serie 05

Kuengjoe_S05/Kuengjoe_S05_Aufg2.py

@@ -0,0 +1,120 @@
+import numpy as np
+import matplotlib.pyplot as plt
+
+
+def _compute_natural_cubic_spline_coefficients(interpolation_nodes_x, interpolation_values_y):
+    interpolation_nodes_x = np.asarray(interpolation_nodes_x, dtype=float).reshape(-1)
+    interpolation_values_y = np.asarray(interpolation_values_y, dtype=float).reshape(-1)
+
+    if interpolation_nodes_x.size != interpolation_values_y.size:
+        raise ValueError("x e y must be same length.")
+
+    if interpolation_nodes_x.size < 2:
+        raise ValueError("at least 2 nodes are required.")
+
+    if np.any(np.diff(interpolation_nodes_x) <= 0):
+        raise ValueError("x values must be strictly increasing.")
+
+    interval_widths = np.diff(interpolation_nodes_x)
+    number_of_intervals = interpolation_nodes_x.size - 1
+
+    system_matrix = np.zeros((number_of_intervals + 1, number_of_intervals + 1), dtype=float)
+    right_hand_side = np.zeros(number_of_intervals + 1, dtype=float)
+
+    system_matrix[0, 0] = 1.0
+    system_matrix[-1, -1] = 1.0
+
+    for internal_node_index in range(1, number_of_intervals):
+        left_interval_width = interval_widths[internal_node_index - 1]
+        right_interval_width = interval_widths[internal_node_index]
+
+        system_matrix[internal_node_index, internal_node_index - 1] = left_interval_width
+        system_matrix[internal_node_index, internal_node_index] = 2.0 * (left_interval_width + right_interval_width)
+        system_matrix[internal_node_index, internal_node_index + 1] = right_interval_width
+
+        right_hand_side[internal_node_index] = 6.0 * (
+            (interpolation_values_y[internal_node_index + 1] - interpolation_values_y[internal_node_index]) / right_interval_width
+            - (interpolation_values_y[internal_node_index] - interpolation_values_y[internal_node_index - 1]) / left_interval_width
+        )
+
+    second_derivative_values = np.linalg.solve(system_matrix, right_hand_side)
+
+    coefficient_a_values = interpolation_values_y[:-1].copy()
+    coefficient_b_values = np.zeros(number_of_intervals, dtype=float)
+    coefficient_c_values = second_derivative_values[:-1] / 2.0
+    coefficient_d_values = np.zeros(number_of_intervals, dtype=float)
+
+    for interval_index in range(number_of_intervals):
+        current_interval_width = interval_widths[interval_index]
+
+        coefficient_b_values[interval_index] = (
+            (interpolation_values_y[interval_index + 1] - interpolation_values_y[interval_index]) / current_interval_width
+            - (current_interval_width / 6.0) * (
+                2.0 * second_derivative_values[interval_index] + second_derivative_values[interval_index + 1]
+            )
+        )
+
+        coefficient_d_values[interval_index] = (
+            (second_derivative_values[interval_index + 1] - second_derivative_values[interval_index])
+            / (6.0 * current_interval_width)
+        )
+
+    return coefficient_a_values, coefficient_b_values, coefficient_c_values, coefficient_d_values
+
+
+def Kuengjoe_S05_Aufg2(x, y, xx, plot_result=True):
+    interpolation_nodes_x = np.asarray(x, dtype=float).reshape(-1)
+    interpolation_values_y = np.asarray(y, dtype=float).reshape(-1)
+    evaluation_points_xx = np.asarray(xx, dtype=float).reshape(-1)
+
+    if np.any(evaluation_points_xx < interpolation_nodes_x[0]) or np.any(evaluation_points_xx > interpolation_nodes_x[-1]):
+        raise ValueError("Tutti i valori di xx devono stare nell'intervallo [x0, xn].")
+
+    (
+        coefficient_a_values,
+        coefficient_b_values,
+        coefficient_c_values,
+        coefficient_d_values,
+    ) = _compute_natural_cubic_spline_coefficients(interpolation_nodes_x, interpolation_values_y)
+
+    number_of_intervals = interpolation_nodes_x.size - 1
+
+    interval_indices_for_evaluation = np.searchsorted(interpolation_nodes_x, evaluation_points_xx, side="right") - 1
+    interval_indices_for_evaluation = np.clip(interval_indices_for_evaluation, 0, number_of_intervals - 1)
+
+    yy = np.zeros_like(evaluation_points_xx, dtype=float)
+
+    for evaluation_point_index, interval_index in enumerate(interval_indices_for_evaluation):
+        local_x_distance = evaluation_points_xx[evaluation_point_index] - interpolation_nodes_x[interval_index]
+
+        yy[evaluation_point_index] = (
+            coefficient_a_values[interval_index]
+            + coefficient_b_values[interval_index] * local_x_distance
+            + coefficient_c_values[interval_index] * local_x_distance ** 2
+            + coefficient_d_values[interval_index] * local_x_distance ** 3
+        )
+
+    if plot_result:
+        plt.figure(figsize=(8, 5))
+        plt.plot(evaluation_points_xx, yy, label="Natürliche kubische Spline")
+        plt.plot(interpolation_nodes_x, interpolation_values_y, "o", label="Stützpunkte")
+        plt.xlabel("x")
+        plt.ylabel("S(x)")
+        plt.title("Natürliche kubische Spline")
+        plt.grid(True)
+        plt.legend()
+        plt.tight_layout()
+        plt.show()
+
+    return yy
+
+
+if __name__ == "__main__":
+    x_test_values = np.array([4, 6, 8, 10], dtype=float)
+    y_test_values = np.array([6, 3, 9, 0], dtype=float)
+    xx_test_values = np.linspace(4, 10, 400)
+
+    yy_test_values = Kuengjoe_S05_Aufg2(x_test_values, y_test_values, xx_test_values, plot_result=True)
+
+    print("value of yy_test_values:")
+    print(Kuengjoe_S05_Aufg2(x_test_values, y_test_values, x_test_values, plot_result=False))

Kuengjoe_S05/Kuengjoe_S05_Aufg3.py

@@ -0,0 +1,119 @@
+import numpy as np
+import matplotlib.pyplot as plt
+from scipy.interpolate import CubicSpline
+
+from Kuengjoe_S05_Aufg2 import Kuengjoe_S05_Aufg2
+
+
+def main():
+    time_values_years = np.array(
+        [1900, 1910, 1920, 1930, 1940, 1950, 1960, 1970, 1980, 1990, 2000, 2010],
+        dtype=float,
+    )
+
+    population_values_millions = np.array(
+        [75.995, 91.972, 105.711, 123.203, 131.669, 150.697, 179.323, 203.212, 226.505, 249.633, 281.422, 308.745],
+        dtype=float,
+    )
+
+    dense_evaluation_time_values = np.linspace(
+        time_values_years[0],
+        time_values_years[-1],
+        1000,
+    )
+
+    # a) Eigene natürliche kubische Spline aus Aufgabe 2
+    spline_values_custom_implementation = Kuengjoe_S05_Aufg2(
+        time_values_years,
+        population_values_millions,
+        dense_evaluation_time_values,
+        plot_result=False,
+    )
+
+    # b) SciPy CubicSpline mit natürlichen Randbedingungen
+    scipy_natural_cubic_spline = CubicSpline(
+        time_values_years,
+        population_values_millions,
+        bc_type="natural",
+    )
+    spline_values_scipy = scipy_natural_cubic_spline(dense_evaluation_time_values)
+
+    # c) Polynom 11. Grades mit verschobener Zeitachse
+    shifted_time_values_years = time_values_years - 1900.0
+    shifted_dense_evaluation_time_values = dense_evaluation_time_values - 1900.0
+
+    polynomial_degree = 11
+    polynomial_coefficients = np.polyfit(
+        shifted_time_values_years,
+        population_values_millions,
+        polynomial_degree,
+    )
+    polynomial_values_degree_eleven = np.polyval(
+        polynomial_coefficients,
+        shifted_dense_evaluation_time_values,
+    )
+
+    # Plot
+    plt.figure(figsize=(10, 6))
+    plt.plot(
+        dense_evaluation_time_values,
+        spline_values_custom_implementation,
+        label="Aufgabe 2: eigene natürliche kubische Spline",
+    )
+    plt.plot(
+        dense_evaluation_time_values,
+        spline_values_scipy,
+        "--",
+        label="SciPy CubicSpline (natural)",
+    )
+    plt.plot(
+        dense_evaluation_time_values,
+        polynomial_values_degree_eleven,
+        ":",
+        label="Polynom 11. Grades",
+    )
+    plt.plot(
+        time_values_years,
+        population_values_millions,
+        "o",
+        label="Messdaten",
+    )
+
+    plt.xlabel("Jahr")
+    plt.ylabel("Bevölkerung (in Mio.)")
+    plt.title("Vergleich der Interpolationen")
+    plt.grid(True)
+    plt.legend()
+    plt.tight_layout()
+    plt.show()
+
+    custom_values_at_nodes = Kuengjoe_S05_Aufg2(
+        time_values_years,
+        population_values_millions,
+        time_values_years,
+        plot_result=False,
+    )
+    scipy_values_at_nodes = scipy_natural_cubic_spline(time_values_years)
+    polynomial_values_at_nodes = np.polyval(
+        polynomial_coefficients,
+        shifted_time_values_years,
+    )
+
+    print("Original data:")
+    print(population_values_millions)
+    print()
+
+    print("Custom spline at nodes:")
+    print(custom_values_at_nodes)
+    print()
+
+    print("SciPy spline at nodes:")
+    print(scipy_values_at_nodes)
+    print()
+
+    print("Degree-11 polynomial at nodes:")
+    print(polynomial_values_at_nodes)
+
+
+if __name__ == "__main__":
+    main()