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Commit 10532367 authored by iliya.saroukha's avatar iliya.saroukha :first_quarter_moon:
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feat: plotting errors on interpolation polynomial

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ISC_421_Controle_4_Saroukhanian_Iliya.py
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View file @ 10532367
......@@ -139,6 +139,8 @@ print()
print(f"valeur de la fonction en x = {SD.Taylor_points} : {
[SD.f(x) for x in SD.Taylor_points]}")
print(f"valeur de la fonction en a et b: {SD.f(SD.a), SD.f(SD.b)}")
# print("===============================")
# print(SD.Maximal_derivatives_values)
# Exemple de graphe de la fonction f.
# t = np.linspace(SD.a, SD.b, Nmbre_pts)
......@@ -205,9 +207,11 @@ def ex2_taylor_poly():
def ex3_lagrange_interpolation_poly():
nb_points = np.linspace(3, 19, 6, dtype=np.uint64)
nb_points = np.linspace(1, 13, 6, dtype=np.uint64)
fig, axes = plt.subplots(2, 3, figsize=(20, 12))
t = np.linspace(SD.a, SD.b, Nmbre_pts)
for i, ax in enumerate(axes.flat):
chebyshev_points = np.cos(
(2 * np.arange(nb_points[i]) + 1) / (2 * nb_points[i]) * np.pi)
......@@ -221,13 +225,15 @@ def ex3_lagrange_interpolation_poly():
l_poly_chebyshev_pts = lagrange(
chebyshev_points_mapped, SD.f(chebyshev_points_mapped))
t = np.linspace(SD.a, SD.b, Nmbre_pts)
ax.plot(t, SD.f(t), color='black', label='f')
ax.plot(t, l_poly_uniform(t), color='red',
label='$L_{f}$, intervalle équidistants')
ax.plot(t, np.abs(SD.f(t) - l_poly_uniform(t)), '--', color='red',
label='$L_{f}$, intervalle équidistants, erreur')
ax.plot(t, l_poly_chebyshev_pts(t), color='blue',
label='$L_{f}$, points de Chebyshev')
ax.plot(t, np.abs(SD.f(t) - l_poly_chebyshev_pts(t)), '--', color='blue',
label='$L_{f}$, points de Chebyshev, erreur')
ax.plot(interpolate_pts, SD.f(interpolate_pts), 'o', color='red',
label='Points équidistants')
ax.plot(chebyshev_points_mapped[::-1],
......@@ -244,85 +250,71 @@ def ex3_lagrange_interpolation_poly():
plt.show()
def ex3_newton_interpolation_poly():
# merce l'ami
def divided_differences(x, y):
n = len(y)
coef = np.zeros([n, n])
coef[:, 0] = y
for j in range(1, n):
for i in range(n - j):
coef[i, j] = (coef[i + 1, j - 1] - coef[i, j - 1]) / \
(x[i + j] - x[i])
return coef[0, :]
def newton_polynomial(x, x_points, coef):
n = len(coef)
p = coef[n - 1]
for k in range(1, n):
p = coef[n - k - 1] + (x - x_points[n - k - 1]) * p
return p
nb_points = np.linspace(3, 19, 6, dtype=np.uint64)
fig, axes = plt.subplots(2, 3, figsize=(20, 12))
t = np.linspace(SD.a, SD.b, Nmbre_pts)
for i, ax in enumerate(axes.flat):
chebyshev_points = np.cos(
(2 * np.arange(nb_points[i]) + 1) / (2 * nb_points[i]) * np.pi)
chebyshev_points_mapped = 0.5 * \
(SD.b - SD.a) * (chebyshev_points + 1) + SD.a
interpolate_pts = np.linspace(SD.a, SD.b, nb_points[i])
y_points_uni = SD.f(interpolate_pts)
coef_uni = divided_differences(interpolate_pts, y_points_uni)
y_plot_uni = newton_polynomial(t, interpolate_pts, coef_uni)
y_points_cheb = SD.f(chebyshev_points_mapped)
coef_cheb = divided_differences(chebyshev_points_mapped, y_points_cheb)
y_plot_cheb = newton_polynomial(t, chebyshev_points_mapped, coef_cheb)
ax.plot(t, SD.f(t), color='black', label='f')
ax.plot(t, y_plot_uni, color='red',
label='$N_{f}$, intervalle équidistants')
ax.plot(t, y_plot_cheb, color='blue',
label='$N_{f}$, points de Chebyshev')
ax.plot(interpolate_pts, SD.f(interpolate_pts), 'o', color='red',
label='Points équidistants')
ax.plot(chebyshev_points_mapped[::-1],
SD.f(chebyshev_points_mapped[::-1]), 'o', color='blue',
label='Points de Chebyshev')
ax.set_title(f'n = {nb_points[i]}')
ax.set_ylim([-1.2, 1.2])
ax.legend()
fig.suptitle(f'Polynôme d\'interpolation de Newton de $f$ avec 2 subdivisions différentes d\'intervalle: Équidistantes (rouge) / Points de Chebyshev (bleu)')
fig.tight_layout()
plt.show()
ex3_newton_interpolation_poly()
# Graphique des polynômes de Taylor
# fig, axes = plt.subplots(1, 3)
# for i, pt in enumerate(SD.Taylor_points):
# ax = axes[i]
ex3_lagrange_interpolation_poly()
# def ex3_newton_interpolation_poly():
# # merce l'ami
# def divided_differences(x, y):
# n = len(y)
# coef = np.zeros([n, n])
# coef[:, 0] = y
#
# for j in range(1, n):
# for i in range(n - j):
# coef[i, j] = (coef[i + 1, j - 1] - coef[i, j - 1]) / \
# (x[i + j] - x[i])
#
# return coef[0, :]
#
# def newton_polynomial(x, x_points, coef):
# n = len(coef)
# p = coef[n - 1]
# for k in range(1, n):
# p = coef[n - k - 1] + (x - x_points[n - k - 1]) * p
#
# return p
#
# nb_points = np.linspace(3, 19, 6, dtype=np.uint64)
# fig, axes = plt.subplots(2, 3, figsize=(20, 12))
#
# t = np.linspace(SD.a, SD.b, Nmbre_pts)
# ax.plot(t, SD.f(t), label="Fonction f(x)")
# taylor_poly = np.poly1d(
# SD.Taylor_derivatives_values[SD.Taylor_points[i]][::-1])
# ax.plot(t, taylor_poly(t), label=f"Polynôme de Taylor en x={pt}")
# ax.legend()
# ax.set_title(f"Polynôme de Taylor en x={pt}")
# plt.show()
#
# for i, ax in enumerate(axes.flat):
# chebyshev_points = np.cos(
# (2 * np.arange(nb_points[i]) + 1) / (2 * nb_points[i]) * np.pi)
#
# chebyshev_points_mapped = 0.5 * \
# (SD.b - SD.a) * (chebyshev_points + 1) + SD.a
#
# interpolate_pts = np.linspace(SD.a, SD.b, nb_points[i])
#
# y_points_uni = SD.f(interpolate_pts)
#
# coef_uni = divided_differences(interpolate_pts, y_points_uni)
# y_plot_uni = newton_polynomial(t, interpolate_pts, coef_uni)
#
# y_points_cheb = SD.f(chebyshev_points_mapped)
#
# coef_cheb = divided_differences(chebyshev_points_mapped, y_points_cheb)
# y_plot_cheb = newton_polynomial(t, chebyshev_points_mapped, coef_cheb)
#
# ax.plot(t, SD.f(t), color='black', label='f')
# ax.plot(t, y_plot_uni, color='red',
# label='$N_{f}$, intervalle équidistants')
# ax.plot(t, y_plot_cheb, color='blue',
# label='$N_{f}$, points de Chebyshev')
# ax.plot(interpolate_pts, SD.f(interpolate_pts), 'o', color='red',
# label='Points équidistants')
# ax.plot(chebyshev_points_mapped[::-1],
# SD.f(chebyshev_points_mapped[::-1]), 'o', color='blue',
# label='Points de Chebyshev')
# ax.set_title(f'n = {nb_points[i]}')
# ax.set_ylim([-1.2, 1.2])
#
# ax.legend()
#
# fig.suptitle(f'Polynôme d\'interpolation de Newton de $f$ avec 2 subdivisions différentes d\'intervalle: Équidistantes (rouge) / Points de Chebyshev (bleu)')
#
# fig.tight_layout()
# plt.show()
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