komplexe Fibonacci-Zahlen

4 files changed, 292 insertions, 1 deletions

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
new 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}
+