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serie 05

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Joe Küng
2026-03-25 14:01:08 +01:00
parent 1e8d68210c
commit 96befc4cb3
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Kuengjoe_S05/Kuengjoe_S05_Aufg1.pdf Normal file
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Kuengjoe_S05/Kuengjoe_S05_Aufg2.py Normal file
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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))

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Kuengjoe_S05/Kuengjoe_S05_Aufg3.py Normal file
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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()