Serie 02

Kuengjoe_s02/Kuengjoe_S02_Aufg2.py

@@ -0,0 +1,88 @@
+import numpy as np
+import matplotlib.pyplot as plt
+
+# a)
+
+def f1(x):
+    return (((((((x - 14)*x + 84)*x - 280)*x + 560)*x - 672)*x + 448)*x - 128)
+
+def f2(x):
+    return (x-2)**7
+
+xa = np.linspace(1.99, 2.01, 501, dtype=np.float64)
+
+ya1 = f1(xa)
+ya2 = f2(xa)
+
+plt.figure()
+plt.plot(xa, ya1, label='f1(x) expanded', linewidth=2)
+plt.plot(xa, ya2, label='f2(x) (x-2)^7', linestyle='--', linewidth=2)
+plt.title("Vergleich zweier äquivalenter Funktionen")
+plt.xlabel("x")
+plt.ylabel("f(x)")
+plt.legend()
+plt.grid(True, alpha=0.3)
+plt.tight_layout()
+
+
+
+with np.errstate(divide='ignore', invalid='ignore'):
+    rel_err= np.abs((ya1 - ya2)/ np.where( ya2!= 0, ya2, 1))
+
+plt.figure()
+plt.plot(xa, rel_err)
+plt.title("Relativer Fehler zwischen f1 und f2")
+plt.xlabel("x")
+plt.ylabel("Relativer Fehler")
+plt.yscale("log")
+plt.grid(True, which="both", alpha=0.3)
+plt.tight_layout()
+
+# b)
+xmin, xmax, h =  -1e-14, 1e-14, 1e-17
+n = int(round((xmax - xmin) / h)) + 1
+xb = np.linspace(xmin, xmax, n, dtype=np.float64)
+
+def g_naive(x):
+    return x / (np.sin(1.0+x) - np.sin(1.0))
+
+g_b = g_naive(xb)
+i0 = np.argmin(np.abs(xb))
+g_b[i0] = np.nan
+
+
+plt.figure()
+plt.plot(xb, g_b)
+plt.title('g(x) = x / (sin(1+x) - sin(1))')
+plt.xlabel('x')
+plt.ylabel('g(x)')
+plt.grid(True, alpha=0.3)
+plt.tight_layout()
+
+
+# c)
+
+def g_stab(x):
+    return x / (2.0 * np.cos(1.0 +0.5*x) * np.sin(0.5*x))
+
+g_c = g_stab(xb)
+g_c[i0] = 1.0 / np.cos(1.0)
+
+plt.figure()
+plt.plot(xb, g_c)
+plt.axhline(1.0 /np.cos(1.0), linestyle='--')
+plt.title('Stabilisierte Berechnung von g(x)')
+plt.xlabel('x')
+plt.ylabel('g(x)')
+plt.grid(True, alpha=0.3)
+plt.tight_layout()
+
+print('theoretischer Grenzwert: g(0) =', 1.0 / np.cos(1.0))
+
+plt.show()
+
+
+
+
+
+