Merge branch 'master' into master

14 files changed, 805 insertions, 0 deletions

diff --git a/buch/papers/multiplikation/images/algo_tab.pdf b/buch/papers/multiplikation/images/algo_tab.pdf
new file mode 100644
index 0000000..7f2bb4f
--- /dev/null
+++ b/buch/papers/multiplikation/images/algo_tab.pdf
diff --git a/buch/papers/multiplikation/images/algo_tab.tex b/buch/papers/multiplikation/images/algo_tab.tex
new file mode 100644
index 0000000..50ce392
--- /dev/null
+++ b/buch/papers/multiplikation/images/algo_tab.tex
@@ -0,0 +1,122 @@
+\documentclass{article}
+\usepackage[left=25mm,right=25mm,top=25mm,bottom=25mm]{geometry}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{times}
+\usepackage{geometry}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{algorithm}
+\usepackage{algpseudocode}
+\usepackage{mathrsfs}
+\usepackage{amsfonts}
+\usepackage{amsthm}
+\usepackage{lipsum}
+\usepackage{amscd}
+\usepackage{graphicx}
+\usepackage{fancyhdr}
+\usepackage{textcomp}
+\usepackage{pgfplots}
+\usepackage{txfonts}
+\usepackage[all]{xy}
+\usepackage{paralist}
+\usepackage[colorlinks=true]{hyperref}
+\usepackage{array}
+\usepackage{tikz}
+\usepackage{slashed}
+\usepackage{pdfpages}
+\usepackage{multicol}
+\usepackage{cite}
+\usepackage{url}
+\usepackage{amsmath,amsfonts,amssymb}
+\usepackage{tikz}
+\usetikzlibrary{arrows,matrix,positioning}
+\usetikzlibrary{overlay-beamer-styles}
+\usetikzlibrary{matrix.skeleton}
+\usetikzlibrary{automata,positioning}
+\usetikzlibrary{decorations.text}
+\usepackage{listings}
+\usepackage{multirow}
+\usepackage{color}
+
+\begin{document}
+
+
+
+\begin{table}[t]
+	\begin{tabular}{ll}
+		\begin{minipage}{0.4\textwidth}
+			\begin{algorithm}[H]\footnotesize\caption{}
+				\label{multiplikation:alg:b1}
+				\setlength{\lineskip}{7pt}
+				\begin{algorithmic}
+					\Function{B1}{$a, b$}
+					\State \textbf{return} $a+b$
+					\EndFunction
+					\State
+					\State
+				\end{algorithmic}
+			\end{algorithm}
+		\end{minipage}	
+		&
+		\begin{minipage}{0.4\textwidth}
+	\begin{algorithm}[H]\footnotesize\caption{}
+	\label{multiplikation:alg:b2}
+	\setlength{\lineskip}{7pt}
+	\begin{algorithmic}
+		\Function{B2}{$a, b$}
+		\State $ x \gets a+b $
+		\State $ y \gets a \cdot b $
+		\State \textbf{return} $x+y$
+		\EndFunction
+	\end{algorithmic}
+\end{algorithm}
+
+		\end{minipage}
+	\end{tabular}
+\end{table}
+
+\begin{table}
+	\begin{tabular}[t]{ll} 
+		\begin{minipage}{0.4\textwidth}
+			\begin{algorithm}[H]\footnotesize\caption{}
+				\setlength{\lineskip}{7pt}
+				\begin{algorithmic}
+					\label{multiplikation:alg:linear}
+					\Function{L}{$\mathbf{a}, \mathbf{b}$,n}
+					\State $ sum \gets 0$
+					\For{$i = 0,1,2 \dots,n$}
+					\State $ sum \gets sum + A[i] \cdot B[i] $
+					\EndFor
+					
+					\State \textbf{return} $sum$
+					
+					\EndFunction
+					\State
+					\State
+				\end{algorithmic}
+			\end{algorithm}
+		\end{minipage}	
+		&
+		\begin{minipage}{0.4\textwidth}
+			\begin{algorithm}[H]\footnotesize\caption{}
+			\label{multiplikation:alg:q1}
+			\setlength{\lineskip}{7pt}
+			\begin{algorithmic}
+				\Function{Q}{$\mathbf{A}, \mathbf{B}$,n}
+				\State $ sum \gets 0$
+				\For{$i = 0,1,2 \dots,n$}
+				\For{$j = 0,1,2 \dots,n$}
+				\State $ sum \gets sum + A[i] \cdot B[j] $
+				\EndFor
+				\EndFor
+				\State \textbf{return} $sum$
+				\EndFunction
+			\end{algorithmic}
+		\end{algorithm}
+		\end{minipage}
+	\end{tabular}
+\end{table}
+
+dhdfh
+\end{document}
diff --git a/buch/papers/multiplikation/images/bigo.pdf b/buch/papers/multiplikation/images/bigo.pdf
new file mode 100644
index 0000000..2519553
--- /dev/null
+++ b/buch/papers/multiplikation/images/bigo.pdf
diff --git a/buch/papers/multiplikation/images/bigo.tex b/buch/papers/multiplikation/images/bigo.tex
new file mode 100644
index 0000000..63fd0fd
--- /dev/null
+++ b/buch/papers/multiplikation/images/bigo.tex
@@ -0,0 +1,112 @@
+\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}[
+    xmode=log, ymode=log,
+    xmin=1e-0, xmax=5000,
+    ymin=10e-1, ymax=1e7,
+    grid=both,
+    major grid style={black!50},
+    xlabel = data input size,
+    ylabel = {time},
+    legend pos=north west,
+    very thick,
+    yticklabels=\empty,
+    xticklabels=\empty,
+    scale only axis=true,
+  width=12cm, height=8cm,
+  legend cell align={left}
+    ]
+\addplot [
+    domain= 1:5000,
+    samples=100,
+    color=red,
+]
+{1};
+\addlegendentry{$\mathcal{O}(1)$}
+\addplot [
+    domain= 1:5000,
+    samples=100,
+    color=green,
+]
+{x};
+\addlegendentry{$\mathcal{O}(n)$}
+\addplot [
+    domain= 1:50000,
+    samples=100,
+    color=blue,
+]
+{x^2};
+\addlegendentry{$\mathcal{O}\left(n^2\right)$}
+\addplot [
+    domain= 1:500,
+    samples=100,
+    color=purple,
+]
+{x^3};
+\addlegendentry{$\mathcal{O}\left(n^3\right)$}
+\addplot [
+    domain= 1:500,
+    samples=100,
+    color=black,
+]
+{exp(x) - 1.7};
+\addlegendentry{$\mathcal{O}\left(e^n\right)$}
+\addplot [
+    domain= 1:5000,
+    samples=100,
+    color=orange,
+]
+{log2(x)+1};
+\addlegendentry{$\mathcal{O}(\log n)$}
+
+\addplot [
+    domain= 1:5000,
+    samples=100,
+    color=gray,
+]
+{x*log2(x)+1};
+\addlegendentry{$\mathcal{O}(n \log n)$}
+\end{axis}
+\end{tikzpicture}
+
+\end{document}
diff --git a/buch/papers/multiplikation/images/c_meas_4096.pdf b/buch/papers/multiplikation/images/c_meas_4096.pdf
new file mode 100644
index 0000000..304015a
--- /dev/null
+++ b/buch/papers/multiplikation/images/c_meas_4096.pdf
diff --git a/buch/papers/multiplikation/images/meas_1024.pdf b/buch/papers/multiplikation/images/meas_1024.pdf
new file mode 100644
index 0000000..70c7ec1
--- /dev/null
+++ b/buch/papers/multiplikation/images/meas_1024.pdf
diff --git a/buch/papers/multiplikation/images/meas_c.pdf b/buch/papers/multiplikation/images/meas_c.pdf
new file mode 100644
index 0000000..521151e
--- /dev/null
+++ b/buch/papers/multiplikation/images/meas_c.pdf
diff --git a/buch/papers/multiplikation/images/meas_c.tex b/buch/papers/multiplikation/images/meas_c.tex
new file mode 100644
index 0000000..12d3527
--- /dev/null
+++ b/buch/papers/multiplikation/images/meas_c.tex
@@ -0,0 +1,150 @@
+
+\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}
+\usepackage{pgfplotstable}
+\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}[
+xmode=log, ymode=log,
+xmin=30, xmax=10000,
+ymin=1e-5, ymax=2e4,
+grid=both,
+major grid style={black!50},
+xlabel = data input ($n$),
+ylabel = {time ($s$)},
+legend pos=north west,
+very thick,
+scale only axis=true,
+width=12cm, height=8cm,
+       log basis x={10},
+       legend cell align={left}
+]
+\addlegendentry{Winograd}
+\addplot[    color=blue,
+        error bars/.cd, y dir=both, y explicit,
+] coordinates {
+%(2,1e-07)
+%(4,5e-07)
+%(8,2.0000000000000003e-06)
+%(16,1.1999999999999999e-05)
+(32,8.329999999999999e-05)
+(64,0.0006479)
+(128,0.0052873)
+(256,0.052674599999999995)
+(512,0.5249752000000001)
+(1024,4.671161)
+(2048,136.6769777)
+(4096,1179.261048)
+(8192,10071.512655)
+};
+\addlegendentry{Strassen}
+\addplot [    color=black,
+]coordinates {
+%(2,1e-07)
+%(4,2.1e-06)
+%(8,1.13e-05)
+%(16,7.07e-05)
+(32,0.0005041)
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+(512,1.2555)
+(1024,8.8302371)
+(2048,61.9018691)
+(4096,414.648901)
+(8192,3014.235467)
+};
+
+\addlegendentry{MM div and conq}
+\addplot[    color=green,
+] coordinates {
+%(2,3e-07)
+%(4,1.1e-06)
+%(8,8.6e-06)
+%(16,7.819999999999999e-05)
+(32,0.0005940000000000001)
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+(128,0.0348443)
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+(512,2.2228507)
+(1024,17.659234500000004)
+(2048,141.6103936)
+(4096,1147.106865)
+(8192,9606.402522)
+};
+
+\addlegendentry{MM}
+\addplot [    color=red,
+]coordinates {
+%(2,0.0)
+%(4,3e-07)
+%(8,1.8000000000000001e-06)
+%(16,1.1999999999999999e-05)
+(32,8.93e-05)
+(64,0.0006923)
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+(512,0.5062468000000001)
+(1024,4.5048086)
+(2048,129.2894619)
+(4096,1111.312696)
+(8192,9376.173434)
+};
+\addlegendentry{BLAS}
+\addplot[    color=purple,
+] coordinates {
+%(2,1e-07)
+%(4,0.0)
+%(8,1e-07)
+%(16,3.9e-06)
+(32,2.1000000000000002e-05)
+(64,0.00018580000000000002)
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+(512,0.0773765)
+(1024,0.7643868)
+(2048,7.6320993999999995)
+(4096,55.845038)
+(8192,478.429957)
+};
+\end{axis}
+\end{tikzpicture}
+
+\end{document}
diff --git a/buch/papers/multiplikation/images/meas_python.pdf b/buch/papers/multiplikation/images/meas_python.pdf
new file mode 100644
index 0000000..fe89773
--- /dev/null
+++ b/buch/papers/multiplikation/images/meas_python.pdf
diff --git a/buch/papers/multiplikation/images/meas_python.tex b/buch/papers/multiplikation/images/meas_python.tex
new file mode 100644
index 0000000..ad43cf6
--- /dev/null
+++ b/buch/papers/multiplikation/images/meas_python.tex
@@ -0,0 +1,145 @@
+
+\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}
+\usepackage{pgfplotstable}
+\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}[
+xmode=log, ymode=log,
+xmin=30, xmax=4200,
+ymin=0.01, ymax=70000,
+grid=both,
+major grid style={black!50},
+xlabel = data input ($n$),
+ylabel = {time ($s$)},
+legend pos=north west,
+very thick,
+scale only axis=true,
+width=12cm, height=8cm,
+       log basis x={10},
+       legend cell align={left}
+]
+\addlegendentry{Winograd}
+\addplot[    color=blue,
+] coordinates {
+% (2,   2.7895e-05 )
+% (4,   0.000104904)
+% (8,   0.000552893)
+% (16,  0.0045557  )
+(32,  0.0187144  )
+(64,  0.153069   )
+(128, 1.19476    )
+(256, 8.29899    )
+(512, 68.3699    )
+(1024,537.374    )
+(2046,4884.61)
+(4096,43597.1)
+};
+\addlegendentry{Strassen}
+\addplot [    color=black,
+]coordinates {
+  %  (2,2.09808e-05 )
+  %  (4,0.000174284 )
+  %  (8,0.000943899 )
+  % (16,0.00475407  )
+  (32,0.0485256   )
+  (64,0.220414    )
+ (128,1.44718    )
+ (256,9.93866    )
+ (512,63.961     )
+(1024,461.494    )
+(2046,3860.57)
+(4096,22904.3)
+};
+
+\addlegendentry{MM div and conq}
+\addplot[    color=green,
+] coordinates {
+  %  (2,8.10623e-06 )
+  %  (4,9.01222e-05 )
+  %  (8,0.000729084 )
+  % (16,0.00497079  )
+  (32,0.02719     )
+  (64,0.26528     )
+ (128,1.77787      )
+ (256,13.27        )
+ (512,105.397      )
+(1024,847.321      )
+(2046,7375.93)
+(4096,58466)
+};
+
+\addlegendentry{MM}
+\addplot [    color=red,
+]coordinates {
+  %  (2,1.85966e-05)
+  %  (4,8.29697e-05 )
+  %  (8,0.000547171)
+  % (16,0.00305367   )
+  (32,  0.0240743 )
+  (64,  0.186895 )
+ (128,  1.56369 )
+ (256,  11.0062 )
+ (512,   85.4768)
+(1024,750.757 )
+(2046,6154.18)
+(4096,46813.3)
+};
+%  \addlegendentry{NumPy}
+%  \addplot[    color=blue,
+%  ] coordinates {
+% %    (2,1.83582e-05 )
+% %    (4,7.86781e-06)
+% %    (8,1.00136e-05)
+% %   (16,5.4121e-05 )
+%    (32,4.26769e-05)
+%  (64,0.000118494)
+%   (128,0.000244141 )
+%   (256,0.000695705  )
+%   (512,0.00221705   )
+%  (1024,0.0188088    )
+% (2046,0.215739)
+% (4096,1.49159)
+%  };
+
+\end{axis}
+\end{tikzpicture}
+
+\end{document}
diff --git a/buch/papers/multiplikation/images/mm_visualisation.pdf b/buch/papers/multiplikation/images/mm_visualisation.pdf
new file mode 100644
index 0000000..9309df1
--- /dev/null
+++ b/buch/papers/multiplikation/images/mm_visualisation.pdf
diff --git a/buch/papers/multiplikation/images/mm_visualisation.tex b/buch/papers/multiplikation/images/mm_visualisation.tex
new file mode 100644
index 0000000..6e8f789
--- /dev/null
+++ b/buch/papers/multiplikation/images/mm_visualisation.tex
@@ -0,0 +1,45 @@
+
+  \begin{tikzpicture}[ampersand replacement=\&]
+
+    \matrix (A)[matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}] at (0,0)
+     {
+     A_{1,1} \& \cdots \& A_{1,k} \& \cdots \& A_{1,n} \\
+     \vdots \&  \& \vdots \&  \& \vdots \\
+     A_{i,1} \& \cdots \& A_{i,k} \& \cdots \& A_{i,n} \\
+     \vdots \&  \& \vdots \& \& \vdots \\
+     A_{m,1} \& \cdots \& A_{m,k} \& \cdots \& A_{m,n} \\
+     };
+ 
+  	 \node [right=0.1 of A] (mul) {$\cdot$};
+
+	 
+     \matrix (B)[right=0.1 of mul, matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}]
+      {
+      B_{1,1} \& \cdots \& B_{1,j} \& \cdots \& B_{1,p} \\
+      \vdots \&  \& \vdots \&  \& \vdots \\
+      B_{k,1} \& \cdots \& B_{k,j} \& \cdots \& B_{k,p} \\
+      \vdots \&  \& \vdots \& \& \vdots \\
+      B_{n,1} \& \cdots \& B_{n,j} \& \cdots \& B_{n,p} \\
+      };
+  
+  	 \node [right=0.1 of B] (eq) {$=$};
+
+     \matrix (C)[right=0.1 of eq, matrix of math nodes, label skeleton, left delimiter=[,right delimiter={]}]
+      {
+      C_{1,1} \& \cdots \& C_{1,j} \& \cdots \& C_{1,p} \\
+      \vdots \&  \& \vdots \&  \& \vdots \\
+      C_{i,1} \& \cdots \& C_{i,j} \& \cdots \& C_{i,p} \\
+      \vdots \&  \& \vdots \& \& \vdots \\
+      C_{m,1} \& \cdots \& C_{m,j} \& \cdots \& C_{m,p} \\
+      };
+
+
+       \node[opacity=0.5, rounded corners=2pt, inner sep=-1pt, fill=green, fit=(A-3-1)(A-3-5)] {};
+       \node[opacity=0.5, rounded corners=2pt, inner sep=-1pt, fill=blue, fit=(B-1-3)(B-5-3)] {};
+       \node[opacity=0.5, rounded corners=2pt, inner sep=-1pt, fill=red, fit=(C-3-3)] {};
+
+
+   \end{tikzpicture}
+
+\end{document}
+
diff --git a/buch/papers/multiplikation/images/strassen.pdf b/buch/papers/multiplikation/images/strassen.pdf
new file mode 100644
index 0000000..d150125
--- /dev/null
+++ b/buch/papers/multiplikation/images/strassen.pdf
diff --git a/buch/papers/multiplikation/images/strassen.tex b/buch/papers/multiplikation/images/strassen.tex
new file mode 100644
index 0000000..b51a9d5
--- /dev/null
+++ b/buch/papers/multiplikation/images/strassen.tex
@@ -0,0 +1,231 @@
+\documentclass[border=10pt]{standalone}
+\usepackage[left=25mm,right=25mm,top=25mm,bottom=25mm]{geometry}
+\usepackage[utf8]{inputenc}
+\usepackage[T1]{fontenc}
+\usepackage{times}
+\usepackage{geometry}
+\usepackage{amsmath}
+\usepackage{amssymb}
+\usepackage{mathrsfs}
+\usepackage{amsfonts}
+\usepackage{amsthm}
+\usepackage{lipsum}
+\usepackage{amscd}
+\usepackage{graphicx}
+\usepackage{fancyhdr}
+\usepackage{textcomp}
+\usepackage{pgfplots}
+\usepackage{txfonts}
+\usepackage[all]{xy}
+\usepackage{paralist}
+\usepackage[colorlinks=true]{hyperref}
+\usepackage{array}
+\usepackage{tikz}
+\usepackage{slashed}
+\usepackage{pdfpages}
+\usepackage{cite}
+\usepackage{url}
+\usepackage{amsmath,amsfonts,amssymb}
+\usepackage{tikz}
+\usetikzlibrary{arrows,matrix,positioning}
+\usetikzlibrary{overlay-beamer-styles}
+\usetikzlibrary{matrix.skeleton}
+\usetikzlibrary{automata,positioning}
+\usetikzlibrary{decorations.text}
+\usepackage{listings}
+\usepackage{multirow}
+\usepackage{color}
+
+\begin{document}
+
+\begin{tikzpicture}[ampersand replacement=\&]
+
+\foreach \i in {1,...,4}
+{
+	\small{
+		\matrix (X\i)[matrix of math nodes,nodes in empty cells,
+		nodes = {draw, minimum size=10mm,
+			anchor=center,
+			inner sep=0pt, outer sep=0pt},
+		column sep=-\pgflinewidth,
+		row sep=-\pgflinewidth,
+		] at (0,-\i*5)
+		{
+			A_{11}B_{11} \& A_{12}B_{11} \& A_{21}B_{11} \& A_{22}B_{11} \\
+			A_{11}B_{21} \& A_{12}B_{21} \& A_{21}B_{21} \& A_{22}B_{21} \\
+			A_{11}B_{11} \& A_{12}B_{12} \& A_{21}B_{12} \& A_{22}B_{12} \\
+			A_{11}B_{22} \& A_{12}B_{22} \& A_{21}B_{22} \& A_{22}B_{22} \\
+		};}
+
+	\foreach \j in {1,...,7}
+	{
+		\matrix(M\i\j)[matrix of math nodes,nodes in empty cells,
+		nodes = {draw, minimum size=10mm,
+			anchor=center,
+			inner sep=0pt, outer sep=0pt},
+		column sep=-\pgflinewidth,
+		row sep=-\pgflinewidth,
+		] at (\j*5,-\i*5)
+		{
+			\& \& \& \\
+			\& \& \& \\
+			\& \& \& \\
+			\& \& \& \\
+		};
+	}
+}
+
+\huge{
+	\node at (-3,-20) {$\mathbf{C}_{22}=$};
+	\node at (-3,-15) {$\mathbf{C}_{21}=$} ;
+	\node at (-3,-10) {$\mathbf{C}_{12}=$} ;
+	\node at (-3,-5)  {$\mathbf{C}_{11}=$} ;
+
+	\node at (5,-2)   {$\mathbf{P}$};
+	\node at (10,-2)  {$\mathbf{Q}$};
+	\node at (15,-2)  {$\mathbf{R}$};
+	\node at (20,-2)  {$\mathbf{S}$};
+	\node at (25,-2)  {$\mathbf{T}$};
+	\node at (30,-2)  {$\mathbf{U}$};
+	\node at (35,-2)  {$\mathbf{V}$};
+}
+
+
+\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)] {};
+
+% P
+\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=gray, fit=(M21-4-1)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M21-1-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M21-4-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M21-1-1)] {};
+
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M31-4-1)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M31-1-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M31-4-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M31-1-1)] {};
+
+\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)] {};
+
+% Q
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M12-1-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M12-1-3)] {};
+
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M22-1-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M22-1-3)] {};
+
+\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=(M42-1-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=red, fit=(M42-1-3)] {};
+
+% R
+
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M13-3-1)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M13-4-1)] {};
+
+\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=gray, fit=(M33-3-1)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M33-4-1)] {};
+
+\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)] {};
+
+% S
+
+\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=gray, fit=(M24-1-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M24-2-4)] {};
+
+\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=gray, fit=(M44-1-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M44-2-4)] {};
+
+%T
+
+\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=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=gray, fit=(M35-4-1)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M35-4-2)] {};
+
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M45-4-1)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M45-4-2)] {};
+
+% U
+
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M16-1-3)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M16-1-1)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M16-3-3)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M16-3-1)] {};
+
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M26-1-3)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M26-1-1)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M26-3-3)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M26-3-1)] {};
+
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M36-1-3)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M36-1-1)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M36-3-3)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M36-3-1)] {};
+
+\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)] {};
+
+%V
+
+\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=gray, fit=(M27-2-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M27-4-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M27-2-2)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M27-4-2)] {};
+
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M37-2-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M37-4-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M37-2-2)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M37-4-2)] {};
+
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M47-2-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M47-4-4)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M47-2-2)] {};
+\node[opacity=0.5, rounded corners=0pt, inner sep=-1pt, fill=gray, fit=(M47-4-2)] {};
+
+
+
+
+
+\end{tikzpicture}
+
+\end{document}