feat: Skript mit Newton Verfahren, einfach und modifiziert. build: sympy zu requirements hinzugefügt

requirements.txt

@@ -2,3 +2,4 @@ matplotlib == 3.10.8
 numpy == 2.4.3
 black == 26.3.1
 ipykernel == 7.2.0
+sympy == 1.14.0

src/optimierung/newton_verfahren.py

@@ -0,0 +1,102 @@
+#%%
+import numpy as np
+import sympy as sp
+import matplotlib.pyplot as plt
+
+#%%
+# "Einfacher" Newton Verfahren
+# Parameter 
+N = 200
+x = sp.Symbol('x')
+
+# Funktion
+f_sym = sp.sin(3*x)+0.02*x**2
+
+# Ableitungen
+f = sp.lambdify(x, f_sym, "numpy")
+f_prime = sp.lambdify(x, sp.diff(f_sym, x), "numpy")
+f_second = sp.lambdify(x, sp.diff(f_sym, x, 2), "numpy")
+
+# Daten
+x_data = np.linspace(0,10,N)
+f_data = f(x_data)
+
+# Plot
+plt.figure(1)
+plt.plot(x_data, f_data)
+plt.xlabel("x")
+plt.ylabel("y")
+plt.grid("on")
+plt.axis("image")
+
+# Newton Verfahren
+# Startwert
+startwerte = [6.6, 3.4, 8.38]
+
+# Iterationsformel
+x_n = lambda x: x - f_prime(x)/f_second(x)
+
+# Iteration
+for x_0 in startwerte:
+    n = 0
+    x_i = x_0
+    f_x_prime = f_prime(x_0)
+    print(f"n: {n}\nf(x): {x_0}\nf'(x): {f_x_prime}\n")
+
+    limit = 4
+
+    while np.abs(f_x_prime) > 1e-16 and limit > 0:
+        n += 1
+        x_i = x_n(x_i)
+        f_x_prime = f_prime(x_i)
+        print(f"n: {n}\nf(x): {x_i}\nf'(x): {f_x_prime}\n")
+        limit -= 1
+
+#%%
+# "Modifizierter" Newton Verfahren
+# Parameter 
+N = 200
+x = sp.Symbol('x')
+
+# Funktion
+f_sym = sp.sin(3*x)+0.02*x**2
+
+# Ableitungen
+f = sp.lambdify(x, f_sym, "numpy")
+f_prime = sp.lambdify(x, sp.diff(f_sym, x), "numpy")
+f_second = sp.lambdify(x, sp.diff(f_sym, x, 2), "numpy")
+
+# Daten
+x_data = np.linspace(0,10,N)
+f_data = f(x_data)
+
+# Plot
+plt.figure(1)
+plt.plot(x_data, f_data)
+plt.xlabel("x")
+plt.ylabel("y")
+plt.grid("on")
+plt.axis("image")
+
+# Newton Verfahren
+# Startwert
+startwerte = [6.6, 3.4, 8.38]
+
+# Iterationsformel
+x_n = lambda x: x - f_prime(x)/np.abs(f_second(x))
+
+# Iteration
+for x_0 in startwerte:
+    n = 0
+    x_i = x_0
+    f_x_prime = f_prime(x_0)
+    print(f"n: {n}\nf(x): {x_0}\nf'(x): {f_x_prime}\n")
+
+    limit = 4
+
+    while np.abs(f_x_prime) > 1e-16 and limit > 0:
+        n += 1
+        x_i = x_n(x_i)
+        f_x_prime = f_prime(x_i)
+        print(f"n: {n}\nf(x): {x_i}\nf'(x): {f_x_prime}\n")
+        limit -= 1