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Diffstat (limited to 'buch/chapters/40-eigenwerte/images')
27 files changed, 1005 insertions, 1 deletions
diff --git a/buch/chapters/40-eigenwerte/images/Makefile b/buch/chapters/40-eigenwerte/images/Makefile index db00dac..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 +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 @@ -14,3 +16,29 @@ sppaths.tex: spbeispiel.m nilpotent.pdf: nilpotent.tex pdflatex nilpotent.tex +kernbild.pdf: kernbild.tex bild2.jpg kern2.jpg + pdflatex kernbild.tex + +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/bild1.jpg b/buch/chapters/40-eigenwerte/images/bild1.jpg Binary files differnew file mode 100644 index 0000000..879fae8 --- /dev/null +++ b/buch/chapters/40-eigenwerte/images/bild1.jpg diff --git a/buch/chapters/40-eigenwerte/images/bild2.jpg b/buch/chapters/40-eigenwerte/images/bild2.jpg Binary files differnew file mode 100644 index 0000000..2597c95 --- /dev/null +++ b/buch/chapters/40-eigenwerte/images/bild2.jpg diff --git a/buch/chapters/40-eigenwerte/images/dimjk.pdf b/buch/chapters/40-eigenwerte/images/dimjk.pdf Binary files differnew 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/drei.jpg b/buch/chapters/40-eigenwerte/images/drei.jpg Binary files differnew file mode 100644 index 0000000..35f9034 --- /dev/null +++ b/buch/chapters/40-eigenwerte/images/drei.jpg diff --git a/buch/chapters/40-eigenwerte/images/jknilp.pdf b/buch/chapters/40-eigenwerte/images/jknilp.pdf Binary files differnew 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/kern1.jpg b/buch/chapters/40-eigenwerte/images/kern1.jpg Binary files differnew file mode 100644 index 0000000..5c99664 --- /dev/null +++ b/buch/chapters/40-eigenwerte/images/kern1.jpg diff --git a/buch/chapters/40-eigenwerte/images/kern2.jpg b/buch/chapters/40-eigenwerte/images/kern2.jpg Binary files differnew file mode 100644 index 0000000..87d18ac --- /dev/null +++ b/buch/chapters/40-eigenwerte/images/kern2.jpg diff --git a/buch/chapters/40-eigenwerte/images/kernbild.pdf b/buch/chapters/40-eigenwerte/images/kernbild.pdf Binary files differnew file mode 100644 index 0000000..2a321b2 --- /dev/null +++ b/buch/chapters/40-eigenwerte/images/kernbild.pdf diff --git a/buch/chapters/40-eigenwerte/images/kernbild.tex b/buch/chapters/40-eigenwerte/images/kernbild.tex new file mode 100644 index 0000000..4eced84 --- /dev/null +++ b/buch/chapters/40-eigenwerte/images/kernbild.tex @@ -0,0 +1,40 @@ +% +% kernbild.tex -- Kern und Bild einer Matrix +% +% (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} +\definecolor{darkgreen}{rgb}{0,0.4,0} +\definecolor{turqoise}{rgb}{0,0.3,0.6} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\begin{scope}[xshift=-3.5cm] +\node at (0,0) {\includegraphics[width=6.8cm]{bild2.jpg}}; + +\fill[color=white,opacity=0.8] (-3,-2.75) rectangle (-2,-2.3); +\node[color=orange] at (-2.5,-2.5) {$\mathcal{J}^1(A)$}; +\node at (3.3,0) {$x_1$}; +\node at (0.3,3.2) {$x_3$}; +\node[color=purple] at (2.3,0.6) [rotate=8] {$\mathcal{J}^2(A)$}; +\end{scope} + +\begin{scope}[xshift=3.5cm] +\node at (0,0) {\includegraphics[width=6.8cm]{kern2.jpg}}; +\node[color=darkgreen] at (1.8,2.2) [rotate=58] {$\mathcal{K}^1(A)$}; +\fill[color=white,opacity=0.8] (-1.5,0.8) rectangle (-0.5,1.2); +\node[color=turqoise] at (-1,1) {$\mathcal{K}^2(A)$}; +\node at (3.3,0) {$x_1$}; +\node at (0.3,3.2) {$x_3$}; +\end{scope} + +\end{tikzpicture} +\end{document} + diff --git a/buch/chapters/40-eigenwerte/images/kernbild1.jpg b/buch/chapters/40-eigenwerte/images/kernbild1.jpg Binary files differnew file mode 100644 index 0000000..87e874e --- /dev/null +++ b/buch/chapters/40-eigenwerte/images/kernbild1.jpg diff --git a/buch/chapters/40-eigenwerte/images/kernbild2.jpg b/buch/chapters/40-eigenwerte/images/kernbild2.jpg Binary files differnew file mode 100644 index 0000000..1160b31 --- /dev/null +++ b/buch/chapters/40-eigenwerte/images/kernbild2.jpg diff --git a/buch/chapters/40-eigenwerte/images/kombiniert.jpg b/buch/chapters/40-eigenwerte/images/kombiniert.jpg Binary files differnew file mode 100644 index 0000000..bebc36f --- /dev/null +++ b/buch/chapters/40-eigenwerte/images/kombiniert.jpg diff --git a/buch/chapters/40-eigenwerte/images/kombiniert.pdf b/buch/chapters/40-eigenwerte/images/kombiniert.pdf Binary files differnew file mode 100644 index 0000000..91cee0b --- /dev/null +++ b/buch/chapters/40-eigenwerte/images/kombiniert.pdf diff --git a/buch/chapters/40-eigenwerte/images/kombiniert.tex b/buch/chapters/40-eigenwerte/images/kombiniert.tex new file mode 100644 index 0000000..d850c64 --- /dev/null +++ b/buch/chapters/40-eigenwerte/images/kombiniert.tex @@ -0,0 +1,48 @@ +% +% kombiniert.tex -- Iterierte Kerne und Bilder +% +% (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} +\definecolor{darkgreen}{rgb}{0,0.4,0} +\definecolor{turqoise}{rgb}{0,0.3,0.6} +\def\skala{1} +\newboolean{showgrid} +\setboolean{showgrid}{false} +\def\breite{7} +\def\hoehe{7} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\node at (0,0) {\includegraphics[width=13.8cm]{kombiniert.jpg}}; + +\node at (6.6,-0.1) {$x_1$}; +\node at (0.3,6.7) {$x_3$}; + +\node[color=purple] at (4.8,1) [rotate=8] {$\mathcal{J}^2(A)$}; +\node[color=darkgreen] at (3.5,4.6) [rotate=58] {$\mathcal{K}^1(A)$}; + +\fill[color=white,opacity=0.8] (-2.3,3.8) rectangle (-1.3,4.2); +\node[color=turqoise] at (-1.8,4) {$\mathcal{K}^2(A)$}; + +\fill[color=white,opacity=0.8] (2.5,-5.75) rectangle (3.5,-5.3); +\node[color=orange] at (3,-5.5) {$\mathcal{J}^1(A)$}; + +%\node at G +% Gitter +\ifthenelse{\boolean{showgrid}}{ +\draw[step=0.1,line width=0.1pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw[step=0.5,line width=0.4pt] (-\breite,-\hoehe) grid (\breite, \hoehe); +\draw (-\breite,-\hoehe) grid (\breite, \hoehe); +\fill (0,0) circle[radius=0.05]; +}{} + +\end{tikzpicture} +\end{document} + diff --git a/buch/chapters/40-eigenwerte/images/minmax.pdf b/buch/chapters/40-eigenwerte/images/minmax.pdf Binary files differnew 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 Binary files differnew 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 Binary files differindex 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 Binary files differnew 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 Binary files differnew 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} + |
