diff options
Diffstat (limited to 'buch/chapters/070-orthogonalitaet/images')
| -rw-r--r-- | buch/chapters/070-orthogonalitaet/images/Makefile | 13 | ||||
| -rw-r--r-- | buch/chapters/070-orthogonalitaet/images/jacobi.cpp | 179 | ||||
| -rw-r--r-- | buch/chapters/070-orthogonalitaet/images/jacobi.pdf | bin | 0 -> 65748 bytes | |||
| -rw-r--r-- | buch/chapters/070-orthogonalitaet/images/jacobi.tex | 104 |
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 Binary files differnew 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} + |