add jacobi illustration

4 files changed, 295 insertions, 1 deletions

diff --git a/buch/chapters/070-orthogonalitaet/images/Makefile b/buch/chapters/070-orthogonalitaet/images/Makefile
index 900ae7f..efe5d36 100644
--- a/buch/chapters/070-orthogonalitaet/images/Makefile
+++ b/buch/chapters/070-orthogonalitaet/images/Makefile
@@ -4,7 +4,7 @@
 #
 # (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 #
-all:	legendre.pdf orthogonal.pdf weight.pdf
+all:	legendre.pdf orthogonal.pdf weight.pdf jacobi.pdf
 
 legendrepaths.tex:	legendre.m
 	octave legendre.m
@@ -19,3 +19,14 @@ weight.pdf:	weight.tex weightfunction.tex
 weightfunction.tex:	weight.m
 	octave weight.m
 
+jacobi:	jacobi.cpp
+	g++ --std=c++11 -O -Wall -g -o jacobi jacobi.cpp
+
+jacobipaths.tex:	jacobi Makefile
+	./jacobi --a=0 --b=0 --prefix=legendre --outfile=jacobipaths.tex
+	./jacobi --a=1 --b=0 --degree=14 --prefix=aone >> jacobipaths.tex
+	./jacobi --a=2 --b=0 --prefix=atwo >> jacobipaths.tex
+	./jacobi --a=-1 --b=0 --degree=14 --prefix=aminusone >> jacobipaths.tex
+	./jacobi --a=-2 --b=0 --prefix=aminustwo >> jacobipaths.tex
+jacobi.pdf:	jacobi.tex jacobipaths.tex
+	pdflatex jacobi.tex
diff --git a/buch/chapters/070-orthogonalitaet/images/jacobi.cpp b/buch/chapters/070-orthogonalitaet/images/jacobi.cpp
new file mode 100644
index 0000000..e010fd6
--- /dev/null
+++ b/buch/chapters/070-orthogonalitaet/images/jacobi.cpp
@@ -0,0 +1,179 @@
+/*
+ * jacobi.cpp
+ *
+ * (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+ */
+#include <cstdlib>
+#include <vector>
+#include <iostream>
+#include <sstream>
+#include <fstream>
+#include <memory>
+#include <getopt.h>
+#include <cmath>
+
+/**
+ * \brief Pfad-Klasse
+ */
+class	pfad {
+	std::ostringstream	*_inhalt;
+	std::string	_name;
+	int	_punkte;
+public:
+	pfad(const pfad&) = delete;
+	pfad(const std::string name) : _name(name), _punkte(0) {
+		_inhalt = new std::ostringstream();
+	}
+	~pfad() {
+		if (_inhalt) {
+			delete _inhalt;
+		}
+	}
+	void	add(const std::pair<double,double>& XY) {
+		if (_punkte == 0) {
+			*_inhalt << "\\def\\" + _name + "{\n\t";
+		} else {
+			*_inhalt << "\n\t-- ";
+		}
+		*_inhalt << "({";
+		*_inhalt << XY.first;
+		*_inhalt << "*\\dx},{";
+		*_inhalt << XY.second;
+		*_inhalt << "*\\dy})";
+		_punkte++;
+	}
+	void	operator()(const std::pair<double,double>& XY) {
+		add(XY);
+	}
+	std::string	macro() const {
+		std::string	result = _inhalt->str();
+		result = result + ";\n}\n";
+		return result;
+	}
+};
+typedef	std::shared_ptr<pfad>	pfad_ptr;
+
+#define	c(k) ((k)+a+b)
+
+std::vector<double>	jacobiP(double a, double b, double x, int n) {
+	std::vector<double>	result;
+	double	p0 = 1.;
+	result.push_back(p0);
+	double	p1 = (a - b)/2. + (1. + (a + b)/2.) * x;
+	result.push_back(p1);
+	int	i = 1;
+	while (i++ < n) {
+		double	p;
+		p = c(2*i-1) * (c(2*i-2) * c(2*i) * x + a*a - b*b) * p1
+			- 2. * (i-1.+a) * (i-1.+b) * c(2*i) * p0;
+		p /= 2 * i * c(i) * c(2*i-2);
+		result.push_back(p);
+		p0 = p1;
+		p1 = p;
+	}
+	return result;
+}
+
+std::string	jacobiPlots(const std::string& prefix,
+			double a, double b, int n, int N) {
+	std::vector<pfad_ptr>	pfade;
+	for (int j = 0; j <= n; j++) {
+		char	c = 'a' + j;
+		std::string	name = prefix;
+		name.append(1, c);
+		pfad_ptr	p(new pfad(name));
+		pfade.push_back(p);
+	}
+	for (int i = -N; i <= N; i++) {
+		double	x = i / (double)N;
+		std::vector<double>	r = jacobiP(a, b, x, n);
+		for (int j = 0; j <= n; j++) {
+			std::pair<double,double>	XY
+				= std::make_pair(x, r[j]);
+			pfade[j]->add(XY);
+		}
+	}
+	std::ostringstream	result;
+	for (int j = 0; j <= n; j++) {
+		result << pfade[j]->macro();
+	}
+	return result.str();
+}
+
+std::string	jacobiWeight(const std::string& prefix,
+			double a, double b, int N) {
+	std::ostringstream	out;
+	out << "\\def\\" << prefix << "weight{" << std::endl;
+	int	_punkte = 0;
+	int	starti = -N;
+	int	endi = N;
+	if (b < 0) {
+		starti += 1;
+	}
+	if (a < 0) {
+		endi -= 1;
+	}
+	for (int i = starti; i <= endi; i++) {
+		double	x = i / (double)N;
+		double	y = pow(1-x, a) * pow(1+x, b);
+		if (_punkte > 0) {
+			out << std::endl << "\t-- ";
+		}
+		out << "({" << x << "*\\dx},{" << y << "*\\dwy})";
+		_punkte++;
+	}
+	out << std::endl << "}" << std::endl;
+	return out.str();
+}
+
+struct option 	options[] = {
+{ "a",		required_argument,	NULL,		'a' },
+{ "b",		required_argument,	NULL,		'b' },
+{ "degree",	required_argument,	NULL,		'n' },
+{ "points",	required_argument,	NULL,		'N' },
+{ "prefix",	required_argument,	NULL,		'p' },
+{ "outfile",	required_argument,	NULL,		'o' },
+{ NULL,		0,			NULL,		 0  }
+};
+
+int	main(int argc, char *argv[]) {
+	int	n = 10;
+	int	N = 100;
+	double	a = 1;
+	double	b = 1;
+	std::string	outfilename;
+	std::string	prefix("prefix");
+	int	c;
+	while (EOF != (c = getopt_long(argc, argv, "a:b:n:N:o:p:", options,
+		NULL)))
+		switch (c) {
+		case 'a':
+			a = std::stod(optarg);
+			break;
+		case 'b':
+			b = std::stod(optarg);
+			break;
+		case 'n':
+			n = std::stoi(optarg);
+			break;
+		case 'N':
+			N = std::stoi(optarg);
+			break;
+		case 'o':
+			outfilename = std::string(optarg);
+			break;
+		case 'p':
+			prefix = std::string(optarg);
+			break;
+		}
+	std::string	result = jacobiPlots(prefix, a, b, n, N);
+	result.append(jacobiWeight(prefix, a, b, N));
+	if (outfilename.size() == 0) {
+		std::cout << result;
+		return EXIT_SUCCESS;
+	}
+	std::ofstream	out(outfilename.c_str());
+	out << result;
+	out.close();
+	return EXIT_SUCCESS;
+}
diff --git a/buch/chapters/070-orthogonalitaet/images/jacobi.pdf b/buch/chapters/070-orthogonalitaet/images/jacobi.pdf
new file mode 100644
index 0000000..8eb4f09
--- /dev/null
+++ b/buch/chapters/070-orthogonalitaet/images/jacobi.pdf
diff --git a/buch/chapters/070-orthogonalitaet/images/jacobi.tex b/buch/chapters/070-orthogonalitaet/images/jacobi.tex
new file mode 100644
index 0000000..7d2853f
--- /dev/null
+++ b/buch/chapters/070-orthogonalitaet/images/jacobi.tex
@@ -0,0 +1,104 @@
+%
+% jacobi.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{1}
+\input{jacobipaths.tex}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\def\dx{6}
+\def\dy{2}
+\def\dwy{1}
+
+\begin{scope}
+	\begin{scope}
+		\clip (-6,-2.55) rectangle (6,2.55);
+		\fill[color=red!20] \legendreweight
+			-- ({1*\dx},0) -- ({-1*\dx},0) -- cycle;
+		%\draw[color=blue] \legendrea;
+		\draw[color=blue] \legendreb;
+		\draw[color=blue] \legendrec;
+		\draw[color=blue] \legendred;
+		\draw[color=blue] \legendree;
+		\draw[color=blue] \legendref;
+		\draw[color=blue] \legendreg;
+		\draw[color=blue] \legendreh;
+		\draw[color=blue] \legendrei;
+		\draw[color=blue] \legendrek;
+		\draw[color=red] \legendreweight;
+	\end{scope}
+	\draw[->] (-6.1,0) -- (6.1,0) coordinate[label={$x$}];
+	\draw[->] (0,-2.55) -- (0,2.6) coordinate[label={right:$y$}];
+	\node at (6,2.3) [left] {$a=0$};
+	\node at (-6,2.3) [right] {$b=0$};
+\end{scope}
+
+\begin{scope}[yshift=-5.6cm]
+	\begin{scope}
+		\clip (-6,-2.55) rectangle (6,2.55);
+		\fill[color=red!20] \aoneweight
+			-- ({1*\dx},0) -- ({-1*\dx},0) -- cycle;
+		%\draw[color=blue] \aonea;
+		\draw[color=blue] \aoneb;
+		\draw[color=blue] \aonec;
+		\draw[color=blue] \aoned;
+		\draw[color=blue] \aonee;
+		\draw[color=blue] \aonef;
+		\draw[color=blue] \aoneg;
+		\draw[color=blue] \aoneh;
+		\draw[color=blue] \aonei;
+		\draw[color=blue] \aonek;
+		\draw[color=blue] \aonel;
+		\draw[color=blue] \aonem;
+		\draw[color=blue] \aonen;
+		\draw[color=blue] \aoneo;
+		\draw[color=red] \aoneweight;
+	\end{scope}
+	\draw[->] (-6.1,0) -- (6.1,0) coordinate[label={$x$}];
+	\draw[->] (0,-2.55) -- (0,2.6) coordinate[label={right:$y$}];
+	\node at (6,2.3) [left] {$a=1$};
+	\node at (-6,2.3) [right] {$b=0$};
+\end{scope}
+
+\begin{scope}[yshift=-11.2cm]
+	\begin{scope}
+		\clip (-6,-2.55) rectangle (6,2.55);
+		\fill[color=red!20] \aminusoneweight
+			-- ({1*\dx},3)
+			-- ({1*\dx},0) -- ({-1*\dx},0) -- cycle;
+		%\draw[color=blue] \aminusonea;
+		\draw[color=blue] \aminusoneb;
+		\draw[color=blue] \aminusonec;
+		\draw[color=blue] \aminusoned;
+		\draw[color=blue] \aminusonee;
+		\draw[color=blue] \aminusonef;
+		\draw[color=blue] \aminusoneg;
+		\draw[color=blue] \aminusoneh;
+		\draw[color=blue] \aminusonei;
+		\draw[color=blue] \aminusonek;
+		\draw[color=blue] \aminusonel;
+		\draw[color=blue] \aminusonem;
+		\draw[color=blue] \aminusonen;
+		\draw[color=blue] \aminusoneo;
+		\draw[color=red] \aminusoneweight;
+	\end{scope}
+	\draw[->] (-6.1,0) -- (6.1,0) coordinate[label={$x$}];
+	\draw[->] (0,-2.55) -- (0,2.6) coordinate[label={right:$y$}];
+	\node at (6,2.3) [left] {$a=-1$};
+	\node at (-6,2.3) [right] {$b=0$};
+\end{scope}
+
+
+
+\end{tikzpicture}
+\end{document}
+