gauss quadratur zeugs

5 files changed, 209 insertions, 0 deletions

diff --git a/buch/chapters/060-integral/gq/Makefile b/buch/chapters/060-integral/gq/Makefile
new file mode 100644
index 0000000..2ce859c
--- /dev/null
+++ b/buch/chapters/060-integral/gq/Makefile
@@ -0,0 +1,22 @@
+#
+# Makefile -- Gauss quadrature experiments
+#
+# (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+#
+all:	kreis gq.pdf
+
+gq.o:	gq.cpp
+	g++ `pkg-config --cflags gsl` -o gq.o -g -Wall -O2 -c gq.cpp 
+
+gq:	gq.o
+	g++ -g -Wall -O2 gq.o `pkg-config --libs gsl` -o gq
+
+kreis:	kreis.cpp
+	g++ `pkg-config --cflags gsl` -g -Wall -O2 kreis.cpp \
+		`pkg-config --libs gsl` -o kreis
+
+gqpaths.tex:	gq
+	./gq > gqpaths.tex
+
+gq.pdf:	gq.tex gqpaths.tex
+	pdflatex gq.tex
diff --git a/buch/chapters/060-integral/gq/gq.cpp b/buch/chapters/060-integral/gq/gq.cpp
new file mode 100644
index 0000000..3eb3ac8
--- /dev/null
+++ b/buch/chapters/060-integral/gq/gq.cpp
@@ -0,0 +1,64 @@
+/*
+ * gq.cpp -- Gauss Quadrature Example
+ *
+ * (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+ */
+#include <iostream>
+#include <iomanip>
+#include <cstdlib>
+#include <cstdio>
+#include <cmath>
+#include <gsl/gsl_integration.h>
+
+int	counter;
+
+double	f(double x, void *) {
+	counter++;
+	return sqrt(1-x*x);
+}
+
+void	do_integration(size_t n, const char *name) {
+	double	scaling = 1e6;
+	// set up integration gauss quadrature
+	counter = 0;
+	gsl_function	func;
+	func.function = f;
+	func.params = NULL;
+
+	// prepare plotting of results
+	int	steps = 200;
+	double	h = 1. / steps;
+
+	printf("\\def\\%s{\n\t(0,0)", name);
+	for (int i = 1; i <= steps; i++) {
+		double	a = i * h;
+		a = 1 - (1 - a) * (1 - a);
+		double	expected = (asin(a) + a * sqrt(1 - a*a)) / 2;
+		gsl_integration_fixed_workspace	*ws
+			= gsl_integration_fixed_alloc(
+				gsl_integration_fixed_legendre,
+				n, 0, a, 0, 0);
+		double	result = 0;
+		gsl_integration_fixed(&func, &result, ws);
+		double	y = fabs(scaling * (result - expected));
+		if (y > 10) {
+			y = 10;
+		}
+		printf("\n\t--({\\dx*%.4f},{\\dy*%.6f})", a, y);
+		gsl_integration_fixed_free(ws);
+	}
+	printf("\n}\n");
+}
+
+int	main(int argc, char *argv[]) {
+	do_integration(2, "two");
+	do_integration(4, "four");
+	do_integration(6, "six");
+	do_integration(8, "eight");
+	do_integration(10, "ten");
+	do_integration(12, "twelve");
+	do_integration(14, "fourteen");
+	do_integration(16, "sixteen");
+
+	return EXIT_SUCCESS;
+}
diff --git a/buch/chapters/060-integral/gq/gq.pdf b/buch/chapters/060-integral/gq/gq.pdf
new file mode 100644
index 0000000..7a0b3c4
--- /dev/null
+++ b/buch/chapters/060-integral/gq/gq.pdf
diff --git a/buch/chapters/060-integral/gq/gq.tex b/buch/chapters/060-integral/gq/gq.tex
new file mode 100644
index 0000000..e5a9c4d
--- /dev/null
+++ b/buch/chapters/060-integral/gq/gq.tex
@@ -0,0 +1,57 @@
+%
+% gq.tex -- Fehler 
+%
+% (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}
+\def\dx{12}
+\def\dy{8}
+\input{gqpaths.tex}
+\begin{tikzpicture}[>=latex,thick,scale=\skala]
+
+\begin{scope}
+\clip (0,0) rectangle ({\dx*1},{\dy*1});
+\draw[color=red,line width=1.4pt] \two;
+\draw[color=red,line width=1.4pt] \four;
+\draw[color=red,line width=1.4pt] \six;
+\draw[color=red,line width=1.4pt] \eight;
+\draw[color=red,line width=1.4pt] \ten;
+\draw[color=red,line width=1.4pt] \twelve;
+\draw[color=red,line width=1.4pt] \fourteen;
+\draw[color=red,line width=1.4pt] \sixteen;
+\end{scope}
+
+% add image content here
+\draw[->] (0,-0.1) -- (0,{\dy+0.5}) coordinate[label={right:$e$}];
+\draw[->] (-0.1,0) -- ({\dx+0.5},0) coordinate[label={$a$}];
+
+\draw ({\dx},-0.1) -- ({\dx},0.1);
+\draw[line width=0.3pt] ({\dx},0) -- ({\dx},{\dy});
+\node at ({\dx},0) [below] {$1$};
+\node at (0,0) [below] {$0$};
+
+\draw (-0.1,\dy) -- (0.1,\dy);
+\node at (0,\dy) [left] {$10^{-6}$};
+
+\node at ({0.227*\dx},{0.7*\dy}) [rotate=83.5] {$n=2$};
+\node at ({0.678*\dx},{0.7*\dy}) [rotate=83] {$n=4$};
+\node at ({0.850*\dx},{0.7*\dy}) [rotate=84] {$n=6$};
+\node at ({0.915*\dx},{0.7*\dy}) [rotate=85] {$n=8$};
+\node at ({1.015*\dx},{0.7*\dy}) [rotate=90] {$n=16$};
+
+\foreach \x in {0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9}{
+	\draw ({\x*\dx},-0.1) -- ({\x*\dx},0.1);
+	\node at ({\x*\dx},0) [below] {$\x$};
+}
+
+\end{tikzpicture}
+\end{document}
+
diff --git a/buch/chapters/060-integral/gq/kreis.cpp b/buch/chapters/060-integral/gq/kreis.cpp
new file mode 100644
index 0000000..c9723cc
--- /dev/null
+++ b/buch/chapters/060-integral/gq/kreis.cpp
@@ -0,0 +1,66 @@
+/*
+ * kreis.cpp -- Gauss Quadrature Example
+ *
+ * (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+ */
+#include <iostream>
+#include <iomanip>
+#include <cstdlib>
+#include <cstdio>
+#include <cmath>
+#include <gsl/gsl_integration.h>
+
+double	xmax = 0.990;
+
+double	f(double x, void *) {
+	return sqrt(1-x*x);
+}
+
+double	do_integration(size_t n) {
+	// set up integration gauss quadrature
+	gsl_function	func;
+	func.function = f;
+	func.params = NULL;
+
+	gsl_integration_fixed_workspace	*ws
+		= gsl_integration_fixed_alloc(
+			gsl_integration_fixed_legendre,
+			n, -xmax, xmax, 0, 0);
+	double	result = 0;
+	gsl_integration_fixed(&func, &result, ws);
+	gsl_integration_fixed_free(ws);
+	return result;
+}
+
+double	do_trapez(size_t n) {
+	double	h = 2 * xmax / n;	
+	double	s = 0;
+	for (int i = 0; i <= n; i++) {
+		double	x = -xmax + i * h;
+		double	a = f(x, NULL);
+		if ((i == 0) || (i == n)) {
+			a = a * 0.5;
+		}
+		s += a;
+	}
+	return h * s;
+}
+
+void	do_table(double limit) {
+	xmax = limit;
+	for (int n = 2; n <= 20; n += 2) {
+		printf("%d & %.16f & %.16f \\\\\n", n, do_integration(n),
+			do_trapez(10 * n));
+	}
+	double	amax = acos(xmax);
+	double	expected = (M_PI / 2 - amax) + sin(amax) * cos(amax);
+	printf("\\infty & %.16f & %.16f \\\\\n", expected, expected);
+}
+
+int	main(int argc, char *argv[]) {
+	do_table(0.5);
+	do_table(0.9);
+	do_table(0.99);
+	do_table(0.999);
+	return EXIT_SUCCESS;
+}