diff --git a/Schenk_Brandenberger_S9_Aufg1.py b/Schenk_Brandenberger_S9_Aufg1.py new file mode 100644 index 0000000..a6b6b95 --- /dev/null +++ b/Schenk_Brandenberger_S9_Aufg1.py @@ -0,0 +1,22 @@ +import numpy as np + + +if __name__ == '__main__': + + A = np.array([[1,0,2], + [0,1,0], + [1e-4,0,1e-4]]) + b_exact = np.array([[1],[1],[0]]) + b_approx = np.array([[1],[1],[1.67e-7]]) + print("Inverse von A:\n", np.linalg.inv(A)) + x_exact = np.linalg.solve(A,b_exact) + print(x_exact) + x_approx = np.linalg.solve(A,b_approx) + print(x_approx) + print(x_exact-x_approx) + + print("Task 1 d:") + + print(60003*(3e-7/3)) + print((60003/(1-0.0060003))*((3e-7/3)+(1.67e-7/1))) + print(((0.01*(1-0.0060003))/60003)-(3e-7/3)) \ No newline at end of file diff --git a/Schenk_Brandenberger_S9_Aufg2.py b/Schenk_Brandenberger_S9_Aufg2.py new file mode 100644 index 0000000..00b06a2 --- /dev/null +++ b/Schenk_Brandenberger_S9_Aufg2.py @@ -0,0 +1,40 @@ +import numpy as np +from math import nan + +def Schenk_Brandenberger_S9_Aufg2(A, A_approx, b, b_approx): + cond_A = np.linalg.cond(A,np.inf) + norm_A_minus_A_approx = np.linalg.norm((A-A_approx),np.inf) + norm_A = np.linalg.norm(A,np.inf) + norm_b_minus_b_approx = np.linalg.norm((b-b_approx),np.inf) + norm_b = np.linalg.norm(b,np.inf) + x = np.linalg.solve(A,b) + x_approx = np.linalg.solve(A_approx,b_approx) + dxmax = (cond_A/(1-(cond_A*(norm_A_minus_A_approx/norm_A))))*((norm_A_minus_A_approx/norm_A)+(norm_b_minus_b_approx/norm_b)) + dxobs = np.linalg.norm((x-x_approx),np.inf)/np.linalg.norm(x,np.inf) + if(cond_A*(norm_A_minus_A_approx/norm_A)) >= 1: + dxmax = nan + return [x, x_approx, dxmax, dxobs] + + + +if __name__ == '__main__': + + # Werte aus Aufgabe 1 + A = np.array([[20, 30, 10], + [10, 17, 6], + [2, 3, 2]]) + A_approx = A - 0.1 + + b = np.array([[5720], + [3300], + [836]]) + b_approx = b + 100 + print(b_approx) + + [x, x_approx, dxmax, dxobs] = Schenk_Brandenberger_S9_Aufg2(A, A_approx, b, b_approx) + print("Value x: ") + print(x) + print("X approx with error: ") + print(x_approx) + print("dx max: ", dxmax) + print("dx obs: ", dxobs) \ No newline at end of file diff --git a/Schenk_Brandenberger_S9_Aufg3.py b/Schenk_Brandenberger_S9_Aufg3.py new file mode 100644 index 0000000..2728b44 --- /dev/null +++ b/Schenk_Brandenberger_S9_Aufg3.py @@ -0,0 +1,33 @@ +import numpy as np +from Schenk_Brandenberger_S9_Aufg2 import * +import matplotlib.pyplot as plt + +all_dxmax = np.array([]) +all_dxobs = np.array([]) +dxmax_dxobs_ratio = np.array([]) +index = np.arange(0,1000,1) + +for i in range(1000): + A = np.random.rand(100,100) + b = np.random.rand(100,1) + A_approx = A + np.random.rand(100,100)/1e5 + b_approx = b + np.random.rand(100,1)/1e5 + [x, x_approx, dxmax, dxobs] = Schenk_Brandenberger_S9_Aufg2(A, A_approx, b, b_approx) + all_dxmax = np.append(all_dxmax, [dxmax], axis=0) + all_dxobs = np.append(all_dxobs, [dxobs], axis=0) + dxmax_dxobs_ratio = np.append(dxmax_dxobs_ratio, [dxmax/dxobs], axis=0) + + +plt.figure(1) +plt.semilogy(all_dxmax) +plt.semilogy(all_dxobs) +plt.semilogy(dxmax_dxobs_ratio) +plt.legend(["dxmax", "dxob", "dxmax/dxobs"]) +plt.grid() +plt.xlabel("x") +plt.ylabel("f(x)") +plt.title("Aufgabe 3 Serie 9") +plt.show() + +#Die obere Schranke dxmax liegt immer dxobs, deshalb ist sie sicher realistisch und korrekt +#Jedoch liegt die obere Schranke immer etwa den Faktor 1e3 über dxobs \ No newline at end of file