Merge branch 'master' of github.com:AndreasFMueller/SeminarSpezielleFunktionen

1 files changed, 415 insertions, 0 deletions

diff --git a/buch/chapters/110-elliptisch/images/rechteck.cpp b/buch/chapters/110-elliptisch/images/rechteck.cpp
new file mode 100644
index 0000000..c65ae0f
--- /dev/null
+++ b/buch/chapters/110-elliptisch/images/rechteck.cpp
@@ -0,0 +1,415 @@
+/*
+ * rechteck.cpp
+ *
+ * (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+ */
+#include <cmath>
+#include <cstdlib>
+#include <cstdio>
+#include <complex>
+#include <iostream>
+#include <fstream>
+#include <list>
+#include <getopt.h>
+#include <gsl/gsl_sf_ellint.h>
+
+double	ast = 1;
+
+/**
+ * \brief Base class for integrands
+ */
+class integrand {
+protected:
+	double	_k;
+public:
+	double	k() const { return _k; }
+	integrand(double k) : _k(k) { }
+	virtual std::complex<double>	operator()(const std::complex<double>& z) const = 0;
+	static double	kprime(double k);
+};
+
+double	integrand::kprime(double k) {
+	return sqrt(1 - k*k);
+}
+
+/**
+ * \brief Elliptic integral of the first kind
+ */
+class integrand1 : public integrand {
+public:
+	integrand1(double k) : integrand(k) { }
+	virtual std::complex<double>	operator()(
+		const std::complex<double>& z) const {
+		std::complex<double>	square = z * z;
+		std::complex<double>	eins(1.0);
+		std::complex<double>	result = eins
+			/ sqrt((eins - square) * (eins - k() * k() * square));
+		return ast * ((result.imag() < 0) ? -result : result);
+	}
+};
+
+/**
+ * \brief A class to trace curves
+ */
+class curvetracer {
+	const integrand&	_f;
+public:
+	typedef std::list<std::complex<double> >	curve_t;
+	curvetracer(const integrand& f) : _f(f) { }
+
+	std::complex<double>	startpoint(const std::complex<double>& z,
+		int n = 100) const;
+
+	std::complex<double>	segment(const std::complex<double>& z1,
+		const std::complex<double>& z2, int n) const;
+
+	std::pair<std::complex<double>,std::complex<double> >	segmentl(
+		const std::complex<double>& start,
+		const std::complex<double>& dir,
+		double stepsize, int n = 10) const;
+
+	curve_t	trace(const std::complex<double>& startz,
+		const std::complex<double>& dir,
+		const std::complex<double>& startw,
+		int maxsteps) const;
+
+	static curve_t	mirrorx(const curve_t& c);
+	static curve_t	mirrory(const curve_t& c);
+	static curve_t	mirror(const curve_t& c);
+};
+
+/**
+ * \brief Perform integration for a 
+ *
+ * \param z1	the start point of the segment in the domain
+ * \param z2	the end point of the segment in the domain
+ * \param n	the number of integration steps to use
+ * \return	the increment along that segment
+ */
+std::complex<double>	curvetracer::segment(const std::complex<double>& z1,
+		const std::complex<double>& z2, int n = 100) const {
+	std::complex<double>	dz = z2 - z1;
+	std::complex<double>	summe(0);
+	double	h = 1. / (2 * n);
+	for (int i = 1; i < 2 * n; i += 2) {
+		double	t = i * h;
+		std::complex<double>	z = (1 - t) * z1 + t * z2;
+		summe += _f(z);
+	}
+	return dz * h * summe * 2.;
+}
+
+/**
+ * \brief Exception thrown when the number of iterations is exceeded
+ */
+class toomanyiterations : public std::runtime_error {
+public:
+	toomanyiterations(const std::string& cause)
+		: std::runtime_error(cause) {
+	}
+};
+
+/**
+ * \brief Perform integration with a given target length
+ *
+ * \param start		the start point
+ * \param dir		the direction in which to integrate
+ * \param stepsize	the required length of the step
+ * \param n		the number of integration steps
+ * \return		the increment in the range by this segment
+ */
+std::pair<std::complex<double>, std::complex<double> >	curvetracer::segmentl(
+	const std::complex<double>& start,
+	const std::complex<double>& dir,
+	double stepsize,
+	int n) const {
+	std::complex<double>	s(0.);
+	std::complex<double>	z = start;
+	int	counter = 100;
+	while (abs(s) < stepsize) {
+		s = s + segment(z, z + dir, n);
+		z = z + dir;
+		if (counter-- == 0) {
+			throw toomanyiterations("too many iterations");
+		}
+	}
+	return std::make_pair(z, s);
+}
+
+/**
+ * \brief Trace a curve from a starting point in the domain
+ *
+ * \param startz	the start point of the curve in the domain
+ * \param dir		the direction of the curve in the domain
+ * \param startw	the initial function value where the curve starts in
+ *			the range
+ * \param maxsteps	the maximum number of dir-steps before giving up
+ * \return		the curve as a list of points
+ */
+curvetracer::curve_t	curvetracer::trace(const std::complex<double>& startz,
+			const std::complex<double>& dir,
+			const std::complex<double>& startw,
+			int maxsteps) const {
+	curve_t	result;
+	std::complex<double>	z = startz;
+	std::complex<double>	w = startw;
+	result.push_back(w);
+	while (maxsteps-- > 0) {
+		try {
+			auto	seg = segmentl(z, dir, abs(dir), 40);
+			z = seg.first;
+			w = w + seg.second;
+			result.push_back(w);
+		} catch (const toomanyiterations& x) {
+			std::cerr << "iterations exceeded after ";
+			std::cerr << result.size();
+			std::cerr << " points";
+			maxsteps = 0;
+		}
+	}
+	return result;
+}
+
+/**
+ * \brief Find the initial point for a coordinate curve
+ *
+ * \param k	the elliptic integral parameter
+ * \param z	the desired starting point argument
+ */
+std::complex<double>	curvetracer::startpoint(const std::complex<double>& z,
+		int n) const {
+	std::cerr << "start at " << z.real() << "," << z.imag() << std::endl;
+	return segment(std::complex<double>(0.), z, n);
+}
+
+
+curvetracer::curve_t	curvetracer::mirrorx(const curve_t& c) {
+	curve_t	result;
+	for (auto z : c) {
+		result.push_back(-std::conj(z));
+	}
+	return result;
+}
+
+curvetracer::curve_t	curvetracer::mirrory(const curve_t& c) {
+	curve_t	result;
+	for (auto z : c) {
+		result.push_back(std::conj(z));
+	}
+	return result;
+}
+
+curvetracer::curve_t	curvetracer::mirror(const curve_t& c) {
+	curve_t	result;
+	for (auto z : c) {
+		result.push_back(-z);
+	}
+	return result;
+}
+
+/**
+ * \brief Class to draw the curves to a file
+ */
+class curvedrawer {
+	std::ostream	*_out;
+	std::string	_color;
+public:
+	curvedrawer(std::ostream *out) : _out(out), _color("red") { }
+	const std::string&	color() const { return _color; }
+	void	color(const std::string& c) { _color = c; }
+	void	operator()(const curvetracer::curve_t& curve);
+	std::ostream	*out() { return _out; }
+};
+
+/**
+ * \brief Operator to draw a curve
+ *
+ * \param curve		the curve to draw
+ */
+void	curvedrawer::operator()(const curvetracer::curve_t& curve) {
+	double	first = true;
+	for (auto z : curve) {
+		if (first) {
+			*_out << "\\draw[color=" << _color << "] ";
+			first = false;
+		} else {
+			*_out << std::endl << "    -- ";
+		}
+		*_out << "({" << z.real() << "*\\dx},{" << z.imag() << "*\\dy})";
+	}
+	*_out << ";" << std::endl;
+}
+
+static struct option	longopts[] = {
+{ "outfile",	required_argument,	NULL,	'o' },
+{ "k",		required_argument,	NULL,	'k' },
+{ "deltax",	required_argument,	NULL,	'd' },
+{ NULL,		0,			NULL,	 0  }
+};
+
+/**
+ * \brief Main function
+ */
+int	main(int argc, char *argv[]) {
+	double	k = 0.625;
+	double	deltax = 0.2;
+
+	int	c;
+	int	longindex;
+	std::string	outfilename;
+	while (EOF != (c = getopt_long(argc, argv, "o:k:d:", longopts,
+		&longindex)))
+		switch (c) {
+		case 'd':
+			deltax = std::stod(optarg);
+			break;
+		case 'o':
+			outfilename = std::string(optarg);
+			break;
+		case 'k':
+			k = std::stod(optarg);
+			break;
+		}
+
+	double	kprime = integrand::kprime(k);
+
+	double	xmax = gsl_sf_ellint_Kcomp(k, GSL_PREC_DOUBLE);
+	double	ymax = gsl_sf_ellint_Kcomp(kprime, GSL_PREC_DOUBLE);
+	std::cout << "xmax = " << xmax << std::endl;
+	std::cout << "ymax = " << ymax << std::endl;
+
+	curvedrawer	*cdp;
+	std::ofstream	*outfile = NULL;
+	if (outfilename.c_str()) {
+		outfile = new std::ofstream(outfilename.c_str());
+	}
+	if (outfile) {
+		cdp = new curvedrawer(outfile);
+	} else {
+		cdp = new curvedrawer(&std::cout);
+	}
+
+	integrand1	f(k);
+	curvetracer	ct(f);
+
+	// fill
+	(*cdp->out()) << "\\fill[color=red!10] ({" << (-xmax) << "*\\dx},0) "
+		<< "rectangle ({" << xmax << "*\\dx},{" << ymax << "*\\dy});"
+		<< std::endl;
+	(*cdp->out()) << "\\fill[color=blue!10] ({" << (-xmax) << "*\\dx},{"
+		<< (-ymax) << "*\\dy}) rectangle ({" << xmax << "*\\dx},0);"
+		<< std::endl;
+
+	// "circles"
+	std::complex<double>	dir(0.01, 0);
+	for (double im = deltax; im < 3; im += deltax) {
+		std::complex<double>	startz(0, im);
+		std::complex<double>	startw = ct.startpoint(startz);
+		curvetracer::curve_t	curve = ct.trace(startz, dir,
+			startw, 1000);
+		cdp->color("red");
+		(*cdp)(curve);
+		(*cdp)(curvetracer::mirrorx(curve));
+		cdp->color("blue");
+		(*cdp)(curvetracer::mirrory(curve));
+		(*cdp)(curvetracer::mirror(curve));
+	}
+
+	// imaginary axis
+	(*cdp->out()) << "\\draw[color=red] (0,0) -- (0,{" << ymax
+		<< "*\\dy});" << std::endl;
+	(*cdp->out()) << "\\draw[color=blue] (0,0) -- (0,{" << (-ymax)
+		<< "*\\dy});" << std::endl;
+
+	// arguments between 0 and 1
+	dir = std::complex<double>(0, 0.01);
+	for (double re = 0.2; re < 1; re += 0.2) {
+		std::complex<double>	startz(re, 0.001);
+		std::complex<double>	startw = ct.startpoint(startz);
+		startw = std::complex<double>(startw.real(), 0);
+		curvetracer::curve_t	curve = ct.trace(startz, dir,
+			startw, 1000);
+		cdp->color("red");
+		(*cdp)(curve);
+		(*cdp)(curvetracer::mirrorx(curve));
+		cdp->color("blue");
+		(*cdp)(curvetracer::mirrory(curve));
+		(*cdp)(curvetracer::mirror(curve));
+	}
+
+	// argument 1 (singularity)
+	{
+		std::complex<double>	startz(1.);
+		std::complex<double>	startw(xmax);
+		curvetracer::curve_t	curve = ct.trace(startz, dir,
+			startw, 1000);
+		cdp->color("red");
+		(*cdp)(curve);
+		(*cdp)(curvetracer::mirrorx(curve));
+		cdp->color("blue");
+		(*cdp)(curvetracer::mirror(curve));
+		(*cdp)(curvetracer::mirrory(curve));
+	}
+
+	// arguments between 1 and 1/k
+	{
+		for (double x0 = 1 + deltax; x0 < 1/k; x0 += deltax) {
+			double	y0 = sqrt(1-1/(x0*x0))/kprime;
+			//std::cout << "y0 = " << y0 << std::endl;
+			double	y = gsl_sf_ellint_F(asin(y0), kprime,
+				GSL_PREC_DOUBLE);
+			std::complex<double>	startz(x0);
+			std::complex<double>	startw(xmax, y);
+			curvetracer::curve_t	curve = ct.trace(startz, dir,
+				startw, 1000);
+			cdp->color("red");
+			(*cdp)(curve);
+			(*cdp)(curvetracer::mirrorx(curve));
+			cdp->color("blue");
+			(*cdp)(curvetracer::mirror(curve));
+			(*cdp)(curvetracer::mirrory(curve));
+		}
+	}
+
+	// argument 1/k
+#if 0
+	{
+		std::complex<double>	startz(1/k);
+		std::complex<double>	startw(xmax, ymax);
+		curvetracer::curve_t	curve = ct.trace(startz, dir,
+			startw, 1000);
+		cdp->color("red");
+		(*cdp)(curve);
+		(*cdp)(curvetracer::mirrorx(curve));
+		cdp->color("blue");
+		(*cdp)(curvetracer::mirror(curve));
+		(*cdp)(curvetracer::mirrory(curve));
+	}
+#endif
+
+	// arguments larger than 1/k
+	{
+		dir = std::complex<double>(0, 0.01);
+		double	x0 = 1;
+		while (x0 <= 1/k + 0.0001) { x0 += deltax; }
+		for (; x0 < 4; x0 += deltax) {
+			std::complex<double>	startz(x0);
+			std::complex<double>	startw(gsl_sf_ellint_F(
+				asin(1/(k*x0)), k, GSL_PREC_DOUBLE), ymax);
+		curvetracer::curve_t	curve = ct.trace(startz, dir,
+			startw, 1000);
+		cdp->color("red");
+		(*cdp)(curve);
+		(*cdp)(curvetracer::mirrorx(curve));
+		cdp->color("blue");
+		(*cdp)(curvetracer::mirror(curve));
+		(*cdp)(curvetracer::mirrory(curve));
+		}
+	}
+
+	// border
+	(*cdp->out()) << "\\def\\xmax{" << xmax << "}" << std::endl;
+	(*cdp->out()) << "\\def\\ymax{" << ymax << "}" << std::endl;
+
+	return EXIT_SUCCESS;
+}