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z@r{gUH+M9>)NgnV$Y&kz^x_JHE@%Akjf)N6^WmGE=I*g5=knx^IzF7+anx*>>i;fN zd)73f&3z;qgp#6L6L>|Nisl+WTNH{GF>#HU2hoDbt^0C8G}RCHwlsu}D4aX}-!=w{ zi-?}B8e2EGh7pkUAC0b>L((R-=||ZTfj-El=BuSYG?*_$1UmQYg_&kag- zD_qm!C#qC1?vlpw0iyI)Qzzu9cQQ#P$wEatEA*S{KxMX4J{H9{R@l30S^0`}^}dj9 zEIu=dRvo~5{es4;RiTD6zw5O7Fv4tk=E3~7KR_x>&SByjy5s!f zI?HD6!lBXN&+ZZ3^=WOurt|g1YOj)1z$MP<5zF#|_ z3f6J0zSankIl0)2n@7$ghOZU!DRv()h^!a z^ba(gPdghoL|_{?f`vSXJkucoeyd-jb)(^@Dc9$c^Ow`|uy%Q>>w_JX@zsJx!0}jb z7Vcsq;1u@^ytgr7a=n?Jq9Jz%!0=&0|Ci+4KM?l+l5>XQe|h%e68~}PC4~Q@a^B#< zZ(QE6O9=m0FAn;TTQ4s8j{x=?cJaS_dvU41e*dah-}v9)L;vk}gSz<7{%sfkufN2N z4E4YDf46gkO#iQZll%L(Uh1#Ck-Pq`AmIOP7nS_a0{_W3-uVAL zA4#ddWzb)Hg9-kx{jZh$zy1D}Zz!n$wuAoG-+<@;Eld3^i~Z*Z`j=4rh8^@T{^Q1q z{SnoRy!m8-lGYM67{UmKHz@F;Ni_ zVKYlnF<}V{a|;m>QAsH)pr{->*xXVKXeKUcZDuY36cM(xwziPCX%iO#nOTVerOZU- zZY0Z+4@g>=~sMcBpI#UTG98Es)nanTTi zM@-Mq*5HIMl1fjX%JW2Vv+hewj~6}H?T*19UL*-k+7Vfgt 1e-8: + candidate_vector = current_vector + damping_factor * newton_step + candidate_function_values = function_values( + candidate_vector[0], + candidate_vector[1], + candidate_vector[2] + ) + + if np.linalg.norm(candidate_function_values, 2) < current_function_norm: + current_vector = candidate_vector + break + + damping_factor = damping_factor / 2 + + final_function_values = function_values( + current_vector[0], + current_vector[1], + current_vector[2] + ) + final_function_norm = np.linalg.norm(final_function_values, 2) + + return current_vector, maximum_iterations, final_function_norm + + +start_vector = [1.5, 3.0, 2.5] + +solution_vector, number_of_iterations, final_norm = damped_newton_method(start_vector) + +print("Start vector =", start_vector) +print("Approximate solution =", solution_vector) +print("Iterations =", number_of_iterations) +print("||f(x^(k))||_2 =", final_norm) \ No newline at end of file diff --git a/Kuengjoe_S03/Kuengjoe_S03_aufg2.py b/Kuengjoe_S03/Kuengjoe_S03_aufg2.py new file mode 100644 index 0000000..c3f60d9 --- /dev/null +++ b/Kuengjoe_S03/Kuengjoe_S03_aufg2.py @@ -0,0 +1,93 @@ +import numpy as np +import sympy as sp + + +def function_vector(x_value, y_value): + first_function_value = (x_value ** 2) / (186 ** 2) - (y_value ** 2) / (300 ** 2 - 186 ** 2) - 1 + second_function_value = ((y_value - 500) ** 2) / (279 ** 2) - ((x_value - 300) ** 2) / (500 ** 2 - 279 ** 2) - 1 + return np.array([first_function_value, second_function_value], dtype=float) + + +def jacobian_matrix(x_value, y_value): + jacobian_11 = (2 * x_value) / (186 ** 2) + jacobian_12 = (-2 * y_value) / (300 ** 2 - 186 ** 2) + + jacobian_21 = (-2 * (x_value - 300)) / (500 ** 2 - 279 ** 2) + jacobian_22 = (2 * (y_value - 500)) / (279 ** 2) + + return np.array([ + [jacobian_11, jacobian_12], + [jacobian_21, jacobian_22] + ], dtype=float) + + +def newton_method_for_system(start_vector, tolerance=1e-5, maximum_number_of_iterations=100): + current_vector = np.array(start_vector, dtype=float) + + for iteration_index in range(maximum_number_of_iterations): + current_function_value = function_vector(current_vector[0], current_vector[1]) + current_function_norm = np.linalg.norm(current_function_value, 2) + + if current_function_norm < tolerance: + return current_vector, iteration_index, current_function_norm + + current_jacobian_matrix = jacobian_matrix(current_vector[0], current_vector[1]) + newton_step = np.linalg.solve(current_jacobian_matrix, -current_function_value) + current_vector = current_vector + newton_step + + final_function_value = function_vector(current_vector[0], current_vector[1]) + final_function_norm = np.linalg.norm(final_function_value, 2) + return current_vector, maximum_number_of_iterations, final_function_norm + + +def plot_implicit_functions(): + x_symbol, y_symbol = sp.symbols('x y') + + first_function_expression = x_symbol ** 2 / 186 ** 2 - y_symbol ** 2 / (300 ** 2 - 186 ** 2) - 1 + second_function_expression = (y_symbol - 500) ** 2 / 279 ** 2 - (x_symbol - 300) ** 2 / (500 ** 2 - 279 ** 2) - 1 + + first_plot = sp.plot_implicit( + sp.Eq(first_function_expression, 0), + (x_symbol, -2000, 2000), + (y_symbol, -2000, 2000), + show=False + ) + + second_plot = sp.plot_implicit( + sp.Eq(second_function_expression, 0), + (x_symbol, -2000, 2000), + (y_symbol, -2000, 2000), + show=False + ) + + first_plot.append(second_plot[0]) + first_plot.show() + + +def main(): + print("Plot of the two hyperbolas is opened...") + plot_implicit_functions() + + # Hier die 4 Startvektoren aus dem Plot eintragen + start_vectors = [ + np.array([-1300, 1600]), + np.array([750, 900]), + np.array([-200, 70]), + np.array([250, 220]) + ] + + print("\nNewton-Verfahren für die 4 Startvektoren:\n") + + for solution_index, current_start_vector in enumerate(start_vectors, start=1): + solution_vector, number_of_iterations, final_norm = newton_method_for_system(current_start_vector) + + print(f"Lösung {solution_index}:") + print(f"startvektor = {current_start_vector}") + print(f" Näaherungslösung = ({solution_vector[0]:.10f}, {solution_vector[1]:.10f})") + print(f" Iterationen = {number_of_iterations}") + print(f" ||f(x^(k))||_2 = {final_norm:.10e}") + print() + + +if __name__ == "__main__": + main() \ No newline at end of file