92 lines
2.8 KiB
Python
92 lines
2.8 KiB
Python
import numpy as np
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def function_values(x1_value, x2_value, x3_value):
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first_function_value = x1_value + x2_value**2 - x3_value**2 - 13
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second_function_value = np.log(x2_value**4 / 4) + np.exp(0.5 * x3_value - 1) - 1
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third_function_value = (x2_value - 3)**2 - x3_value**3 + 7
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return np.array([
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first_function_value,
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second_function_value,
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third_function_value
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], dtype=float)
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def jacobian_matrix(x1_value, x2_value, x3_value):
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derivative_f1_x1 = 1
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derivative_f1_x2 = 2 * x2_value
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derivative_f1_x3 = -2 * x3_value
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derivative_f2_x1 = 0
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derivative_f2_x2 = 4 / x2_value
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derivative_f2_x3 = 0.5 * np.exp(0.5 * x3_value - 1)
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derivative_f3_x1 = 0
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derivative_f3_x2 = 2 * (x2_value - 3)
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derivative_f3_x3 = -3 * x3_value**2
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return np.array([
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[derivative_f1_x1, derivative_f1_x2, derivative_f1_x3],
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[derivative_f2_x1, derivative_f2_x2, derivative_f2_x3],
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[derivative_f3_x1, derivative_f3_x2, derivative_f3_x3]
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], dtype=float)
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def damped_newton_method(start_vector, tolerance=1e-5, maximum_iterations=100):
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current_vector = np.array(start_vector, dtype=float)
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for iteration_index in range(maximum_iterations):
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current_function_values = function_values(
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current_vector[0],
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current_vector[1],
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current_vector[2]
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)
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current_function_norm = np.linalg.norm(current_function_values, 2)
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if current_function_norm < tolerance:
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return current_vector, iteration_index, current_function_norm
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current_jacobian_matrix = jacobian_matrix(
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current_vector[0],
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current_vector[1],
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current_vector[2]
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)
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newton_step = np.linalg.solve(current_jacobian_matrix, -current_function_values)
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damping_factor = 1.0
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while damping_factor > 1e-8:
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candidate_vector = current_vector + damping_factor * newton_step
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candidate_function_values = function_values(
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candidate_vector[0],
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candidate_vector[1],
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candidate_vector[2]
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)
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if np.linalg.norm(candidate_function_values, 2) < current_function_norm:
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current_vector = candidate_vector
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break
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damping_factor = damping_factor / 2
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final_function_values = function_values(
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current_vector[0],
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current_vector[1],
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current_vector[2]
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)
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final_function_norm = np.linalg.norm(final_function_values, 2)
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return current_vector, maximum_iterations, final_function_norm
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start_vector = [1.5, 3.0, 2.5]
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solution_vector, number_of_iterations, final_norm = damped_newton_method(start_vector)
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print("Start vector =", start_vector)
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print("Approximate solution =", solution_vector)
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print("Iterations =", number_of_iterations)
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print("||f(x^(k))||_2 =", final_norm) |