Merge remote-tracking branch 'upstream/master'

15 files changed, 912 insertions, 1 deletions

diff --git a/buch/chapters/40-eigenwerte/images/Makefile b/buch/chapters/40-eigenwerte/images/Makefile
index 753153d..54b36d5 100644
--- a/buch/chapters/40-eigenwerte/images/Makefile
+++ b/buch/chapters/40-eigenwerte/images/Makefile
@@ -3,7 +3,9 @@
 #
 # (c) 2020 Prof Dr Andreas Müller, Hochschule Rappersil
 #
-all:	sp.pdf   nilpotent.pdf kernbild.pdf kombiniert.pdf
+all:	sp.pdf nilpotent.pdf kernbild.pdf kombiniert.pdf \
+	wurzelapprox.pdf wurzel.pdf dimjk.pdf jknilp.pdf \
+	normalform.pdf minmax.pdf
 
 sp.pdf:	sp.tex sppaths.tex
 	pdflatex sp.tex
@@ -19,3 +21,24 @@ kernbild.pdf:	kernbild.tex bild2.jpg kern2.jpg
 
 kombiniert.pdf:	kombiniert.tex kombiniert.jpg
 	pdflatex kombiniert.tex
+
+wurzelapprox.pdf:	wurzelapprox.tex wa.tex
+	pdflatex wurzelapprox.tex
+
+wa.tex:	wa.m
+	octave wa.m
+
+wurzel.pdf:	wurzel.tex
+	pdflatex wurzel.tex
+
+dimjk.pdf:	dimjk.tex
+	pdflatex dimjk.tex
+
+jknilp.pdf:	jknilp.tex
+	pdflatex jknilp.tex
+
+normalform.pdf:	normalform.tex
+	pdflatex normalform.tex
+
+minmax.pdf:	minmax.tex
+	pdflatex minmax.tex
diff --git a/buch/chapters/40-eigenwerte/images/dimjk.pdf b/buch/chapters/40-eigenwerte/images/dimjk.pdf
new file mode 100644
index 0000000..fcfe4da
--- /dev/null
+++ b/buch/chapters/40-eigenwerte/images/dimjk.pdf
diff --git a/buch/chapters/40-eigenwerte/images/dimjk.tex b/buch/chapters/40-eigenwerte/images/dimjk.tex
new file mode 100644
index 0000000..28e0f9f
--- /dev/null
+++ b/buch/chapters/40-eigenwerte/images/dimjk.tex
@@ -0,0 +1,78 @@
+%
+% dimjk.tex -- dimensionen von K^l und J^l
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\def\skala{1.2}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+\def\pfad{
+        ({0*\sx},{6-6}) --
+        ({1*\sx},{6-4.5}) --
+        ({2*\sx},{6-3.5}) --
+        ({3*\sx},{6-2.9}) --
+        ({4*\sx},{6-2.6}) --
+        ({5*\sx},{6-2.4}) --
+        ({8*\sx},{6-2.4})
+}
+\def\sx{1.2}
+
+\fill[color=orange!20] \pfad -- ({6*\sx},6) -- (0,6) -- cycle;
+\fill[color=darkgreen!20] \pfad -- ({6*\sx},0) -- cycle;
+
+\fill[color=orange!40] ({5*\sx},6) rectangle ({8*\sx},{6-2.4});
+\fill[color=darkgreen!40] ({5*\sx},0) rectangle ({8*\sx},{6-2.4});
+
+\draw[color=darkgreen,line width=2pt] ({3*\sx},{6-6}) -- ({3*\sx},{6-2.9});
+\node[color=darkgreen] at ({3*\sx},{6-4.45}) [rotate=90,above] {$\dim\mathcal{K}^k(A)$};
+\draw[color=orange,line width=2pt] ({3*\sx},{6-0}) -- ({3*\sx},{6-2.9});
+\node[color=orange] at ({3*\sx},{6-1.45}) [rotate=90,above] {$\dim\mathcal{J}^k(A)$};
+
+\node[color=orange] at ({6.5*\sx},{6-1.2}) {bijektiv};
+\node[color=darkgreen] at ({6.5*\sx},{6-4.2}) {konstant};
+
+\fill ({0*\sx},{6-6}) circle[radius=0.08];
+\fill ({1*\sx},{6-4.5}) circle[radius=0.08];
+\fill ({2*\sx},{6-3.5}) circle[radius=0.08];
+\fill ({3*\sx},{6-2.9}) circle[radius=0.08];
+\fill ({4*\sx},{6-2.6}) circle[radius=0.08];
+\fill ({5*\sx},{6-2.4}) circle[radius=0.08];
+\fill ({6*\sx},{6-2.4}) circle[radius=0.08];
+\fill ({7*\sx},{6-2.4}) circle[radius=0.08];
+\fill ({8*\sx},{6-2.4}) circle[radius=0.08];
+
+\draw \pfad;
+
+\draw[->] (-0.5,0) -- ({8*\sx+0.5},0) coordinate[label={$k$}];
+\draw[->] (-0.5,6) -- ({8*\sx+0.5},6);
+
+\foreach \x in {0,...,8}{
+        \draw ({\x*\sx},-0.05) -- ({\x*\sx},0.05);
+}
+\foreach \x in {0,...,3}{
+        \node at ({\x*\sx},-0.05) [below] {$\x$};
+}
+\node at ({4*\sx},-0.05) [below] {$\dots\mathstrut$};
+\node at ({5*\sx},-0.05) [below] {$l$};
+\node at ({6*\sx},-0.05) [below] {$l+1$};
+\node at ({7*\sx},-0.05) [below] {$l+2$};
+\node at ({8*\sx},-0.05) [below] {$l+3$};
+
+\node[color=orange] at ({1.2*\sx},5.6)
+	{$\mathcal{J}^k(A)\supset\mathcal{J}^{k+1}(A)$};
+\node[color=darkgreen] at ({1.2*\sx},0.4)
+	{$\mathcal{K}^k(A)\subset\mathcal{K}^{k+1}(A)$};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/40-eigenwerte/images/jknilp.pdf b/buch/chapters/40-eigenwerte/images/jknilp.pdf
new file mode 100644
index 0000000..9293263
--- /dev/null
+++ b/buch/chapters/40-eigenwerte/images/jknilp.pdf
diff --git a/buch/chapters/40-eigenwerte/images/jknilp.tex b/buch/chapters/40-eigenwerte/images/jknilp.tex
new file mode 100644
index 0000000..e8e8e14
--- /dev/null
+++ b/buch/chapters/40-eigenwerte/images/jknilp.tex
@@ -0,0 +1,181 @@
+%
+% jknilp.tex -- Dimensionen von K^l und J^l für nilpotente Matrizen
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+\def\s{0.15}
+\def\punkt#1#2{({#1*\s},{#2*\s})}
+
+\def\vektor#1{
+	\fill[color=darkgreen!30] \punkt{#1}{0} rectangle \punkt{(#1+1)}{12};
+}
+\def\feld#1#2{
+	\fill[color=orange!60] ({#1*\s},{(12-#2)*\s}) rectangle
+		({(#1+1)*\s},{(11-#2)*\s});
+}
+
+\def\quadrat#1{
+	\draw \punkt{0}{0} rectangle \punkt{12}{12};
+
+	\draw \punkt{0}{11} -- \punkt{2}{11} -- \punkt{2}{9} -- \punkt{4}{9}
+		-- \punkt{4}{6} -- \punkt{12}{6};
+
+	\draw \punkt{1}{12} -- \punkt{1}{10} -- \punkt{3}{10}
+		-- \punkt{3}{8} -- \punkt{6}{8} -- \punkt{6}{0};
+	\node at ({6*\s},0) [below] {#1\strut};
+}
+
+\begin{scope}[xshift=-0.9cm,yshift=-3cm]
+\foreach \n in {0,...,11}{
+	\feld{\n}{\n}
+}
+\quadrat{$A^0=I$}
+\end{scope}
+
+\begin{scope}[xshift=1.1cm,yshift=-3cm]
+\vektor{0}
+\vektor{1}
+\vektor{2}
+\vektor{3}
+\vektor{4}
+\vektor{6}
+\feld{5}{4}
+\feld{7}{6}
+\feld{8}{7}
+\feld{9}{8}
+\feld{10}{9}
+\feld{11}{10}
+\quadrat{$A$}
+\end{scope}
+
+\begin{scope}[xshift=3.1cm,yshift=-3cm]
+\vektor{0}
+\vektor{1}
+\vektor{2}
+\vektor{3}
+\vektor{4}
+\vektor{5}
+\vektor{6}
+\vektor{7}
+\feld{8}{6}
+\feld{9}{7}
+\feld{10}{8}
+\feld{11}{9}
+\quadrat{$A^2$}
+\end{scope}
+
+\begin{scope}[xshift=5.1cm,yshift=-3cm]
+\vektor{0}
+\vektor{1}
+\vektor{2}
+\vektor{3}
+\vektor{4}
+\vektor{5}
+\vektor{6}
+\vektor{7}
+\vektor{8}
+\feld{9}{6}
+\feld{10}{7}
+\feld{11}{8}
+\quadrat{$A^3$}
+\end{scope}
+
+\begin{scope}[xshift=7.1cm,yshift=-3cm]
+\vektor{0}
+\vektor{1}
+\vektor{2}
+\vektor{3}
+\vektor{4}
+\vektor{5}
+\vektor{6}
+\vektor{7}
+\vektor{8}
+\vektor{9}
+\feld{10}{6}
+\feld{11}{7}
+\quadrat{$A^4$}
+\end{scope}
+
+\begin{scope}[xshift=9.1cm,yshift=-3cm]
+\vektor{0}
+\vektor{1}
+\vektor{2}
+\vektor{3}
+\vektor{4}
+\vektor{5}
+\vektor{6}
+\vektor{7}
+\vektor{8}
+\vektor{9}
+\vektor{10}
+\feld{11}{6}
+\quadrat{$A^5$}
+\end{scope}
+
+\begin{scope}[xshift=11.1cm,yshift=-3cm]
+\vektor{0}
+\vektor{1}
+\vektor{2}
+\vektor{3}
+\vektor{4}
+\vektor{5}
+\vektor{6}
+\vektor{7}
+\vektor{8}
+\vektor{9}
+\vektor{10}
+\vektor{11}
+\quadrat{$A^6$}
+\end{scope}
+
+\def\pfad{
+	(0,0) -- (2,3) -- (4,4) -- (6,4.5) -- (8,5) -- (10,5.5) -- (12,6)
+}
+
+
+\fill[color=orange!20] \pfad -- (-1,6) -- (-1,0) -- cycle;
+\fill[color=darkgreen!20] \pfad -- (13,6) -- (13,0) -- cycle;
+\draw[line width=1.3pt] \pfad;
+
+\fill (0,0) circle[radius=0.08];
+\fill (2,3) circle[radius=0.08];
+\fill (4,4) circle[radius=0.08];
+\fill (6,4.5) circle[radius=0.08];
+\fill (8,5) circle[radius=0.08];
+\fill (10,5.5) circle[radius=0.08];
+\fill (12,6) circle[radius=0.08];
+
+\foreach \y in {0.5,1,...,5.5}{
+	\draw[line width=0.3pt] (-1.1,\y) -- (13.0,\y);
+}
+\foreach \y in {0,2,4,...,12}{
+	\node at (-1.1,{\y*0.5}) [left] {$\y$};
+}
+\foreach \x in {0,...,6}{
+	\draw ({2*\x},0) -- ({2*\x},-1.2);
+	\node at ({2*\x},-0.6) [above,rotate=90] {$k=\x$};
+}
+
+\draw[->] (-1.1,0) -- (13.4,0) coordinate[label={$k$}];
+\draw[->] (-1.1,6) -- (13.4,6);
+\draw[->] (-1.0,0) -- (-1.0,6.5);
+
+\node[color=darkgreen] at (8,1.95) [above] {$\dim \mathcal{K}^k(A)$};
+\node[color=orange] at (2,4.95) [above] {$\dim \mathcal{J}^k(A)$};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/40-eigenwerte/images/minmax.pdf b/buch/chapters/40-eigenwerte/images/minmax.pdf
new file mode 100644
index 0000000..46ed28a
--- /dev/null
+++ b/buch/chapters/40-eigenwerte/images/minmax.pdf
diff --git a/buch/chapters/40-eigenwerte/images/minmax.tex b/buch/chapters/40-eigenwerte/images/minmax.tex
new file mode 100644
index 0000000..f661d5b
--- /dev/null
+++ b/buch/chapters/40-eigenwerte/images/minmax.tex
@@ -0,0 +1,134 @@
+%
+% minmax.tex -- minimum und maximum
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\definecolor{darkgreen}{rgb}{0,0.5,0}
+
+\def\mittellinie{
+	plot[domain=0:6.2832,samples=400]
+		({\x},{0.5*(sin(180*\x/3.14159)+cos(180*\x/3.14159))})
+}
+
+\begin{scope}
+	\fill[color=darkgreen!20]
+		plot[domain=0:6.2832,samples=360]
+			({\x},{sin(180*\x/3.1415)})
+		--
+		plot[domain=6.2832:0,samples=360]
+			({\x},{cos(180*\x/3.1415)})
+		-- cycle;
+	\foreach \x in {0.5,1,...,6}{
+		\draw[color=darkgreen]
+			({\x},{sin(180*\x/3.1415)})
+			--
+			({\x},{cos(180*\x/3.1415)});
+	}
+
+	\node[color=darkgreen] at (2,-0.8) [left] {$|f(x)-g(x)|$};
+	\draw[color=darkgreen,line width=0.3pt] (2,-0.8) -- (2.5,-0.7);
+
+	\draw[color=blue,line width=1.4pt] plot[domain=0:6.29,samples=360]
+		({\x},{sin(180*\x/3.1415)});
+	\draw[color=red,line width=1.4pt] plot[domain=0:6.29,samples=360]
+		({\x},{cos(180*\x/3.1415)});
+	\draw[color=purple!50,line width=1.4pt] \mittellinie;
+	\node[color=purple!50] at (6.2832,0.5) [right] {$\frac12(f(x)+g(x))$};
+
+	\draw[->] (-0.1,0) -- (6.5,0) coordinate[label={below:$x$}];
+	\draw[->] (0,-1.1) -- (0,1.3) coordinate[label={right:$y$}];
+
+
+	\xdef\x{2}
+	\node[color=blue] at (\x,{sin(180*\x/3.1415)}) [above right] {$f(x)$};
+	\pgfmathparse{2.5*3.14159-\x}
+	\xdef\x{\pgfmathresult}
+	\node[color=red] at (\x,{cos(180*\x/3.1415)}) [above left] {$g(x)$};
+
+\end{scope}
+
+\draw[->,line width=4pt,color=gray!40] ({3.1415-1},-1.3) -- ({3.1415-2.3},-3);
+\draw[->,line width=4pt,color=gray!40] ({3.1415+1},-1.3) -- ({3.1415+2.3},-3);
+
+\node at ({3.1415-1.75},-2.15) [left] {$\frac12(f(x)+g(x))+\frac12|f(x)-g(x)|$};
+\node at ({3.1415+1.75},-2.15) [right] {$\frac12(f(x)+g(x))-\frac12|f(x)-g(x)|$};
+
+\def\s{(-0.1)}
+
+\begin{scope}[xshift=-3.4cm,yshift=-4.6cm]
+	\fill[color=darkgreen!20]
+		\mittellinie
+		--
+		plot[domain=6.2832:0,samples=400]
+			({\x},{0.5*(sin(180*\x/3.14159)+cos(180*\x/3.14159)+abs(sin(180*\x/3.14159)-cos(180*\x/3.14159)))})
+		-- cycle;
+	\foreach \x in {0.5,1,...,6}{
+		\draw[color=darkgreen]
+			({\x},{0.5*(sin(180*\x/3.14159)+cos(180*\x/3.14159)+abs(sin(180*\x/3.14159)-cos(180*\x/3.14159)))})
+			--
+			({\x},{0.5*(sin(180*\x/3.14159)+cos(180*\x/3.14159))});
+	}
+	\draw[color=darkgreen,line width=1.4pt]
+		plot[domain=6.2832:0,samples=400]
+			({\x},{0.5*(sin(180*\x/3.14159)+cos(180*\x/3.14159)+abs(sin(180*\x/3.14159)-cos(180*\x/3.14159)))});
+
+	\node[color=darkgreen] at (2,-0.3) [left] {$|f(x)-g(x)|$};
+	\draw[color=darkgreen,line width=0.3pt] (2,-0.3) -- (2.5,0.2);
+
+	\draw[color=purple!50,line width=1.4pt] \mittellinie;
+	\pgfmathparse{0.75*3.1415+\s}
+	\xdef\x{\pgfmathresult}
+	\node[color=darkgreen] at (\x,{sin(180*\x/3.1415)}) [above right]
+		{$\max(f(x),g(x))$};
+	\node[color=purple!50] at ({1.25*3.1415},-0.7) [below]
+		{$\frac12(f(x)+g(x))$};
+	\draw[->] (-0.1,0) -- (6.5,0) coordinate[label={$x$}];
+	\draw[->] (0,-1.1) -- (0,1.3) coordinate[label={right:$y$}];
+\end{scope}
+
+
+\begin{scope}[xshift=+3.4cm,yshift=-4.6cm]
+	\fill[color=darkgreen!20]
+		\mittellinie
+		--
+		plot[domain=6.2832:0,samples=400]
+			({\x},{0.5*(sin(180*\x/3.14159)+cos(180*\x/3.14159)-abs(sin(180*\x/3.14159)-cos(180*\x/3.14159)))})
+		-- cycle;
+	\foreach \x in {0.5,1,...,6}{
+		\draw[color=darkgreen]
+			({\x},{0.5*(sin(180*\x/3.14159)+cos(180*\x/3.14159)-abs(sin(180*\x/3.14159)-cos(180*\x/3.14159)))})
+			--
+			({\x},{0.5*(sin(180*\x/3.14159)+cos(180*\x/3.14159))});
+	}
+	\draw[color=darkgreen,line width=1.4pt]
+		plot[domain=6.2832:0,samples=400]
+			({\x},{0.5*(sin(180*\x/3.14159)+cos(180*\x/3.14159)-abs(sin(180*\x/3.14159)-cos(180*\x/3.14159)))});
+
+	\node[color=darkgreen] at (3,0.3) [right] {$|f(x)-g(x)|$};
+	\draw[color=darkgreen,line width=0.3pt] (3,0.3) -- (2.5,-0.4);
+
+	\draw[color=purple!50,line width=1.4pt] \mittellinie;
+	\pgfmathparse{0.75*3.1415-\s}
+	\xdef\x{\pgfmathresult}
+	\node[color=darkgreen] at (\x,{cos(180*\x/3.1415)}) [below left]
+		{$\min(f(x),g(x))$};
+	\node[color=purple!50] at ({0.25*3.1415},0.7) [above right]
+		{$\frac12(f(x)+g(x))$};
+	\draw[->] (-0.1,0) -- (6.5,0) coordinate[label={$x$}];
+	\draw[->] (0,-1.1) -- (0,1.3) coordinate[label={right:$y$}];
+\end{scope}
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/40-eigenwerte/images/normalform.pdf b/buch/chapters/40-eigenwerte/images/normalform.pdf
new file mode 100644
index 0000000..c5bdb61
--- /dev/null
+++ b/buch/chapters/40-eigenwerte/images/normalform.pdf
diff --git a/buch/chapters/40-eigenwerte/images/normalform.tex b/buch/chapters/40-eigenwerte/images/normalform.tex
new file mode 100644
index 0000000..f3cb532
--- /dev/null
+++ b/buch/chapters/40-eigenwerte/images/normalform.tex
@@ -0,0 +1,214 @@
+%
+% normalform.tex -- Normalform einer Matrix ermitteln
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\def\skala{1}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+\def\b{0.025}
+
+\def\s{2.5}
+\def\t{0.7}
+\def\T{0.5}
+
+\fill[color=darkgreen!20]
+	({-3*\s-0.5*\t},{8*\t})
+	--
+	({-3*\s+0.5*\t},{8*\t})
+	--
+	({-2*\s-0.5*\t},{7*\t})
+	--
+	({-2*\s+0.5*\t},{7*\t})
+	--
+	({-1*\s-0.5*\t},{4*\t})
+	--
+	({-1*\s+0.5*\t},{4*\t})
+	--
+	({-0.5*\t},0)
+	--
+	({-3*\s-0.5*\t},{0*\t})
+	-- cycle;
+
+
+\fill[color=white,rounded corners=3pt]
+	({-0.5*\t-\b},{-\b}) rectangle ({0.5*\t+\b},{\b+0.15});
+\draw[rounded corners=3pt]
+	({-0.5*\t-\b},{-\b}) rectangle ({0.5*\t+\b},{\b+0.15});
+\node at (0,0) [below] {$\mathcal{K}^0(A)$};
+
+\fill[color=white,rounded corners=3pt]
+	({-1*\s-0.5*\t-\b},{-\b}) rectangle ({-1*\s+0.5*\t+\b},{4*\t+\b});
+\draw[rounded corners=3pt]
+	({-1*\s-0.5*\t-\b},{-\b}) rectangle ({-1*\s+0.5*\t+\b},{4*\t+\b});
+\fill[color=blue!20,rounded corners=2pt] 
+	({-1*\s-0.5*\t+\b},{1*\t+\b}) rectangle ({-1*\s+0.5*\t-\b},{3*\t-\b});
+\draw[color=blue!40,rounded corners=2pt] 
+	({-1*\s-0.5*\t+\b},{1*\t+\b}) rectangle ({-1*\s+0.5*\t-\b},{3*\t-\b});
+\fill[color=blue!20,rounded corners=2pt] 
+	({-1*\s-0.5*\t+\b},{3*\t+\b}) rectangle ({-1*\s+0.5*\t-\b},{4*\t-\b});
+\draw[color=blue!40,rounded corners=2pt] 
+	({-1*\s-0.5*\t+\b},{3*\t+\b}) rectangle ({-1*\s+0.5*\t-\b},{4*\t-\b});
+\fill[color=red!20,rounded corners=2pt]
+	({-1*\s-0.5*\t+\b},{\b}) rectangle ({-1*\s+0.5*\t-\b},{1*\t-\b});
+\draw[color=red,rounded corners=2pt]
+	({-1*\s-0.5*\t+\b},{\b}) rectangle ({-1*\s+0.5*\t-\b},{1*\t-\b});
+\fill[color=red] ({-1*\s},{0.5*\t}) circle[radius=0.1];
+\fill[color=red,opacity=0.5] ({-1*\s},{1.5*\t}) circle[radius=0.1];
+\fill[color=red,opacity=0.5] ({-1*\s},{2.5*\t}) circle[radius=0.1];
+\fill[color=red,opacity=0.5] ({-1*\s},{3.5*\t}) circle[radius=0.1];
+\node at ({-1*\s},0) [below] {$\mathcal{K}^1(A)$};
+
+\fill[color=white,rounded corners=3pt]
+	({-2*\s-0.5*\t-\b},{-\b}) rectangle ({-2*\s+0.5*\t+\b},{7*\t+\b});
+\draw[rounded corners=3pt]
+	({-2*\s-0.5*\t-\b},{-\b}) rectangle ({-2*\s+0.5*\t+\b},{7*\t+\b});
+\fill[color=gray!20,rounded corners=2pt]
+	({-2*\s-0.5*\t+\b},{+\b}) rectangle ({-2*\s+0.5*\t-\b},{4*\t-\b});
+\draw[color=gray,rounded corners=2pt]
+	({-2*\s-0.5*\t+\b},{+\b}) rectangle ({-2*\s+0.5*\t-\b},{4*\t-\b});
+\node[color=black!70] at ({-2*\s},{2*\t}) [rotate=90] {$\mathcal{K}^1(A)$};
+\fill[color=red!20,rounded corners=2pt]
+	({-2*\s-0.5*\t+\b},{4*\t+\b}) rectangle ({-2*\s+0.5*\t-\b},{6*\t-\b});
+\draw[color=red,rounded corners=2pt]
+	({-2*\s-0.5*\t+\b},{4*\t+\b}) rectangle ({-2*\s+0.5*\t-\b},{6*\t-\b});
+\fill[color=blue!20,rounded corners=2pt] 
+	({-2*\s-0.5*\t+\b},{6*\t+\b}) rectangle ({-2*\s+0.5*\t-\b},{7*\t-\b});
+\draw[color=blue!40,rounded corners=2pt] 
+	({-2*\s-0.5*\t+\b},{6*\t+\b}) rectangle ({-2*\s+0.5*\t-\b},{7*\t-\b});
+\fill[color=red] ({-2*\s},{4.5*\t}) circle[radius=0.1];
+\fill[color=red] ({-2*\s},{5.5*\t}) circle[radius=0.1];
+\fill[color=red,opacity=0.5] ({-2*\s},{6.5*\t}) circle[radius=0.1];
+\draw[->,color=blue,line width=1.2pt,shorten >= 0.15cm,shorten <= 0.15cm]
+	({-2*\s},{6.5*\t}) -- ({-1*\s},{3.5*\t});
+\draw[->,color=blue,line width=1.2pt,shorten >= 0.15cm,shorten <= 0.15cm]
+	({-2*\s},{5.5*\t}) -- ({-1*\s},{2.5*\t});
+\draw[->,color=blue,line width=1.2pt,shorten >= 0.15cm,shorten <= 0.15cm]
+	({-2*\s},{4.5*\t}) -- ({-1*\s},{1.5*\t});
+\node at ({-2*\s},0) [below] {$\mathcal{K}^2(A)$};
+
+\fill[color=white,rounded corners=3pt]
+	({-3*\s-0.5*\t-\b},{-\b}) rectangle ({-3*\s+0.5*\t+\b},{8*\t+\b});
+\draw[rounded corners=3pt]
+	({-3*\s-0.5*\t-\b},{-\b}) rectangle ({-3*\s+0.5*\t+\b},{8*\t+\b});
+\fill[color=gray!20,rounded corners=2pt]
+	({-3*\s-0.5*\t+\b},{+\b}) rectangle ({-3*\s+0.5*\t-\b},{7*\t-\b});
+\draw[color=gray,rounded corners=2pt]
+	({-3*\s-0.5*\t+\b},{+\b}) rectangle ({-3*\s+0.5*\t-\b},{7*\t-\b});
+\node[color=black!70] at ({-3*\s},{3.5*\t}) [rotate=90] {$\mathcal{K}^2(A)$};
+\fill[color=red!20,rounded corners=2pt]
+	({-3*\s-0.5*\t+\b},{7*\t+\b}) rectangle ({-3*\s+0.5*\t-\b},{8*\t-\b});
+\draw[color=red,rounded corners=2pt]
+	({-3*\s-0.5*\t+\b},{7*\t+\b}) rectangle ({-3*\s+0.5*\t-\b},{8*\t-\b});
+\fill[color=red] ({-3*\s},{7.5*\t}) circle[radius=0.1];
+\draw[->,color=blue,line width=1.2pt,shorten >= 0.15cm,shorten <= 0.15cm]
+	({-3*\s},{7.5*\t}) -- ({-2*\s},{6.5*\t});
+\node at ({-3*\s},0) [below] {$\mathcal{K}^3(A)$};
+
+\def\xo{1}
+\def\yo{-1}
+
+\def\punkt#1#2{
+	({\xo+(#1)*\T},{\yo-(#2)*\T})
+}
+
+\fill[color=red!20] \punkt{0}{0} rectangle \punkt{1}{8};
+\fill[color=red!20] \punkt{2}{0} rectangle \punkt{3}{8};
+\fill[color=red!20] \punkt{4}{0} rectangle \punkt{5}{8};
+\fill[color=red!20] \punkt{7}{0} rectangle \punkt{8}{8};
+
+\fill[color=blue!20] \punkt{2}{1} rectangle \punkt{3}{2};
+\fill[color=blue!20] \punkt{4}{3} rectangle \punkt{5}{4};
+\fill[color=blue!20] \punkt{6}{5} rectangle \punkt{7}{6};
+\fill[color=blue!20] \punkt{7}{6} rectangle \punkt{8}{7};
+
+\draw \punkt{0}{0} rectangle \punkt{8}{8};
+
+\draw[color=gray] \punkt{0}{1} -- \punkt{3}{1} -- \punkt{3}{5} -- \punkt{8}{5};
+\draw[color=gray] \punkt{1}{0} -- \punkt{1}{3} -- \punkt{5}{3} -- \punkt{5}{8};
+
+\draw[->,color=red]
+	({-3*\s+0.5*\t+\b},{7.5*\t})
+	--
+	({0*\s+0.5*\t},{7.5*\t})
+	to[out=0,in=90]
+	({\xo+7.5*\T},{\yo});
+
+\draw[->,color=blue]
+	({-2*\s+0.5*\t+\b},{6.5*\t})
+	--
+	({0*\s+0.5*\t},{6.5*\t})
+	to[out=0,in=90]
+	({\xo+6.5*\T},{\yo});
+
+\draw[->,color=blue]
+	({-1*\s+0.5*\t+\b},{3.5*\t})
+	--
+	({0*\s+0.5*\t},{3.5*\t})
+	to[out=0,in=90]
+	({\xo+5.5*\T},{\yo});
+
+\draw[->,color=red]
+	({-2*\s+0.5*\t+\b},{5.5*\t})
+	--
+	({0*\s+0.5*\t},{5.5*\t})
+	to[out=0,in=90]
+	({\xo+4.5*\T},{\yo});
+
+\draw[->,color=red]
+	({-2*\s+0.5*\t+\b},{4.5*\t})
+	--
+	({0*\s+0.5*\t},{4.5*\t})
+	to[out=0,in=90]
+	({\xo+2.5*\T},{\yo});
+
+\draw[->,color=blue]
+	({-1*\s+0.5*\t+\b},{2.5*\t})
+	--
+	({0*\s+0.5*\t},{2.5*\t})
+	to[out=0,in=90]
+	({\xo+3.5*\T},{\yo});
+
+\draw[->,color=red]
+	({-1*\s+0.5*\t+\b},{0.5*\t})
+	--
+	({0*\s+0.5*\t},{0.5*\t})
+	to[out=0,in=90]
+	({\xo+0.5*\T},{\yo});
+
+\draw[->,color=blue]
+	({-1*\s+0.5*\t+\b},{1.5*\t})
+	--
+	({0*\s+0.5*\t},{1.5*\t})
+	to[out=0,in=90]
+	({\xo+1.5*\T},{\yo});
+
+\node at \punkt{0.5}{0.5} {$0$};
+\node at \punkt{1.5}{1.5} {$0$};
+\node at \punkt{2.5}{2.5} {$0$};
+\node at \punkt{3.5}{3.5} {$0$};
+\node at \punkt{4.5}{4.5} {$0$};
+\node at \punkt{5.5}{5.5} {$0$};
+\node at \punkt{6.5}{6.5} {$0$};
+\node at \punkt{7.5}{7.5} {$0$};
+
+\node[color=blue] at \punkt{2.5}{1.5} {$1$};
+\node[color=blue] at \punkt{4.5}{3.5} {$1$};
+\node[color=blue] at \punkt{6.5}{5.5} {$1$};
+\node[color=blue] at \punkt{7.5}{6.5} {$1$};
+
+\node at \punkt{-0.5}{4} [left] {$A=$};
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/40-eigenwerte/images/sp.pdf b/buch/chapters/40-eigenwerte/images/sp.pdf
index d4de984..b93b890 100644
--- a/buch/chapters/40-eigenwerte/images/sp.pdf
+++ b/buch/chapters/40-eigenwerte/images/sp.pdf
diff --git a/buch/chapters/40-eigenwerte/images/wa.m b/buch/chapters/40-eigenwerte/images/wa.m
new file mode 100644
index 0000000..3d6d2c3
--- /dev/null
+++ b/buch/chapters/40-eigenwerte/images/wa.m
@@ -0,0 +1,80 @@
+#
+# wa.m -- Wurzelapproximation
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+global u;
+global N;
+global t;
+global s;
+
+N = 100;
+n = 10;
+s = 1;
+
+u = zeros(N + 2, n);
+t = (0:N+1)' / N;
+t = t.^2;
+
+for i = (2:n)
+	u(:,i) = u(:,i-1) + 0.5 * (t-u(:,i-1).^2);
+end
+
+u
+
+global f;
+f = fopen("wa.tex", "w");
+fprintf(f, "%%\n");
+fprintf(f, "%% Approximation der Wurzelfunktion\n");
+fprintf(f, "%%\n");
+
+function pfad(i, name)
+	global f;
+	global u;
+	global t;
+	global N;
+	fprintf(f, "\\def\\pfad%s{\n", name);
+	fprintf(f, "(%.4f,%.4f)\n", t(1,1), u(1,i));
+	for j = (2:N+1)
+		fprintf(f, "--(%.4f,%.4f)\n", t(j,1), u(j,i));
+	end
+	fprintf(f, "}\n");
+end
+
+pfad( 1, "a")
+pfad( 2, "b")
+pfad( 3, "c")
+pfad( 4, "d")
+pfad( 5, "e")
+pfad( 6, "f")
+pfad( 7, "g")
+pfad( 8, "h")
+pfad( 9, "i")
+pfad(10, "j")
+
+function fehler(i, name)
+	global f;
+	global u;
+	global t;
+	global N;
+	global s;
+	fprintf(f, "\\def\\fehler%s{\n", name);
+	fprintf(f, "(%.4f,%.4f)\n", t(1,1), s*(sqrt(t(1,1))-u(1,i)));
+	for j = (2:N+2)
+		fprintf(f, "--(%.4f,%.4f)\n", t(j,1), s*(sqrt(t(j,1))-u(j,i)));
+	end
+	fprintf(f, "}\n");
+end
+
+fehler( 1, "a")
+fehler( 2, "b")
+fehler( 3, "c")
+fehler( 4, "d")
+fehler( 5, "e")
+fehler( 6, "f")
+fehler( 7, "g")
+fehler( 8, "h")
+fehler( 9, "i")
+fehler(10, "j")
+
+fclose(f);
diff --git a/buch/chapters/40-eigenwerte/images/wurzel.pdf b/buch/chapters/40-eigenwerte/images/wurzel.pdf
new file mode 100644
index 0000000..751cf33
--- /dev/null
+++ b/buch/chapters/40-eigenwerte/images/wurzel.pdf
diff --git a/buch/chapters/40-eigenwerte/images/wurzel.tex b/buch/chapters/40-eigenwerte/images/wurzel.tex
new file mode 100644
index 0000000..ca2825a
--- /dev/null
+++ b/buch/chapters/40-eigenwerte/images/wurzel.tex
@@ -0,0 +1,94 @@
+%
+% wurzel.tex -- Wurzel
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\def\skala{10}
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\def\a{0.8}
+\def\U{0}
+
+\fill[color=blue!20] (0,\a) rectangle (1.00,1.03);
+\draw[line width=0.4pt] (0,1) -- (1,1) -- (1,0);
+
+\draw[->] (0,{-0.01}) -- (0,{1.06}) coordinate[label={right:$y$}];
+
+\begin{scope}
+\clip (0,0) rectangle (1,1);
+\draw[color=blue,line width=1.4pt] plot[domain=0:1.01,samples=100]
+	({\x},{\x*\x});
+\end{scope}
+
+%\draw[color=purple,line width=0.5pt] (0.48,-0.01) -- (1,1);
+%\fill[color=purple] (1,1) circle[radius=0.008];
+
+\node[color=blue] at (0,{\a}) [left] {$a$};
+
+\def\schritt#1#2{
+	\xdef\u{\U}
+	\pgfmathparse{0.5*(\a-\u*\u)}
+	\xdef\d{\pgfmathresult}
+	\pgfmathparse{\u+\d}
+	\xdef\U{\pgfmathresult}
+
+	\fill[color=purple!10] (\u,{\u*\u}) -- (\U,\a) -- (\u,\a) -- cycle;
+
+	\node[color=darkgreen] at (\u,0) [below] {$u_#1$};
+	\draw[color=darkgreen,line width=0.1pt] (\u,0)--(\u,\a);
+
+	\fill[color=darkgreen] (\u,{\u*\u}) circle[radius=0.006];
+
+	\draw[<->,color=darkgreen] (\u,{\u*\u}) -- (\u,\a);
+
+	\draw[color=purple,shorten <= 0.6mm]
+		(\u,{\u*\u}) -- (\U,\a);
+}
+\def\marke#1#2{
+	\node[color=orange] at ({0.5*(\u+\U)},\a) [#2] {$\frac12(a-u_#1^2)$};
+	\draw[<->,color=orange,shorten >= 0mm,shorten <= 0mm]
+		(\u,\a) -- (\U,\a);
+}
+
+\def\hoehe#1{
+	\node[color=darkgreen] at ({\u+0.01},{\a-\d-0.01})
+		[above,rotate=90] {$a-u_#1^2$};
+}
+
+\schritt{0}{1}
+\hoehe{0}
+\marke{0}{above}
+
+\schritt{1}{2}
+\hoehe{1}
+\marke{1}{above}
+\node[color=darkgreen] at (\u,{\u*\u-0.02}) [above left] {$u_1^2$};
+
+\schritt{2}{3}
+\hoehe{2}
+%\marke{2}{right,rotate=90}
+\marke{2}{above}
+\node[color=darkgreen] at (\u,{\u*\u-0.02}) [above left] {$u_2^2$};
+
+\schritt{3}{4}
+
+\draw[color=blue] ({sqrt(\a)},-0.01) -- ({sqrt(\a)},\a);
+\node[color=blue] at ({sqrt(\a)-0.02},0) [below right] {$\sqrt{a}$};
+
+\draw[->] (-0.01,0) -- (1.05,0) coordinate[label={$u$}];
+\node at (1,0) [below] {$1$};
+\node at (0,1) [left] {$1$};
+\draw (1,-0.01) -- (1,0.01);
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/40-eigenwerte/images/wurzelapprox.pdf b/buch/chapters/40-eigenwerte/images/wurzelapprox.pdf
new file mode 100644
index 0000000..01fa714
--- /dev/null
+++ b/buch/chapters/40-eigenwerte/images/wurzelapprox.pdf
diff --git a/buch/chapters/40-eigenwerte/images/wurzelapprox.tex b/buch/chapters/40-eigenwerte/images/wurzelapprox.tex
new file mode 100644
index 0000000..676c7e9
--- /dev/null
+++ b/buch/chapters/40-eigenwerte/images/wurzelapprox.tex
@@ -0,0 +1,107 @@
+%
+% wurzelapprox.tex -- template for standalon tikz images
+%
+% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+%
+\documentclass[tikz]{standalone}
+\usepackage{amsmath}
+\usepackage{times}
+\usepackage{txfonts}
+\usepackage{pgfplots}
+\usepackage{csvsimple}
+\usetikzlibrary{arrows,intersections,math}
+\begin{document}
+\def\skala{5.7}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\definecolor{darkgreen}{rgb}{0,0.6,0}
+
+\input{wa.tex}
+
+\begin{scope}[xshift=-0.63cm]
+
+\draw[->] (-0.01,0) -- (1.05,0) coordinate[label={$t$}];
+
+\begin{scope}
+	\clip (0,0) rectangle (1,1.01);
+	\draw[color=blue,line width=1.6pt] \pfada;
+	\draw[color=blue,line width=1.2pt] \pfadb;
+	\draw[color=blue,line width=1.2pt] \pfadc;
+	\draw[color=blue,line width=1.2pt] \pfadd;
+	\draw[color=blue,line width=1.2pt] \pfade;
+	\draw[color=blue,line width=1.2pt] \pfadf;
+	\draw[color=blue,line width=1.2pt] \pfadg;
+	\draw[color=blue,line width=1.2pt] \pfadh;
+	\draw[color=blue,line width=1.2pt] \pfadi;
+	\draw[color=blue,line width=1.2pt] \pfadj;
+
+	\draw[color=red,line width=1.6pt]
+		plot[domain=0:1.01,samples=100] ({\x*\x},{\x});
+\end{scope}
+
+\node[color=red] at (0.5,0.707) [above,rotate={atan(0.5)}] {$\sqrt{t}$};
+
+\draw[->] (0,-0.01) -- (0,1.05) coordinate[label={right:$u_n(t)$}];
+
+\foreach \x in {2,4,...,8}{
+	\draw ({0.1*\x},-0.01) -- ({0.1*\x},0.01);
+	\node at ({0.1*\x},-0.01) [below] {0.\x\strut};
+	\draw (-0.01,{0.1*\x}) -- (0.01,{0.1*\x});
+	\node at (-0.01,{0.1*\x}) [left] {0.\x\strut};
+}
+\draw (1,-0.01) -- (1,0.01);
+\node at (1,-0.01) [below] {1.0\strut};
+\node at (0,-0.01) [below] {0\strut};
+
+\draw (-0.01,1) -- (0.01,1);
+\node at (-0.01,1) [left] {1.0\strut};
+
+\node[color=blue] at (1.01,0) [above left] {$u_0(t)$};
+\node[color=blue] at (1,0.51) [below left,rotate={atan(0.5)}] {$u_1(t)$};
+\node[color=blue] at (1,{0.86+0.03}) [below left,rotate={atan(0.86)}] {$u_2(t)$};
+\node[color=blue] at (1,1.00) [below left,rotate={atan(0.5)}] {$u_3(t)$};
+
+\end{scope}
+
+\begin{scope}[xshift=0.63cm]
+
+\begin{scope}
+	\clip (0,0) rectangle (1,1.01);
+	\draw[color=darkgreen,line width=1.2pt] \fehlera;
+	\draw[color=darkgreen,line width=1.2pt] \fehlerb;
+	\draw[color=darkgreen,line width=1.2pt] \fehlerc;
+	\draw[color=darkgreen,line width=1.2pt] \fehlerd;
+	\draw[color=darkgreen,line width=1.2pt] \fehlere;
+	\draw[color=darkgreen,line width=1.2pt] \fehlerf;
+	\draw[color=darkgreen,line width=1.2pt] \fehlerg;
+	\draw[color=darkgreen,line width=1.2pt] \fehlerh;
+	\draw[color=darkgreen,line width=1.2pt] \fehleri;
+	\draw[color=darkgreen,line width=1.2pt] \fehlerj;
+\end{scope}
+
+\draw[->] (0,-0.01) -- (0,1.05) coordinate[label={right:${\color{red}\sqrt{t}}-{\color{blue}u_n(t)}$}];
+\draw[->] (-0.01,0) -- (1.05,0) coordinate[label={$t$}];
+
+\foreach \x in {2,4,...,9}{
+	\draw ({0.1*\x},-0.01) -- ({0.1*\x},0.01);
+	\node at ({0.1*\x},-0.01) [below] {0.\x\strut};
+	\draw (-0.01,{0.1*\x}) -- (0.01,{0.1*\x});
+	\node at (-0.01,{0.1*\x}) [left] {0.\x\strut};
+}
+\draw (1,-0.01) -- (1,0.01);
+\node at (1,-0.01) [below] {1.0\strut};
+\node at (0,-0.01) [below] {0\strut};
+
+\draw (-0.01,1) -- (0.01,1);
+\node at (-0.01,1) [left] {1.0\strut};
+
+\node[color=darkgreen] at (1,1) [below left,rotate={atan(0.5)}] {$n=0$};
+\node[color=darkgreen] at (1,0.5) [above left] {$n=1$};
+\node[color=darkgreen] at (1,0.13) [above left,rotate=-13] {$n=2$};
+\node[color=darkgreen] at (1,0.00) [above left,rotate=-9] {$n=3$};
+
+\end{scope}
+
+\end{tikzpicture}
+\end{document}
+