4 files changed, 118 insertions, 2 deletions

diff --git a/buch/chapters/110-elliptisch/agm/Makefile b/buch/chapters/110-elliptisch/agm/Makefile
index e7975e1..8dab511 100644
--- a/buch/chapters/110-elliptisch/agm/Makefile
+++ b/buch/chapters/110-elliptisch/agm/Makefile
@@ -3,6 +3,11 @@
 #
 # (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
 #
+all:	sn
+
+sn:	sn.cpp
+	g++ -O -Wall -g -std=c++11 sn.cpp -o sn `pkg-config --cflags gsl` `pkg-config --libs gsl`
+
 
 agm:	agm.cpp
 	g++ -O -Wall -g -std=c++11 agm.cpp -o agm `pkg-config --cflags gsl` `pkg-config --libs gsl`
diff --git a/buch/chapters/110-elliptisch/agm/agm.cpp b/buch/chapters/110-elliptisch/agm/agm.cpp
index fdb0441..8abb4b2 100644
--- a/buch/chapters/110-elliptisch/agm/agm.cpp
+++ b/buch/chapters/110-elliptisch/agm/agm.cpp
@@ -9,23 +9,54 @@
 #include <iostream>
 #include <gsl/gsl_sf_ellint.h>
 
+inline long double	sqrl(long double x) {
+	return x * x;
+}
 
+long double	Xn(long double a, long double b, long double x) {
+	double long	epsilon = fabsl(a - b);
+	if (epsilon > 0.001) {
+		return (a - sqrtl(sqrl(a) - sqrl(x) * (a + b) * (a - b)))
+			/ (x * (a - b));
+	}
+	long double	d = a + b;
+	long double	x1 = 0;
+	long double	y2 = sqrl(x/a);
+	long double	c = 1;
+	long double	s = 0;
+	int	k = 1;
+	while (c > 0.0000000000001) {
+		c *= (0.5 - (k - 1)) / k;
+		c *= (d - epsilon) * y2;
+		s += c;
+		c *= epsilon;
+		c = -c;
+		k++;
+	}
+	return s * a / x;
+}
 
 int	main(int argc, char *argv[]) {
 	long double	a = 1;
 	long double	b = sqrtl(2.)/2;
+	long double	x = 0.7;
 	if (argc >= 3) {
 		a = std::stod(argv[1]);
 		b = std::stod(argv[2]);
 	}
+	if (argc >= 4) {
+		x = std::stod(argv[3]);
+	}
 
 	{
 		long double	an = a;
 		long double	bn = b;
+		long double	xn = x;
 		for (int i = 0; i < 10; i++) {
-			printf("%d  %24.18Lf  %24.18Lf  %24.18Lf\n",
-				i, an, bn, a * M_PI / (2 * an));
+			printf("%d  %24.18Lf  %24.18Lf  %24.18Lf  %24.18Lf\n",
+				i, an, bn, xn, a * asin(xn) / an);
 			long double A = (an + bn) / 2;
+			xn = Xn(an, bn, xn);
 			bn = sqrtl(an * bn);
 			an = A;
 		}
@@ -36,6 +67,8 @@ int	main(int argc, char *argv[]) {
 		k = sqrt(1 - k*k);
 		double	K = gsl_sf_ellint_Kcomp(k, GSL_PREC_DOUBLE);
 		printf("                             %24.18f  %24.18f\n", k, K);
+		double	F = gsl_sf_ellint_F(asinl(x), k, GSL_PREC_DOUBLE);
+		printf("                             %24.18f  %24.18f\n", k, F);
 	}
 
 	return EXIT_SUCCESS;
diff --git a/buch/chapters/110-elliptisch/agm/agm.maxima b/buch/chapters/110-elliptisch/agm/agm.maxima
new file mode 100644
index 0000000..c7facd4
--- /dev/null
+++ b/buch/chapters/110-elliptisch/agm/agm.maxima
@@ -0,0 +1,26 @@
+/*
+ * agm.maxima
+ *
+ * (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule 
+ */
+
+S:  2*a*sin(theta1) / (a+b+(a-b)*sin(theta1)^2);
+
+C2: ratsimp(diff(S, theta1)^2 / (1 - S^2));
+C2: ratsimp(subst(sqrt(1-sin(theta1)^2), cos(theta1), C2));
+C2: ratsimp(subst(S,                     sin(theta),  C2));
+C2: ratsimp(subst(sqrt(1-S^2),           cos(theta),  C2));
+
+D2: (a^2 * cos(theta)^2 + b^2 * sin(theta)^2)
+	/
+    (a1^2 * cos(theta1)^2 + b1^2 * sin(theta1)^2);
+D2: subst((a+b)/2,   a1, D2);
+D2: subst(sqrt(a*b), b1, D2);
+D2: ratsimp(subst(1-S^2,  cos(theta)^2, D2));
+D2: ratsimp(subst(S,      sin(theta),   D2));
+D2: ratsimp(subst(1-sin(theta1)^2, cos(theta1)^2, D2));
+
+Q: D2/C2;
+Q: ratsimp(subst(x, sin(theta1), Q));
+
+Q: ratsimp(expand(ratsimp(Q)));
diff --git a/buch/chapters/110-elliptisch/agm/sn.cpp b/buch/chapters/110-elliptisch/agm/sn.cpp
new file mode 100644
index 0000000..ff2ed17
--- /dev/null
+++ b/buch/chapters/110-elliptisch/agm/sn.cpp
@@ -0,0 +1,52 @@
+/*
+ * ns.cpp
+ *
+ * (c) 2022 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
+ */
+#include <cstdlib>
+#include <cstdio>
+#include <cmath>
+#include <iostream>
+#include <gsl/gsl_sf_ellint.h>
+#include <gsl/gsl_sf_elljac.h>
+
+static const int	N = 10;
+
+inline long double	sqrl(long double x) {
+	return x * x;
+}
+
+int	main(int argc, char *argv[]) {
+	long double	u = 0.6;
+	long double	k = 0.9;
+	long double	kprime = sqrt(1 - sqrl(k));
+	
+	long double	a[N], b[N], x[N+1];
+	a[0] = 1;
+	b[0] = kprime;
+
+	for (int n = 0; n < N-1; n++) {
+		printf("a[%d] = %22.18Lf  b[%d] = %22.18Lf\n", n, a[n], n, b[n]);
+		a[n+1] = (a[n] + b[n]) / 2;
+		b[n+1] = sqrtl(a[n] * b[n]);
+	}
+
+	x[N] = sinl(u * a[N-1]);
+	printf("x[%d] = %22.18Lf\n", N, x[N]);
+
+	for (int n = N - 1; n >= 0; n--) {
+		x[n] = 2 * a[n] * x[n+1] / (a[n] + b[n] + (a[n] - b[n]) * sqrl(x[n+1]));
+		printf("x[%2d] = %22.18Lf\n", n, x[n]);
+	}
+	
+	printf("sn(%7.4Lf, %7.4Lf) = %20.24Lf\n", u, k, x[0]);
+
+	double	sn, cn, dn;
+	double	m = sqrl(k);
+	gsl_sf_elljac_e((double)u, m, &sn, &cn, &dn);
+	printf("sn(%7.4Lf, %7.4Lf) = %20.24f\n", u, k, sn);
+	printf("cn(%7.4Lf, %7.4Lf) = %20.24f\n", u, k, cn);
+	printf("dn(%7.4Lf, %7.4Lf) = %20.24f\n", u, k, dn);
+
+	return EXIT_SUCCESS;
+}