Serie 03

Kuengjoe_S03/Kuengjoe_S03_Aufg3b.py

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+
+import numpy as np
+import matplotlib.pyplot as plt
+
+def f(x):
+    return 5.0 * (2.0 * x**2)**(-1.0/3.0)
+
+def g(x):
+    return 10.0**5 * (2.0 * np.e)**(-x/100.0)
+
+
+def h_exp(x):
+    return (625.0/64.0)**x
+
+# (i)
+x1 = np.logspace(-3, 2, 4000)  # 0.001 ... 100
+y1 = f(x1)
+plt.figure()
+plt.loglog(x1, y1)
+plt.title("Aufg. 3b (i) – f(x) as straight line in log-log")
+plt.xlabel("x")
+plt.ylabel("f(x)")
+plt.grid(True, which="both")
+
+# (ii)
+x2 = np.linspace(1e-6, 100.0, 4000)
+y2 = g(x2)
+plt.figure()
+plt.semilogy(x2, y2)
+plt.title("Aufg. 3b (ii) – g(x) as straight line in semilog-y")
+plt.xlabel("x")
+plt.ylabel("g(x)")
+plt.grid(True, which="both")
+
+# (iii)
+x3 = np.linspace(1e-6, 100.0, 4000)
+y3 = h_exp(x3)
+plt.figure()
+plt.semilogy(x3, y3)
+plt.title("Aufg. 3b (iii) – h(x) as straight line in semilog-y")
+plt.xlabel("x")
+plt.ylabel("h(x)")
+plt.grid(True, which="both")
+
+plt.show()

Kuengjoe_S03/Kuengjoe_S03_Aufg4.py

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+import numpy as np
+import matplotlib.pyplot as plt
+
+def g(x):
+    # Naive Form des Polynoms 100 x² − 200 x + 99
+    return 100*x**2 - 200*x + 99
+
+def g_fact(x):
+    # Faktorisierte Form 100 (x − 1.1)(x − 0.9) – vermeidet Auslöschung
+    return 100*(x-1.1)*(x-0.9)
+
+def h(x, factored=False):
+    # Liefert h(x) = sqrt(g(x)); bei factored=True wird die stabile Variante
+    # g_fact(x) verwendet
+    return np.sqrt(g_fact(x) if factored else g(x))
+
+def kappa_h(x):
+    # Konditionszahl κ_h(x) = |x * h'(x) / h(x)|
+    # mit h'(x) = 100 (x − 1) / h(x)
+    return np.abs(x * 100*(x-1) / (h(x)**2))
+
+# a) Vergleich der Auswertungen
+x_test = 1.1 + np.array([1e-8, 1e-7, 1e-6, 1e-5])
+print("x           h_naiv           h_fakt          relFehler")
+for x in x_test:
+    # instabile Auswertung
+    h_naiv = h(x)
+    # stabile Auswertungm
+    h_fakt = h(x, factored=True)
+    relerr = abs(h_naiv - h_fakt)/abs(h_fakt)
+    print(f"{x:.10f}  {h_naiv:.12e}  {h_fakt:.12e}  {relerr:.2e}")
+
+# b) Plot der Konditionszahl κ_h(x) auf [1.1, 1.3]
+dx = 1e-7
+x_vals = np.arange(1.1 + dx, 1.3 + dx, dx)
+plt.semilogy(x_vals, kappa_h(x_vals))
+plt.xlabel("x")
+plt.ylabel(r"$\kappa_h(x)$")
+plt.title("Konditionszahl von $h(x)$ auf [1.1, 1.3]")
+plt.grid(True, which="both", ls="--", alpha=0.6)
+plt.tight_layout()
+plt.show()