112 lines
2.6 KiB
Python
112 lines
2.6 KiB
Python
import numpy as np
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from scipy.optimize import least_squares
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REFERENCE_K1 = 0.577
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REFERENCE_K2 = 0.422
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TIMES = np.linspace(0.0, 10.0, 101)
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SUBSTEPS = 10
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def analytical(t, k1, k2):
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x1 = np.exp(-k1 * t)
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if k1 == k2:
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x2 = k1 * t * np.exp(-k1 * t)
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else:
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x2 = k1 / (k2 - k1) * (
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np.exp(-k1 * t) - np.exp(-k2 * t)
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)
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x3 = 1.0 - x1 - x2
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return np.column_stack((x1, x2, x3))
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def solve_rk4(k1, k2):
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def rhs(x):
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x1, x2, x3 = x
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return np.array([
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-k1 * x1,
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k1 * x1 - k2 * x2,
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k2 * x2,
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])
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state = np.array([1.0, 0.0, 0.0])
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solution = np.empty((len(TIMES), 3))
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solution[0] = state
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for j in range(1, len(TIMES)):
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h = (TIMES[j] - TIMES[j - 1]) / SUBSTEPS
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for _ in range(SUBSTEPS):
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q1 = rhs(state)
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q2 = rhs(state + h * q1 / 2.0)
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q3 = rhs(state + h * q2 / 2.0)
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q4 = rhs(state + h * q3)
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state = state + h / 6.0 * (
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q1 + 2.0 * q2 + 2.0 * q3 + q4
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)
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solution[j] = state
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return solution
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def residuals(parameters, reference):
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k1, k2 = parameters
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calculated = solve_rk4(k1, k2)
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return (calculated - reference).ravel()
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def main():
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reference = analytical(TIMES, REFERENCE_K1, REFERENCE_K2)
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result = least_squares(
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fun=residuals,
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x0=[0.3, 0.7],
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args=(reference,),
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bounds=(
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[1e-5, 0.01],
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[100.0, 1.0],
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),
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method="trf",
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max_nfev=200,
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verbose=1,
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)
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k1_found, k2_found = result.x
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absolute_error_k1 = abs(k1_found - REFERENCE_K1)
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absolute_error_k2 = abs(k2_found - REFERENCE_K2)
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relative_error_k1 = absolute_error_k1 / REFERENCE_K1 * 100
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relative_error_k2 = absolute_error_k2 / REFERENCE_K2 * 100
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sum_squared_errors = np.sum(result.fun ** 2)
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rmse = np.sqrt(np.mean(result.fun ** 2))
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print(f"\nk1 = {k1_found:.4f}")
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print(f"k2 = {k2_found:.4f}")
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print(f"\nАбсолютное отклонение k1: {absolute_error_k1:.6e}")
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print(f"Абсолютное отклонение k2: {absolute_error_k2:.6e}")
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print(f"\nОтносительное отклонение k1: {relative_error_k1:.6e}%")
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print(f"Относительное отклонение k2: {relative_error_k2:.6e}%")
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print(f"\nСумма квадратов отклонений: {sum_squared_errors:.6e}")
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print(f"RMSE: {rmse:.6e}")
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if not result.success:
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print("\nПоиск завершён без подтверждённой сходимости.")
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if __name__ == "__main__":
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main()
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