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Diffstat (limited to '')
| -rw-r--r-- | buch/chapters/040-rekursion/beta.tex | 5 | ||||
| -rw-r--r-- | buch/chapters/040-rekursion/images/Makefile | 8 | ||||
| -rw-r--r-- | buch/chapters/040-rekursion/images/fibonacci.m | 136 | ||||
| -rw-r--r-- | buch/chapters/040-rekursion/images/fibonacci.pdf | bin | 0 -> 712633 bytes | |||
| -rw-r--r-- | buch/chapters/040-rekursion/images/fibonacci.tex | 149 | ||||
| -rw-r--r-- | buch/chapters/040-rekursion/linear.tex | 36 |
6 files changed, 324 insertions, 10 deletions
diff --git a/buch/chapters/040-rekursion/beta.tex b/buch/chapters/040-rekursion/beta.tex index 24d6ac5..f244d18 100644 --- a/buch/chapters/040-rekursion/beta.tex +++ b/buch/chapters/040-rekursion/beta.tex @@ -256,7 +256,7 @@ Für $x=\frac12$ wird der Ausdruck besonders einfach: = \int_0^1 t^{-\frac12}(1-t)^{-\frac12}\,dt = -\int_0^1 \frac{1}{\sqrt{t(1-t)}}\,dt. +\int_0^1 \frac{1}{\sqrt{t(1-t)\mathstrut}}\,dt. \] Mit der Substition $t=\sin^2 s$ wird daraus \[ @@ -475,7 +475,8 @@ Setzt man $x=\frac12$ in die Verdoppelungsformel ein, erhält man \qquad\Rightarrow\qquad \Gamma({\textstyle\frac12}) = \sqrt{\pi}, \] -in Übereinstimmung mit dem bereits bekannten Wert. +in Übereinstimmung mit dem aus \eqref{buch:rekursion:gamma:gamma12} +bereits bekannten Wert. \subsubsection{Beta-Funktion und Binomialkoeffizienten} Die Binomialkoeffizienten können mit Hilfe der Fakultät als diff --git a/buch/chapters/040-rekursion/images/Makefile b/buch/chapters/040-rekursion/images/Makefile index 58f79b8..9608a94 100644 --- a/buch/chapters/040-rekursion/images/Makefile +++ b/buch/chapters/040-rekursion/images/Makefile @@ -3,10 +3,16 @@ # # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule # -all: gammaplot.pdf +all: gammaplot.pdf fibonacci.pdf gammaplot.pdf: gammaplot.tex gammapaths.tex pdflatex gammaplot.tex gammapaths.tex: gammaplot.m octave gammaplot.m + +fibonaccigrid.tex: fibonacci.m + octave fibonacci.m + +fibonacci.pdf: fibonacci.tex fibonaccigrid.tex + pdflatex fibonacci.tex diff --git a/buch/chapters/040-rekursion/images/fibonacci.m b/buch/chapters/040-rekursion/images/fibonacci.m new file mode 100644 index 0000000..4c9754e --- /dev/null +++ b/buch/chapters/040-rekursion/images/fibonacci.m @@ -0,0 +1,136 @@ +# +# fibonacci.m +# +# (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +# +global phi1; +phi1 = (1+sqrt(5)) / 2; +phi2 = (1-sqrt(5)) / 2; +global logphi1; +logphi1 = log(phi1) +global logphi2; +logphi2 = log(phi2) + +global s; +s = 0.1; + +A = [ 1, 1; phi1, phi2 ]; +b = [ 0; 1 ]; +global a; +a = A \ b + +global xmin; +xmin = 0; +global xmax; +xmax = 10; +global ylim; +ylim = 10; + +global N; +N = 200; + +function retval = fibonacci(n) + global a; + global logphi1; + global logphi2; + retval = a(1,1) * exp(n * logphi1) + a(2,1) * exp(n * logphi2); +endfunction + +for n = (0:10) + fibonacci(n) +endfor + +function punkt(fn, z) + if (abs(z) > 100) + z = 100 * z / abs(z); + endif + fprintf(fn, "(%.5f,%.5f)", real(z), imag(z)); +endfunction + +function drawline(fn, p, color,lw) + global N; + fprintf(fn, "\\draw[color=%s,line width=%.1fpt] ", color, lw); + punkt(fn, p(1)); + for i = (2:N+1) + fprintf(fn, "\n\t--"); + punkt(fn, p(i)); + endfor + fprintf(fn, ";\n"); +endfunction + +function realline(fn, x, ymin, ymax, color, lw) + global N; + h = (ymax - ymin) / N; + fprintf(fn, "%% real line for x = %f, h = %f\n", x, h); + count = 1; + for y = ymin + (0:N) * h + z(count) = fibonacci(x + i * y); + count = count + 1; + endfor + drawline(fn, z, color, lw); +endfunction + +function imaginaryline(fn, y, xmin, xmax, color, lw) + global N; + h = (xmax - xmin) / N; + fprintf(fn, "%% imaginary line for y = %f, h = %f\n", y, h); + count = 1; + for x = xmin + (0:N) * h + z(count) = fibonacci(x + i * y); + count = count + 1; + endfor + drawline(fn, z, color, lw); +endfunction + +function fibmapping(fn, n, name, lw) + global s; + fprintf(fn, "\\def\\%s{\n", name); + for x = n + s*(-5:5) + realline(fn, x, -5*s, 5*s, "red", lw); + endfor + for y = s*(-5:5) + imaginaryline(fn, y, n-5*s, n+5*s, "blue", lw); + endfor + fprintf(fn, "}\n"); +endfunction + +function fibgrid(fn, lw) + global s; + fprintf(fn, "\\def\\fibgrid{\n"); + for y = s*(-5:5) + imaginaryline(fn, y, -0.5, 6.5, "gray", lw); + endfor + for x = s*(-5:65) + realline(fn, x, -0.5, 0.5, "gray", lw); + endfor + fprintf(fn, "}\n"); +endfunction + +function fibcurve(fn, lw) + fprintf(fn, "\\def\\fibcurve{\n"); + imaginaryline(fn, 0, 0, 6.5, "white", 1.2*lw); + imaginaryline(fn, 0, 0, 6.5, "darkgreen", lw); + for n = (0:6) + z = fibonacci(n); + fprintf(fn, "\\fill[color=darkgreen] "); + punkt(fn, z); + fprintf(fn, " circle[radius=0.08];\n"); + fprintf(fn, "\\fill[color=white] "); + punkt(fn, z); + fprintf(fn, " circle[radius=0.04];\n"); + endfor + fprintf(fn, "}\n"); +endfunction + +fn = fopen("fibonaccigrid.tex", "w"); +fibmapping(fn, 0, "fibzero", 1); +fibmapping(fn, 1, "fibone", 1); +fibmapping(fn, 2, "fibtwo", 1); +fibmapping(fn, 3, "fibthree", 1); +fibmapping(fn, 4, "fibfour", 1); +fibmapping(fn, 5, "fibfive", 1); +fibmapping(fn, 6, "fibsix", 1); +fibgrid(fn, 0.3); +fibcurve(fn, 1.4); + +fclose(fn); diff --git a/buch/chapters/040-rekursion/images/fibonacci.pdf b/buch/chapters/040-rekursion/images/fibonacci.pdf Binary files differnew file mode 100644 index 0000000..e745b73 --- /dev/null +++ b/buch/chapters/040-rekursion/images/fibonacci.pdf diff --git a/buch/chapters/040-rekursion/images/fibonacci.tex b/buch/chapters/040-rekursion/images/fibonacci.tex new file mode 100644 index 0000000..3bd8b63 --- /dev/null +++ b/buch/chapters/040-rekursion/images/fibonacci.tex @@ -0,0 +1,149 @@ +% +% fibonacci.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} +\input{fibonaccigrid.tex} +\def\skala{1} +\definecolor{darkgreen}{rgb}{0,0.6,0} +\begin{tikzpicture}[>=latex,thick,scale=\skala] + +\def\achsen{ + \draw[->] (-2.1,0) -- (10.8,0) + coordinate[label={$\operatorname{Re}F(z)$}]; + \draw[->] (0,-2.6) -- (0,2.7) + coordinate[label={right:$\operatorname{Im}F(z)$}]; + \node at (10.8,2.7) {$\mathbb{C}$}; +} + +\def\zahl#1#2{ + \fill[color=white,opacity=0.8] + ({#1-0.65},-0.6) rectangle ({#1+0.65},-0.2); + \node[color=darkgreen] at (#1,-0.43) {$#2\mathstrut$}; +} + +\draw[color=gray!20,line width=2.0pt] (-2,3.2) -- (11.3,3.2); + +\def\topskala{1.55} + +\draw[->,color=darkgreen!30,line width=5pt] + ({-0.2*\topskala},4.3) to[out=-120,in=120] (-0.1,0.2); +\draw[->,color=darkgreen!30,line width=5pt] + ({0.9*\topskala},4.3) to[out=-110,in=70] (0.9,-5.1); +\draw[->,color=darkgreen!30,line width=5pt] + ({2.0*\topskala},4.3) to[out=-90,in=76] (1.0,-10.6); + +\draw[->,color=darkgreen!30,line width=5pt] + ({3.0*\topskala},4.3) to[out=-90,in=90] (2.0,0.2); +\draw[->,color=darkgreen!30,line width=5pt] + ({3.9*\topskala},4.3) to[out=-110,in=80] (3.0,-5.3); +\draw[->,color=darkgreen!30,line width=5pt] + ({4.8*\topskala},4.3) to[out=-120,in=90] (5.0,-10.8); + +\draw[->,color=darkgreen!30,line width=5pt] + ({6.0*\topskala},4.3) to[out=-90,in=70] (8.1,0.2); + + +\begin{scope}[yshift=4.8cm,scale=\topskala] + + \draw[->] (-0.7,0) -- (6.8,0) + coordinate[label={$\operatorname{Re}z$}]; + \draw[->] (0,-0.7) -- (0,0.8) + coordinate[label={right:$\operatorname{Im}z$}]; + \foreach \x in {-0.5,-0.4,...,6.501}{ + \draw[color=gray,line width=0.1pt] (\x,-0.5) -- (\x,0.5); + } + \foreach \y in {-0.5,-0.4,...,0.501}{ + \draw[color=gray,line width=0.1pt] (-0.5,\y) -- (6.5,\y); + } + + \foreach \n in {0,3,6}{ + \foreach \x in {-0.5,-0.4,...,0.501}{ + \draw[color=blue,line width=0.7pt] + ({\n+\x},-0.5) -- ({\n+\x},0.5); + } + \foreach \y in {-0.5,-0.4,...,0.501}{ + \draw[color=red,line width=0.7pt] + ({\n-0.5},\y) -- ({\n+0.5},\y); + } + } + \foreach \n in {0,...,6}{ + \fill[color=white,opacity=0.8] + ({\n-0.1},-0.28) rectangle ({\n+0.1},-0.05); + \node[color=darkgreen] at (\n,0) [below] {$\n$}; + \fill[color=darkgreen] (\n,0) circle[radius=0.05]; + \fill[color=white] (\n,0) circle[radius=0.02]; + } + + \node at (6.8,0.8) {$\mathbb{C}$}; + +\end{scope} + +\begin{scope}[scale=1] + \begin{scope} + \clip (-2,-2.6) rectangle (10.5,2.6); + \fibgrid + \end{scope} + \achsen + \begin{scope} + \clip (-2,-2.6) rectangle (10.5,2.6); + \fibzero + \fibthree + \fibsix + \end{scope} + \fibcurve + \node[color=magenta] at (-1.3,-1.8) {$n=0$}; + \node[color=magenta] at (1.9,0.8) {$n=3$}; + \node[color=magenta] at (8,2.3) {$n=6$}; + \zahl{0}{F(0)=0} + \zahl{2}{F(3)=2} + \zahl{8}{F(6)=8} +\end{scope} + +\begin{scope}[yshift=-5.5cm,scale=1] + \begin{scope} + \clip (-2,-2.6) rectangle (10.5,2.6); + \fibgrid + \end{scope} + \achsen + \begin{scope} + \clip (-2,-2.6) rectangle (10.5,2.6); + \fibone + \fibfour + \end{scope} + \fibcurve + \node[color=magenta] at (1,1.2) {$n=1$}; + \node[color=magenta] at (3,1.1) {$n=4$}; + \zahl{1}{F(1)=1} + \zahl{4}{F(4)=3} +\end{scope} + +\begin{scope}[yshift=-11cm,scale=1] + \begin{scope} + \clip (-2,-2.6) rectangle (10.5,2.6); + \fibgrid + \end{scope} + \achsen + \begin{scope} + \clip (-2,-2.6) rectangle (10.5,2.6); + \fibtwo + \fibfive + \end{scope} + \fibcurve + \node[color=magenta] at (0.7,1.1) {$n=2$}; + \node[color=magenta] at (5,1.5) {$n=5$}; + \zahl{2}{F(2)=1} + \zahl{5}{F(5)=5} +\end{scope} + +\end{tikzpicture} +\end{document} + diff --git a/buch/chapters/040-rekursion/linear.tex b/buch/chapters/040-rekursion/linear.tex index a3ff0c2..33b8043 100644 --- a/buch/chapters/040-rekursion/linear.tex +++ b/buch/chapters/040-rekursion/linear.tex @@ -110,25 +110,47 @@ Die Nullstellen des charakteristischen Polynoms \qquad \lambda_\pm = \begin{cases} \displaystyle -\frac{\sqrt{5}+1}{2}=\varphi +\frac{1+\sqrt{5}}{2}=\varphi \\[8pt] \displaystyle -\frac{\sqrt{5}-1}{2}=\frac{1}{\varphi}, +\frac{1-\sqrt{5}}{2}=-\frac{1}{\varphi}, \end{cases} \] dabei ist $\varphi$ das Verhältnis des goldenen Schnittes. Die Anfangsbedingungen $F(0)=0$ und $F(1)=1$ bedeutet, dass -\[ -F(z) = \varphi^z - \frac{1}{\varphi^z} -\] -Dies ist die Funktion, die Matt Parker visualisiert hat. +\begin{equation} +F(z) = \frac{1}{\sqrt{5}}\varphi^z - \frac{1}{\sqrt{5}}\frac{1}{(-\varphi)^z} +\label{buch:rekursion:linear:fibonaccifunktion} +\end{equation} +Dies ist die Funktion, die Matt Parker in seinem Video visualisiert hat. +\begin{figure} +\centering +\includegraphics[width=0.82\textwidth]{chapters/040-rekursion/images/fibonacci.pdf} +\caption{Komplexe Fibonacci-Zahlen-Funktion $F(z)$ von +\eqref{buch:rekursion:linear:fibonaccifunktion} +dargestellt als Abbildung $\mathbb{C}\to\mathbb{C}$. +Die ganzzahligen $z$ werden auf die wohlbekannten Fibonacci-Zahlen +abgebildet. +Zur besseren Lesbarkeit wird der Wertebereich dreimal dargestellt, +damit die Bilder der einzelnen reellen Teilintervalle in verschiedene +Wertebereich-Bilder verteilt werden können. +$x$-Werte zwischen $3n-\frac12$ und $3n+\frac12$ werden im obersten +Bildbereich dargestellt, solche zwischen $3n+\frac12$ und $3n+\frac32$ +im mittleren und schliesslich solche zwischen $3n+\frac32$ und $3n+\frac52$ +im untersten. +Die reelle Achse wird auf die grüne Kurve abgebildet. +\label{buch:rekursion:linear:fibonaccigraph}} +\end{figure} +Abbildung~\eqref{buch:rekursion:linear:fibonaccigraph} zeigt die Abbildung +$z\mapsto F(z)$. Allerdings sind die Funktionen \[ F_{kl}(z) = +\frac{1}{\sqrt{5}} \varphi^ze^{2k\pi iz} - -\frac{1}{\varphi^z} e^{2l\pi z} +\frac{1}{\sqrt{5}(-\varphi)^z} e^{2l\pi z} \] ebenfalls Lösungen der Differenzengleichung mit den gleichen Anfangswerten. |