}}) \\
+\text{\textbf{IV}} &= \mathbf{A_{22}} \cdot (-\mathbf{B_{11}}+\mathbf{B_{21}}) \\
+\text{\textbf{V}} &= (\mathbf{A_{11}} + \mathbf{A_{12}}) \cdot \mathbf{B_{22}} \\
+\text{\textbf{VI}} &= (-\mathbf{A_{11}} + \mathbf{A_{21}}) \cdot (\mathbf{B_{11}} + \mathbf{B_{12}}) \\
+\text{\textbf{VII}} &= (\mathbf{A_{12}} - \mathbf{A_{22}}) \cdot (\mathbf{B_{21}} + \mathbf{B_{22}}) \\
+\end{aligned}
+\end{math}
+
+\end{column}
+
+\begin{column}{0.5\textwidth}
+ \large
+ \begin{math}
+ \begin{aligned}
+ \mathbf{C_{11}} &= \text{\textbf{I}} + \text{\textbf{IV}} - \text{\textbf{V}} + \text{\textbf{VII}} \\
+ \mathbf{C_{21}} &= \text{\textbf{II}} + \text{\textbf{IV}} \\
+ \mathbf{C_{12}} &= \text{\textbf{III}} + \text{\textbf{V}}\\
+ \mathbf{C_{22}} &= \text{\textbf{I}} + \text{\textbf{III}} - \text{\textbf{II}} + \text{\textbf{VI}} \\
+ \end{aligned}
+ \end{math}
+
+\end{column}
+\end{columns}
+
+\end{frame}
+
+\begin{frame}
+ \frametitle{Algorithm}
+ \onslide<1->{
+
+ \scalebox{0.45}{\parbox{\linewidth}{
+ \begin{algorithm}[H]\caption{Strassen Matrix Multiplication}
+ \setlength{\lineskip}{7pt}
+ \begin{algorithmic}[1]
+ \Function{strassen}{$\textbf{A}, \textbf{B}, n$}
+ \If{$n = 2$}
+ \State $ \mathbf{C} \gets zeros((n, n))$
+ \State $P \gets (A[0][0]+A[1][1])\cdot( B[0][0]+B[1][1])$
+ \State $Q \gets (A[1][0]+A[1][1])\cdot B[0][0]$
+ \State $R \gets A[0][0]\cdot (B[0][1]-B[1][1])$
+ \State $S \gets A[1][1]\cdot (B[1][0]-B[0][0])$
+ \State $T \gets (A[0][0]+A[0][1])\cdot B[1][1]$
+ \State $U \gets (A[1][0]-A[0][0])\cdot (B[0][0]+B[0][1])$
+ \State $V \gets (A[0][1]-A[1][1])\cdot (B[1][0]+B[1][1])$
+ \State $C[0][0] \gets P+S-T+V$
+ \State $C[0][1] \gets R+T$
+ \State $C[1][0] \gets Q+S$
+ \State $C[1][1] \gets P+R-Q+U$
+ \Else
+ \State $ m \gets n/2$
+ \State $\mathbf{A11}, \mathbf{A12}, \mathbf{A21}, \mathbf{A22} \gets \mathbf{A}[:m][:m], \mathbf{A}[:m][m:], \mathbf{A}[m:][:m], \mathbf{A}[m:][m:]$
+ \State $\mathbf{B11}, \mathbf{B12}, \mathbf{B21}, \mathbf{B22} \gets \mathbf{B}[:m][:m], \mathbf{B}[:m][m:], \mathbf{B}[m:][:m], \mathbf{B}[m:][m:]$
+
+ \State $ \mathbf{P} \gets \text{strassen}((\mathbf{A11}+ \mathbf{A22}),(\mathbf{B11}+\mathbf{B22}), m)$
+ \State $ \mathbf{Q} \gets \text{strassen}((\mathbf{A21}+ \mathbf{A22}), \mathbf{B11},m)$
+ \State $ \mathbf{R} \gets \text{strassen}( \mathbf{A11},(\mathbf{B12}- \mathbf{B22}),m)$
+ \State $ \mathbf{S} \gets \text{strassen}( \mathbf{A22},(\mathbf{B21}- \mathbf{B11}),m)$
+ \State $ \mathbf{T} \gets \text{strassen}((\mathbf{A11}+ \mathbf{A12}), \mathbf{B22},m)$
+ \State $ \mathbf{U} \gets \text{strassen}((\mathbf{A21}- \mathbf{A11}),(\mathbf{B11}+\mathbf{B12}),m)$
+ \State $ \mathbf{V} \gets \text{strassen}((\mathbf{A12}- \mathbf{A22}),(\mathbf{B21}+\mathbf{B22}),m)$
+
+
+
+ \State $\mathbf{C11} \gets \mathbf{P+S-T+V}$
+ \State $\mathbf{C12} \gets \mathbf{R+T}$
+ \State $\mathbf{C21} \gets \mathbf{Q+S}$
+ \State $\mathbf{C22} \gets \mathbf{P+R-Q+U}$
+ \State $ C \gets vstack((hstack((C11, C12)), hstack((C21, C22))))$
+
+ \EndIf
+ \State \textbf{return} $\textbf{C}$
+
+ \EndFunction
+ \end{algorithmic}
+ \end{algorithm}
+ }}}
+% \[
+% \mathcal{T}(n) = \left\{\begin{array}{lr}
+% 1, & \text{if} n \leq 2\\
+% 7 \mathcal{T}(\frac{n}{2}) + n^2, & \text{if} n > 2\\
+% \end{array}\right\}
+% \]
+\only<2>{
+ $
+ \mathcal{T}(n) =
+ \begin{cases}
+ 1 & \text{if } n \leq 2\\
+ 7 \cdot \mathcal{T}(\frac{n}{2}) + n^2 & \text{if } n > 2
+ \end{cases} = \mathcal{O}(n^{\log_2 7})$
+
+}
+\only<3>{
+ $
+ \mathcal{T}(n) =
+ \begin{cases}
+ 1 & \text{if } n \leq 2\\
+ 7 \cdot \mathcal{T}(\frac{n}{2}) + n^2 & \text{if } n > 2
+ \end{cases} = \mathcal{O}(n^{2.81})$
+
+}
+
+\end{frame}
+
+\begin{frame}
+ \frametitle{Algorithm}
+ \onslide<1->{
+
+ \scalebox{0.45}{\parbox{\linewidth}{
+ \begin{algorithm}[H]\caption{Strassen Matrix Multiplication}
+ \setlength{\lineskip}{7pt}
+ \begin{algorithmic}[1]
+ \Function{MM}{$\textbf{A}, \textbf{B}, n$}
+ \If{$n = 2$}
+ \State $ \mathbf{C} \gets zeros((n, n))$
+ \State $C[0, 0] \gets A[0][0]*B[0][0]+A[0][1]*B[1][0]$
+ \State $C[0, 1] \gets A[0][0]*B[0][1]+A[0][1]*B[1][1]$
+ \State $C[1, 0] \gets A[1][0]*B[0][0]+A[1][1]*B[1][0]$
+ \State $C[1, 1] \gets A[1][0]*B[0][1]+A[1][1]*B[1][1]$
+ \Else
+ \State $ m \gets n/2$
+ \State $\mathbf{A11}, \mathbf{A12}, \mathbf{A21}, \mathbf{A22} \gets \mathbf{A}[:m][:m], \mathbf{A}[:m][m:], \mathbf{A}[m:][:m], \mathbf{A}[m:][m:]$
+ \State $\mathbf{B11}, \mathbf{B12}, \mathbf{B21}, \mathbf{B22} \gets \mathbf{B}[:m][:m], \mathbf{B}[:m][m:], \mathbf{B}[m:][:m], \mathbf{B}[m:][m:]$
+
+ \State $\mathbf{C11} \gets \text{MM}(\mathbf{A11}, \mathbf{B11}) + \text{MM}(\mathbf{A12}, \mathbf{B21})$
+ \State $\mathbf{C12} \gets \text{MM}(\mathbf{A11},\mathbf{B12}) + \text{MM}(\mathbf{A12},\mathbf{B22})$
+ \State $\mathbf{C21} \gets \text{MM}(\mathbf{A21}, \mathbf{B11}) + \text{MM}(\mathbf{A22}, \mathbf{B21})$
+ \State $\mathbf{C22} \gets \text{MM}(\mathbf{A21}, \mathbf{B12}) + \text{MM}(\mathbf{A22}, \mathbf{B22})$
+ \State $ C \gets vstack((hstack((C11, C12)), hstack((C21, C22))))$
+
+ \EndIf
+ \State \textbf{return} $\textbf{C}$
+
+ \EndFunction
+ \end{algorithmic}
+ \end{algorithm}
+ \bigskip
+ \bigskip
+ \bigskip
+ \bigskip
+ \bigskip
+ }}}
+
+\only<2>{
+
+
+ $
+ \mathcal{T}(n) =
+ \begin{cases}
+ 1 & \text{if } n \leq 2\\
+ 8 \cdot \mathcal{T}(\frac{n}{2}) + n^2 & \text{if } n > 2
+ \end{cases} = \mathcal{O}(n^{\log_2 8})$
+
+}
+\only<3>{
+ $
+ \mathcal{T}(n) =
+ \begin{cases}
+ 1 & \text{if } n \leq 2\\
+ 8 \cdot \mathcal{T}(\frac{n}{2}) + n^2 & \text{if } n > 2
+ \end{cases} = \mathcal{O}(n^{3})$
+
+}
+
+\end{frame}
diff --git a/buch/papers/multiplikation/presentation/tikz/algo.pdf b/buch/papers/multiplikation/presentation/tikz/algo.pdf
new file mode 100644
index 0000000..752f42e
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/tikz/algo.pdf
Binary files differ
diff --git a/buch/papers/multiplikation/presentation/tikz/algo.tex b/buch/papers/multiplikation/presentation/tikz/algo.tex
new file mode 100644
index 0000000..0b2c567
--- /dev/null
+++ b/buch/papers/multiplikation/presentation/tikz/algo.tex
@@ -0,0 +1,52 @@
+\documentclass[border=10pt]{article}
+\usepackage[left=25mm,right=25mm,top=25mm,bottom=25mm]{geometry}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{times}
+\usepackage{geometry}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{mathrsfs}
+\usepackage{amsfonts}
+\usepackage{amsthm}
+\usepackage{lipsum}
+\usepackage{amscd}
+\usepackage{graphicx}
+\usepackage{fancyhdr}
+\usepackage{textcomp}
+\usepackage{txfonts}
+\usepackage[all]{xy}
+\usepackage{paralist}
+\usepackage[colorlinks=true]{hyperref}
+\usepackage{array}
+\usepackage{tikz}
+\usepackage{slashed}
+\usepackage{pdfpages}
+\usepackage{cite}
+\usepackage{url}
+\usepackage{algorithm}
+\usepackage[noend]{algpseudocode}
+\usepackage{listings}
+\usepackage{multirow}
+\usepackage{color}
+
+\begin{document}
+
+\begin{algorithm}[H]\caption{Square Matrix Multiplication}
+ \setlength{\lineskip}{7pt}
+ \begin{algorithmic}[1]
+ \Function{MM}{$\textbf{A}, \textbf{B}, \textbf{C}, n$}
+ \State $sum \gets 0$
+ \For{$i = 0,1,2 \dots,n-1$}
+ \For{$j = 0,1,2 \dots,n-1$}
+ \State $sum \gets 0$
+ \For{$k = 0,1,2 \dots,n-1$}
+ \State $sum \gets sum + \textbf{A}[i][k] \cdot \textbf{B}[k][j]$
+ \EndFor
+ \State $\textbf{C}[i][j] \gets sum $
+ \EndFor
+ \EndFor
+ \EndFunction
+ \end{algorithmic}
+\end{algorithm}
+\end{document}
diff --git a/buch/papers/multiplikation/problemstellung.tex b/buch/papers/multiplikation/problemstellung.tex
new file mode 100755
index 0000000..b20a791
--- /dev/null
+++ b/buch/papers/multiplikation/problemstellung.tex
@@ -0,0 +1,104 @@
+%
+% teil1.tex -- Beispiel-File für das Paper
+%
+% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
+%
+\section{Problemstellung}
+\rhead{Problemstellung}
+Dank der breiten Anwendung der Matrizenmultiplikation ist eine effiziente L\"osung dieser Operation von grosser Bedeutung.
+Das Ziel dieses Papers ist verschiedenen Algorithmen der Matrizenmultiplikation vorzustellen.
+Wobei gezielt auf Algorithmen, welche das Problem schneller als der Standard Algorithmus L\"osen eingegangen wird.
+
+\subsection{Big $\mathcal{O}$ Notation}
+Die Big $\mathcal{O}$ Notation beschreibt die Laufzeitkomplexit\"at eines Algorithmus \cite{multiplikation:bigo}.
+$f(x) \in \mathcal{O}(g(x))$ besagt das die Funktion $f$ nicht wesentlich schneller w\"achst als $g$ wenn $x \rightarrow \infty$.
+Vereinfacht werden f\"ur Algorithmen die folgende Notation verwendet:
+\begin{itemize}
+ \item $f \in \mathcal{O}(1) \rightarrow f$ ist beschr\"ankt
+ \item $f \in \mathcal{O}(n) \rightarrow f$ w\"achst linear
+ \item $f \in \mathcal{O}(n^2) \rightarrow f$ w\"achst quadratisch
+ \item $f \in \mathcal{O}(\log n) \rightarrow f$ w\"achst logarithmisch
+ \item $f \in \mathcal{O}(n \log n) \rightarrow f$ hat super-lineares Wachstum
+ \item $f \in \mathcal{O}(e^n) \rightarrow f$ w\"achst exponentiell
+ \item usw.
+\end{itemize}
+
+In der Abbildung \ref{multiplikation:fig:bigo} k\"onnen die Verschiedenen Laufzeiten miteinander verglichen werden.
+
+\begin{figure}
+ \center
+ \includegraphics[]{papers/multiplikation/images/bigo}
+ \caption{Verschiedene Laufzeiten}
+ \label{multiplikation:fig:bigo}
+\end{figure}
+
+\subsubsection{Beispiel Algorithmen}
+\paragraph{Beschr\"ankter Algorithmus}
+
+Ein Beispiel eines Beschr\"ankter Verhalten $\mathcal{O}(1)$, kann im Algorithmus \ref{multiplikation:alg:b1} entnommen werden.
+
+\begin{algorithm}\caption{}
+ \label{multiplikation:alg:b1}
+ \setlength{\lineskip}{7pt}
+ \begin{algorithmic}
+ \Function{B1}{$a, b$}
+ \State \textbf{return} $a+b$
+ \EndFunction
+ \end{algorithmic}
+\end{algorithm}
+
+Wobei Konstanten nicht beachtet werden, der Algorithmus \ref{multiplikation:alg:b2} f\"uhrt ebenso zu $\mathcal{O}(1)$ und nicht zu $\mathcal{O}(2)$.
+
+\begin{algorithm}\caption{}
+ \label{multiplikation:alg:b2}
+ \setlength{\lineskip}{7pt}
+ \begin{algorithmic}
+ \Function{B2}{$a, b$}
+ \State $ x \gets a+b $
+ \State $ y \gets a \cdot b $
+ \State \textbf{return} $x+y$
+ \EndFunction
+ \end{algorithmic}
+\end{algorithm}
+
+\paragraph{Linearer Algorithmus}
+
+Folgender Algorithmus \ref{multiplikation:alg:l1} hat ein lineares $\mathcal{O}(n)$ Verhalten.
+
+\begin{algorithm}\caption{}
+ \setlength{\lineskip}{7pt}
+ \begin{algorithmic}
+ \label{multiplikation:alg:l1}
+ \Function{L}{$\mathbf{A}, \mathbf{B}$,n}
+ \State $ sum \gets 0$
+ \For{$i = 0,1,2 \dots,n$}
+ \State $ sum \gets sum + A[i] \cdot B[i] $
+ \EndFor
+
+ \State \textbf{return} $sum$
+
+ \EndFunction
+ \end{algorithmic}
+\end{algorithm}
+
+\paragraph{Quadratischer Algorithmus}
+
+Folgender Algorithmus \ref{multiplikation:alg:q1} hat ein quadratisches $\mathcal{O}(n^2)$ Verhalten.
+
+\begin{algorithm}[H]\caption{}
+ \label{multiplikation:alg:q1}
+ \setlength{\lineskip}{7pt}
+ \begin{algorithmic}
+ \Function{Q}{$\mathbf{A}, \mathbf{B}$,n}
+ \State $ sum \gets 0$
+ \For{$i = 0,1,2 \dots,n$}
+ \For{$j = 0,1,2 \dots,n$}
+ \State $ sum \gets sum + A[i] \cdot B[j] $
+ \EndFor
+ \EndFor
+ \State \textbf{return} $sum$
+ \EndFunction
+ \end{algorithmic}
+\end{algorithm}
+
+
diff --git a/buch/papers/multiplikation/references.bib b/buch/papers/multiplikation/references.bib
index 7149fb1..9d76e8e 100644..100755
--- a/buch/papers/multiplikation/references.bib
+++ b/buch/papers/multiplikation/references.bib
@@ -33,3 +33,33 @@
url = {https://doi.org/10.1016/j.acha.2017.11.004}
}
+@article{multiplikation:winograd_1968,
+ title={A New Algorithm for Inner Product},
+ volume={C-17},
+ DOI={10.1109/tc.1968.227420},
+ number={7},
+ journal={IEEE Transactions on Computers},
+ author={Winograd, S.},
+ year={1968},
+ pages={693–694}
+}
+
+@article{multiplikation:strassen_1969,
+ title={Gaussian elimination is not optimal},
+ volume={13},
+ DOI={10.1007/bf02165411},
+ number={4},
+ journal={Numerische Mathematik},
+ author={Strassen, Volker},
+ year={1969},
+ pages={354–356}
+}
+
+@online{multiplikation:bigo,
+ title = {Big O notation},
+ url = {https://en.wikipedia.org/wiki/Big_O_notation},
+ date = {2021-07-27},
+ year = {2021},
+ month = {7},
+ day = {27}
+}
diff --git a/buch/papers/multiplikation/teil0.tex b/buch/papers/multiplikation/teil0.tex
deleted file mode 100644
index 082b7f5..0000000
--- a/buch/papers/multiplikation/teil0.tex
+++ /dev/null
@@ -1,22 +0,0 @@
-%
-% einleitung.tex -- Beispiel-File für die Einleitung
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 0\label{multiplikation:section:teil0}}
-\rhead{Teil 0}
-Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam
-nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam
-erat, sed diam voluptua \cite{multiplikation:bibtex}.
-At vero eos et accusam et justo duo dolores et ea rebum.
-Stet clita kasd gubergren, no sea takimata sanctus est Lorem ipsum
-dolor sit amet.
-
-Lorem ipsum dolor sit amet, consetetur sadipscing elitr, sed diam
-nonumy eirmod tempor invidunt ut labore et dolore magna aliquyam
-erat, sed diam voluptua.
-At vero eos et accusam et justo duo dolores et ea rebum. Stet clita
-kasd gubergren, no sea takimata sanctus est Lorem ipsum dolor sit
-amet.
-
-
diff --git a/buch/papers/multiplikation/teil1.tex b/buch/papers/multiplikation/teil1.tex
deleted file mode 100644
index 0a6903a..0000000
--- a/buch/papers/multiplikation/teil1.tex
+++ /dev/null
@@ -1,55 +0,0 @@
-%
-% teil1.tex -- Beispiel-File für das Paper
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 1
-\label{multiplikation:section:teil1}}
-\rhead{Problemstellung}
-Sed ut perspiciatis unde omnis iste natus error sit voluptatem
-accusantium doloremque laudantium, totam rem aperiam, eaque ipsa
-quae ab illo inventore veritatis et quasi architecto beatae vitae
-dicta sunt explicabo.
-Nemo enim ipsam voluptatem quia voluptas sit aspernatur aut odit
-aut fugit, sed quia consequuntur magni dolores eos qui ratione
-voluptatem sequi nesciunt
-\begin{equation}
-\int_a^b x^2\, dx
-=
-\left[ \frac13 x^3 \right]_a^b
-=
-\frac{b^3-a^3}3.
-\label{multiplikation:equation1}
-\end{equation}
-Neque porro quisquam est, qui dolorem ipsum quia dolor sit amet,
-consectetur, adipisci velit, sed quia non numquam eius modi tempora
-incidunt ut labore et dolore magnam aliquam quaerat voluptatem.
-
-Ut enim ad minima veniam, quis nostrum exercitationem ullam corporis
-suscipit laboriosam, nisi ut aliquid ex ea commodi consequatur?
-Quis autem vel eum iure reprehenderit qui in ea voluptate velit
-esse quam nihil molestiae consequatur, vel illum qui dolorem eum
-fugiat quo voluptas nulla pariatur?
-
-\subsection{De finibus bonorum et malorum
-\label{multiplikation:subsection:finibus}}
-At vero eos et accusamus et iusto odio dignissimos ducimus qui
-blanditiis praesentium voluptatum deleniti atque corrupti quos
-dolores et quas molestias excepturi sint occaecati cupiditate non
-provident, similique sunt in culpa qui officia deserunt mollitia
-animi, id est laborum et dolorum fuga \eqref{000tempmlate:equation1}.
-
-Et harum quidem rerum facilis est et expedita distinctio
-\ref{multiplikation:section:loesung}.
-Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil
-impedit quo minus id quod maxime placeat facere possimus, omnis
-voluptas assumenda est, omnis dolor repellendus
-\ref{multiplikation:section:folgerung}.
-Temporibus autem quibusdam et aut officiis debitis aut rerum
-necessitatibus saepe eveniet ut et voluptates repudiandae sint et
-molestiae non recusandae.
-Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis
-voluptatibus maiores alias consequatur aut perferendis doloribus
-asperiores repellat.
-
-
diff --git a/buch/papers/multiplikation/teil2.tex b/buch/papers/multiplikation/teil2.tex
deleted file mode 100644
index efbf31a..0000000
--- a/buch/papers/multiplikation/teil2.tex
+++ /dev/null
@@ -1,40 +0,0 @@
-%
-% teil2.tex -- Beispiel-File für teil2
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 2
-\label{multiplikation:section:teil2}}
-\rhead{Teil 2}
-Sed ut perspiciatis unde omnis iste natus error sit voluptatem
-accusantium doloremque laudantium, totam rem aperiam, eaque ipsa
-quae ab illo inventore veritatis et quasi architecto beatae vitae
-dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit
-aspernatur aut odit aut fugit, sed quia consequuntur magni dolores
-eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam
-est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci
-velit, sed quia non numquam eius modi tempora incidunt ut labore
-et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima
-veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam,
-nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure
-reprehenderit qui in ea voluptate velit esse quam nihil molestiae
-consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla
-pariatur?
-
-\subsection{De finibus bonorum et malorum
-\label{multiplikation:subsection:bonorum}}
-At vero eos et accusamus et iusto odio dignissimos ducimus qui
-blanditiis praesentium voluptatum deleniti atque corrupti quos
-dolores et quas molestias excepturi sint occaecati cupiditate non
-provident, similique sunt in culpa qui officia deserunt mollitia
-animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis
-est et expedita distinctio. Nam libero tempore, cum soluta nobis
-est eligendi optio cumque nihil impedit quo minus id quod maxime
-placeat facere possimus, omnis voluptas assumenda est, omnis dolor
-repellendus. Temporibus autem quibusdam et aut officiis debitis aut
-rerum necessitatibus saepe eveniet ut et voluptates repudiandae
-sint et molestiae non recusandae. Itaque earum rerum hic tenetur a
-sapiente delectus, ut aut reiciendis voluptatibus maiores alias
-consequatur aut perferendis doloribus asperiores repellat.
-
-
diff --git a/buch/papers/multiplikation/teil3.tex b/buch/papers/multiplikation/teil3.tex
deleted file mode 100644
index f58508b..0000000
--- a/buch/papers/multiplikation/teil3.tex
+++ /dev/null
@@ -1,40 +0,0 @@
-%
-% teil3.tex -- Beispiel-File für Teil 3
-%
-% (c) 2020 Prof Dr Andreas Müller, Hochschule Rapperswil
-%
-\section{Teil 3
-\label{multiplikation:section:teil3}}
-\rhead{Teil 3}
-Sed ut perspiciatis unde omnis iste natus error sit voluptatem
-accusantium doloremque laudantium, totam rem aperiam, eaque ipsa
-quae ab illo inventore veritatis et quasi architecto beatae vitae
-dicta sunt explicabo. Nemo enim ipsam voluptatem quia voluptas sit
-aspernatur aut odit aut fugit, sed quia consequuntur magni dolores
-eos qui ratione voluptatem sequi nesciunt. Neque porro quisquam
-est, qui dolorem ipsum quia dolor sit amet, consectetur, adipisci
-velit, sed quia non numquam eius modi tempora incidunt ut labore
-et dolore magnam aliquam quaerat voluptatem. Ut enim ad minima
-veniam, quis nostrum exercitationem ullam corporis suscipit laboriosam,
-nisi ut aliquid ex ea commodi consequatur? Quis autem vel eum iure
-reprehenderit qui in ea voluptate velit esse quam nihil molestiae
-consequatur, vel illum qui dolorem eum fugiat quo voluptas nulla
-pariatur?
-
-\subsection{De finibus bonorum et malorum
-\label{multiplikation:subsection:malorum}}
-At vero eos et accusamus et iusto odio dignissimos ducimus qui
-blanditiis praesentium voluptatum deleniti atque corrupti quos
-dolores et quas molestias excepturi sint occaecati cupiditate non
-provident, similique sunt in culpa qui officia deserunt mollitia
-animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis
-est et expedita distinctio. Nam libero tempore, cum soluta nobis
-est eligendi optio cumque nihil impedit quo minus id quod maxime
-placeat facere possimus, omnis voluptas assumenda est, omnis dolor
-repellendus. Temporibus autem quibusdam et aut officiis debitis aut
-rerum necessitatibus saepe eveniet ut et voluptates repudiandae
-sint et molestiae non recusandae. Itaque earum rerum hic tenetur a
-sapiente delectus, ut aut reiciendis voluptatibus maiores alias
-consequatur aut perferendis doloribus asperiores repellat.
-
-
diff --git a/buch/papers/multiplikation/tikz_formulas/algo.fdb_latexmk b/buch/papers/multiplikation/tikz_formulas/algo.fdb_latexmk
new file mode 100644
index 0000000..5f14129
--- /dev/null
+++ b/buch/papers/multiplikation/tikz_formulas/algo.fdb_latexmk
@@ -0,0 +1,254 @@
+# Fdb version 3
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diff --git a/buch/papers/multiplikation/tikz_formulas/algo.pdf b/buch/papers/multiplikation/tikz_formulas/algo.pdf
new file mode 100644
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--- /dev/null
+++ b/buch/papers/multiplikation/tikz_formulas/algo.pdf
Binary files differ
diff --git a/buch/papers/multiplikation/tikz_formulas/algo.tex b/buch/papers/multiplikation/tikz_formulas/algo.tex
new file mode 100755
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+++ b/buch/papers/multiplikation/tikz_formulas/algo.tex
@@ -0,0 +1,131 @@
+\documentclass[border=10pt,varwidth]{standalone}
+\usepackage[left=25mm,right=25mm,top=25mm,bottom=25mm]{geometry}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{times}
+\usepackage{geometry}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{mathrsfs}
+\usepackage{amsfonts}
+\usepackage{amsthm}
+\usepackage{lipsum}
+\usepackage{amscd}
+\usepackage{graphicx}
+\usepackage{fancyhdr}
+\usepackage{textcomp}
+\usepackage{txfonts}
+\usepackage[all]{xy}
+\usepackage{paralist}
+\usepackage[colorlinks=true]{hyperref}
+\usepackage{array}
+\usepackage{tikz}
+\usepackage{slashed}
+\usepackage{pdfpages}
+\usepackage{cite}
+\usepackage{url}
+\usepackage{amsmath,amsfonts,amssymb}
+\usepackage{tikz}
+\usetikzlibrary{arrows,matrix,positioning}
+\usetikzlibrary{overlay-beamer-styles}
+\usetikzlibrary{matrix.skeleton}
+\usetikzlibrary{automata,positioning}
+\usepackage{listings}
+\usepackage{multirow}
+\usepackage{color}
+
+\begin{document}
+
+$
+A=
+\begin{bmatrix}
+A_{11} & A_{12}\\
+A_{21} & A_{22}
+\end{bmatrix},
+B=
+\begin{bmatrix}
+B_{11} & B_{12}\\
+B_{21} & B_{22}
+\end{bmatrix},
+C=
+\begin{bmatrix}
+C_{11} & C_{12}\\
+C_{21} & C_{22}
+\end{bmatrix}
+$
+
+\medskip
+$
+A \cdot B = C
+$
+
+\medskip
+$
+C_{11} = A_{11} \cdot B_{11} + A_{12} \cdot B_{21}\\
+C_{12} = A_{11} \cdot B_{12} + A_{12} \cdot B_{22}\\
+C_{21} = A_{21} \cdot B_{11} + A_{22} \cdot B_{21}\\
+C_{22} = A_{21} \cdot B_{12} + A_{22} \cdot B_{22}
+$
+
+\medskip
+\begin{math}
+\begin{aligned}
+\text{I} &= (A_{11} + A_{22}) \cdot (B_{11} + B_{22}) \\
+\text{II} &= (A_{21} + A_{22}) \cdot B_{11} \\
+\text{III} &= A_{11} \cdot (B_{12}-B_{22}) \\
+\text{IV} &= A_{22} \cdot (-B_{11}+B_{21}) \\
+\text{V} &= (A_{11} + A_{12}) \cdot B_{22} \\
+\text{VI} &= (-A_{11} + A_{21}) \cdot (B_{11} + B_{12})) \\
+\text{VII} &= (A_{12} - A_{22}) \cdot (B_{21} + B_{22}) \\
+\end{aligned}
+\end{math}
+
+
+\medskip
+\begin{math}
+\begin{aligned}
+C_{11} &= \text{I} + \text{IV} - \text{V} + \text{VII} \\
+C_{21} &= \text{II} + \text{IV} \\
+C_{12} &= \text{III} + \text{V}\\
+C_{22} &= \text{I} + \text{III} - \text{II} + \text{VI} \\
+\end{aligned}
+\end{math}
+
+
+\medskip
+\begin{math}
+\begin{aligned}
+C_{11} &= \text{II} + \text{IV} \\
+C_{11} &= (A_{11} + A_{22}) \cdot (B_{11} + B_{22}) + A_{22} \cdot (-B_{11}+B_{21}) - (A_{11} + A_{12}) \cdot B_{22} + (A_{12} - A_{22}) \cdot (B_{21} + B_{22})C_{21} \\
+C_{11} &= A_{11}B_{11} + A_{11}B_{22} + A_{22}B_{11} + A_{22}B_{22} -A_{22}B_{11}+A_{22}B_{21} - A_{11}B_{22} - A_{12}B_{22}+ A_{12}B_{21} + A_{12}B_{22} - A_{22}B_{21} - A_{22}B_{22} \\
+C_{11} &= A_{11}B_{11} + A_{12}B_{21}
+\end{aligned}
+\end{math}
+
+\section{Winograd}
+
+$
+x_1 y_1 + x_2 y_2 = (x_1 +y_2)(y_1 + x_2)-x_1 x_2 - y_1 y_2
+$
+
+$
+x = (x_1, \cdots, x_n), y=(y_1, \cdots, y_n)
+$
+
+\[
+\xi = \sum_{j=1}^{ \lfloor n/2 \rfloor} x_{2j-1} \cdot x_{2j}
+\]
+
+\[
+\eta = \sum_{j=1}^{ \lfloor n/2 \rfloor} y_{2j-1} \cdot y_{2j}
+\]
+
+\[
+\langle x,y \rangle =
+\begin{cases}
+ \displaystyle \sum_{j=1}^{ \lfloor n/2 \rfloor} (x_{2j-1} + y_{2j})(x_{2j}+y_{2j-1})-\xi - \eta & \text{if $n$ is even}\\
+\displaystyle \sum_{j=1}^{ \lfloor n/2 \rfloor} (x_{2j-1} + y_{2j})(x_{2j}+y_{2j-1})-\xi - \eta + x_n y_n & \text{if $n$ is odd}
+\end{cases}
+\]
+
+\end{document}
diff --git a/buch/papers/multiplikation/tikz_formulas/algo_graph.fdb_latexmk b/buch/papers/multiplikation/tikz_formulas/algo_graph.fdb_latexmk
new file mode 100644
index 0000000..ddfa880
--- /dev/null
+++ b/buch/papers/multiplikation/tikz_formulas/algo_graph.fdb_latexmk
@@ -0,0 +1,245 @@
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diff --git a/buch/papers/multiplikation/tikz_formulas/algo_graph.pdf b/buch/papers/multiplikation/tikz_formulas/algo_graph.pdf
new file mode 100755
index 0000000..7f5a984
--- /dev/null
+++ b/buch/papers/multiplikation/tikz_formulas/algo_graph.pdf
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diff --git a/buch/papers/multiplikation/tikz_formulas/algo_graph.tex b/buch/papers/multiplikation/tikz_formulas/algo_graph.tex
new file mode 100755
index 0000000..ad4228b
--- /dev/null
+++ b/buch/papers/multiplikation/tikz_formulas/algo_graph.tex
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+\usepackage[left=25mm,right=25mm,top=25mm,bottom=25mm]{geometry}
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+\usepackage[T1]{fontenc}
+\usepackage{times}
+\usepackage{geometry}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{mathrsfs}
+\usepackage{amsfonts}
+\usepackage{amsthm}
+\usepackage{lipsum}
+\usepackage{amscd}
+\usepackage{graphicx}
+\usepackage{fancyhdr}
+\usepackage{textcomp}
+\usepackage{txfonts}
+\usepackage[all]{xy}
+\usepackage{paralist}
+\usepackage[colorlinks=true]{hyperref}
+\usepackage{array}
+\usepackage{tikz}
+\usepackage{slashed}
+\usepackage{pdfpages}
+\usepackage{cite}
+\usepackage{url}
+\usepackage{amsmath,amsfonts,amssymb}
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+\usetikzlibrary{overlay-beamer-styles}
+\usetikzlibrary{matrix.skeleton}
+\usetikzlibrary{automata,positioning}
+\usepackage{listings}
+\usepackage{multirow}
+\usepackage{color}
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+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M43-3-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M43-4-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M46-1-3)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M46-1-1)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=green, fit=(M46-3-3)] {};
+ \node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M46-3-1)] {};
+\end{tikzpicture}
+
+
+
+\end{document}