From 374d8cdedeee28335ac59a31f105d7879d44cffa Mon Sep 17 00:00:00 2001
From: "iliya.saroukha" <iliya.saroukhanian@etu.hesge.ch>
Date: Tue, 25 Jun 2024 15:27:50 +0200
Subject: [PATCH] feat: wrote taylor_errmax and updated chebyshev_points
function
---
ISC_421_Controle_4_Saroukhanian_Iliya.py | 48 +++++++++++++++++-------
1 file changed, 34 insertions(+), 14 deletions(-)
diff --git a/ISC_421_Controle_4_Saroukhanian_Iliya.py b/ISC_421_Controle_4_Saroukhanian_Iliya.py
index 3c862c4..8f9aa3c 100644
--- a/ISC_421_Controle_4_Saroukhanian_Iliya.py
+++ b/ISC_421_Controle_4_Saroukhanian_Iliya.py
@@ -155,17 +155,30 @@ print(f"valeur de la fonction en a et b: {SD.f(SD.a), SD.f(SD.b)}")
#### Exercices #####
###########################################################################
+
+def taylor_err_max(plot_range, eval_pt):
+ return (SD.Maximal_derivatives_values[len(SD.Maximal_derivatives_values) - 1] /
+ math.factorial(len(SD.Maximal_derivatives_values) + 1)) * \
+ (plot_range - eval_pt)**(len(SD.Maximal_derivatives_values) + 1)
+
+
def ex2_taylor_poly():
t = np.linspace(SD.a, SD.b, Nmbre_pts)
y0 = compute_taylor_series(
t, SD.Taylor_points[0], SD.Taylor_derivatives_values[SD.Taylor_points[0]])
+ err_y0 = taylor_err_max(t, SD.Taylor_points[0])
+
y1 = compute_taylor_series(
t, SD.Taylor_points[1], SD.Taylor_derivatives_values[SD.Taylor_points[1]])
+ err_y1 = taylor_err_max(t, SD.Taylor_points[1])
+
y2 = compute_taylor_series(
t, SD.Taylor_points[2], SD.Taylor_derivatives_values[SD.Taylor_points[2]])
+ err_y2 = taylor_err_max(t, SD.Taylor_points[2])
+
fig, axs = plt.subplots(2, 2, figsize=(18, 12))
axs[0, 0].plot(t, SD.f(t), color='black')
@@ -174,6 +187,8 @@ def ex2_taylor_poly():
axs[0, 1].plot(t, SD.f(t), color='black', label='$f$')
axs[0, 1].plot(t, y0, color='orange', label='$T_{f}$')
+ axs[0, 1].plot(t, err_y0, '--', color='orange',
+ label='Erreur maximal de $T_{f}$')
axs[0, 1].plot(SD.Taylor_points[0], SD.f(
SD.Taylor_points[0]), "-o", color='red', label=f'a = {SD.Taylor_points[0]}')
axs[0, 1].set_title(
@@ -184,6 +199,8 @@ def ex2_taylor_poly():
axs[1, 0].plot(t, SD.f(t), color='black', label='$f$')
axs[1, 0].plot(t, y1, color='blue', label='$T_{f}$')
+ axs[1, 0].plot(t, err_y1, '--', color='blue',
+ label='ErrMax de $T_{f}$')
axs[1, 0].plot(SD.Taylor_points[1], SD.f(
SD.Taylor_points[1]), "-o", color='red', label=f'a = {SD.Taylor_points[1]}')
axs[1, 0].set_title(
@@ -194,6 +211,8 @@ def ex2_taylor_poly():
axs[1, 1].plot(t, SD.f(t), color='black', label='$f$')
axs[1, 1].plot(t, y2, color='violet', label='$T_{f}$')
+ axs[1, 1].plot(t, err_y2, '--', color='violet',
+ label='ErrMax de $T_{f}$')
axs[1, 1].plot(SD.Taylor_points[2], SD.f(
SD.Taylor_points[2]), "-o", color='red', label=f'a = {SD.Taylor_points[2]}')
axs[1, 1].set_title(
@@ -223,6 +242,11 @@ def polerr(nb_points, interpolation_pts, plot_range):
return errs_range
+def chebyshev_pts(start, stop, nb_points):
+ return (((start + stop) / 2) + ((stop - start) / 2) *
+ np.cos(((2 * np.arange(nb_points) + 1) * np.pi) / (2 * (nb_points))))
+
+
def ex3_lagrange_interpolation_poly():
nb_points = np.linspace(1, 12, 6, dtype=np.uint8)
fig, axes = plt.subplots(2, 3, figsize=(20, 12))
@@ -230,20 +254,16 @@ def ex3_lagrange_interpolation_poly():
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
+ cheb_pts = chebyshev_pts(SD.a, SD.b, nb_points[i])
interpolate_pts = np.linspace(SD.a, SD.b, nb_points[i])
l_poly_uniform = lagrange(interpolate_pts, SD.f(interpolate_pts))
l_poly_chebyshev_pts = lagrange(
- chebyshev_points_mapped, SD.f(chebyshev_points_mapped))
+ cheb_pts, SD.f(cheb_pts))
uniform_err = polerr(nb_points[i], interpolate_pts, t)
- chebyshev_err = polerr(nb_points[i], chebyshev_points_mapped, t)
+ chebyshev_err = polerr(nb_points[i], cheb_pts, t)
ax.plot(t, SD.f(t), color='black', label='f')
ax.plot(t, l_poly_uniform(t), color='red',
@@ -251,17 +271,17 @@ def ex3_lagrange_interpolation_poly():
# ax.plot(t, np.abs(SD.f(t) - l_poly_uniform(t)), '--', color='red',
# label='$L_{f}$, intervalle équidistants, erreur')
ax.plot(t, uniform_err, '--', color='red',
- label='$L_{f}$, intervalle équidistants, erreur')
+ label='ErrMax de $L_{f}$, intervalle équidistants')
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(t, chebyshev_err, '--', color='blue',
- label='$L_{f}$, points de Chebyshev, erreur')
+ label='ErrMax de $L_{f}$, 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',
+ ax.plot(cheb_pts[::-1],
+ SD.f(cheb_pts[::-1]), 'o', color='blue',
label='Points de Chebyshev')
ax.set_title(f'n = {nb_points[i]}')
# ax.set_ylim([-1.2, 1.2])
@@ -274,12 +294,12 @@ def ex3_lagrange_interpolation_poly():
plt.show()
-def caca():
- print(len(SD.Maximal_derivatives_values))
+# def caca():
+# print(len(SD.Maximal_derivatives_values))
+# ex2_taylor_poly()
ex3_lagrange_interpolation_poly()
-# caca()
# def ex3_newton_interpolation_poly():
--
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