Serie 09

Kuengjoe_S09/Kuengjoe_S09_Aufg2.py

@@ -0,0 +1,49 @@
+import numpy as np
+
+
+def Kuengjoe_S09_Aufg2(A: np.ndarray, A_tilde: np.ndarray,
+                   b: np.ndarray, b_tilde: np.ndarray):
+    try:
+        x = np.linalg.solve(A, b)
+    except np.linalg.LinAlgError as e:
+        raise ValueError(f"Ungestörte Matrix A ist singulär oder schlecht konditioniert: {e}")
+
+    try:
+        x_tilde = np.linalg.solve(A_tilde, b_tilde)
+    except np.linalg.LinAlgError as e:
+        raise ValueError(f"Gestörte Matrix A_tilde ist singulär oder schlecht konditioniert: {e}")
+
+    norm_A_inf = np.linalg.norm(A, np.inf)
+    norm_AtAtilde = np.linalg.norm(A - A_tilde, np.inf)
+    norm_b_inf = np.linalg.norm(b, np.inf)
+    norm_bbtilde = np.linalg.norm(b - b_tilde, np.inf)
+
+    cond_A_inf = np.linalg.cond(A, np.inf)
+
+    ratio_A = norm_AtAtilde / norm_A_inf
+    ratio_b = norm_bbtilde / norm_b_inf
+
+    if cond_A_inf * ratio_A < 1.0:
+        dx_max = (cond_A_inf / (1 - cond_A_inf * ratio_A)) \
+                 * (ratio_A + ratio_b)
+    else:
+        dx_max = np.nan
+
+    # 4. Beobachteter Fehler
+    dx_obs = np.linalg.norm(x - x_tilde, np.inf) / np.linalg.norm(x, np.inf)
+
+    return x, x_tilde, dx_max, dx_obs
+
+
+if __name__ == "__main__":
+    A = np.array([[1, 0, 2],
+                  [0, 1, 0],
+                  [1e-4, 0, 1e-4]])
+    At = A.copy()
+    At[0, 2] += 1e-6
+    b = np.array([1, 1, 0])
+    bt = b.copy();
+    bt[2] += 1e-7
+
+    x, xt, dx_max, dx_obs = Kuengjoe_S09_Aufg2(A, At, b, bt)
+    print("dx_obs =", dx_obs, "   dx_max =", dx_max)