feat: working on ex2

.gitignore

+.venv/
+*.pdf

ISC_421_Controle_4_Saroukhanian_Iliya.py

+import numpy as np
+import math
+from matplotlib import pyplot as plt
+from scipy.interpolate import lagrange
+
+
+#############################################################################
+######################### Variables globales ################################
+#############################################################################
+
+Nmbre_derivees = 14
+Nmbre_pts = 201
+
+
+def Base_function(x): return np.sin(np.pi*np.cos(4*x*x))
+
+
+Taylor_points = np.array([-0.8, 0.25, 0.8])
+
+Borne_a = 1
+Borne_b = 4
+
+
+def sigma(a, b, x): return (b-a)/2*x + (b+a)/2
+
+
+def amgis(a, b, x): return -1 + 2*(x-a)/(b-a)
+
+
+#############################################################################
+############ Fonctions necessaires a la production des donnees ##############
+#############################################################################
+
+def deltas2(lstx, lsty, depth=1):
+    N = len(lstx)
+    deltas_dic = {tuple([l]): y for l, y in enumerate(lsty)}
+    for k in range(1, min(depth, N)):
+        for l in range(N - k):
+            indices_now = tuple(range(l, k+l+1))
+            deltas_dic[indices_now] = (
+                deltas_dic[indices_now[1:]] - deltas_dic[indices_now[:-1]])/(lstx[k+l]-lstx[l])
+    return deltas_dic
+
+
+def approx_der_lst(lstx, lsty, a, depth):
+    D = deltas2(lstx, lsty, depth=depth)
+    indices = [l for l, x in enumerate(lstx) if x >= a]
+    index = indices[0]
+    lst_return = []
+    l = 0
+    while (index - l >= 0) and (index + l < len(lstx)) and (l < depth):
+        der_temp = []
+        indices_now = range(index - l, index + 1)
+        for m in indices_now:
+            approx_der_temp = D[tuple(range(m, m+l+1))]*np.math.factorial(l)
+            der_temp.append(approx_der_temp)
+        approx_der = sum(der_temp)/(len(der_temp))
+        lst_return.append(approx_der)
+        l += 1
+
+    return lst_return
+
+
+#############################################################################
+##### Classe regroupant toutes les donnees pertinentes d'une fonction #######
+#############################################################################
+
+class Function_data:
+
+    def __init__(self, f=lambda x: x):
+        self.function = f
+
+        def approx_der_temp(pt, width=.1, number_points=Nmbre_derivees):
+            t = np.linspace(pt - width, pt + width, 2*number_points + 1)
+            return approx_der_lst(t, f(t), pt, number_points)
+
+        def max_der(a, b, width=.1, number_points=12, total_number=Nmbre_derivees):
+            ts = np.linspace(a, b, total_number)
+            max_ret = [0]*(number_points-1)
+            for p in ts:
+                der = approx_der_temp(p)
+                max_ret = [max(oldmax, abs(der[l]))
+                           for l, oldmax in enumerate(max_ret)]
+            return max_ret
+        self.approx_derivatives = approx_der_temp
+        self.values = lambda t: f(t)
+        self.max_der = max_der
+
+
+#############################################################################
+######## Classe regroupant toutes les donnees fournies a l'etudiant #########
+#############################################################################
+
+class Student_data:
+
+    def __init__(self):
+
+        self.a, self.b = Borne_a, Borne_b
+        def f_ab(x): return Base_function(amgis(Borne_a, Borne_b, x))
+        F = Function_data(f_ab)
+        self.Maximal_derivatives_values = F.max_der(
+            Borne_a, Borne_b, number_points=Nmbre_derivees, total_number=Nmbre_pts)
+        self.Taylor_points = [sigma(Borne_a, Borne_b, p)
+                              for p in Taylor_points]
+        self.Taylor_derivatives_values = {pt: F.approx_derivatives(
+            pt, number_points=Nmbre_derivees) for pt in self.Taylor_points}
+        self.f = f_ab
+
+
+def compute_taylor_series(x, pt, derivs):
+    sum = 0
+
+    for nth_deg, nth_deriv in enumerate(derivs):
+        sum += (nth_deriv / math.factorial(nth_deg)) * (x - pt)**nth_deg
+
+    return sum
+
+###########################################################################
+#### Exemple d'utilisation de la classe, avec explication des donnees #####
+###########################################################################
+
+
+SD = Student_data()
+
+print(f"bornes de l'intervalle considere: [{SD.a},{SD.b}]")
+print()
+print(f"Points ou on connait la derivee : {SD.Taylor_points}")
+print(f"Valeurs des derivees 0 a {Nmbre_derivees-1}:")
+for x in SD.Taylor_points:
+    print(f"x = {x}, dérivées: {SD.Taylor_derivatives_values[x]}")
+print()
+print(f"Valeurs maximales des derivees 0 a {
+      Nmbre_derivees-1} sur l'intervalle [{SD.a},{SD.b}]: {SD.Maximal_derivatives_values}")
+print()
+print("La fonction est dans la variable SD.f, si on doit l'évaluer en certains points")
+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)}")
+
+# Exemple de graphe de la fonction f.
+# t = np.linspace(SD.a, SD.b, Nmbre_pts)
+# y = compute_taylor_series(
+#     t, SD.Taylor_points[1], SD.Taylor_derivatives_values[SD.Taylor_points[1]])
+
+# plt.plot(t, SD.f(t), "k")
+# plt.show()
+
+
+###########################################################################
+#### Exercices #####
+###########################################################################
+
+def ex2():
+    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]])
+
+    y1 = compute_taylor_series(
+        t, SD.Taylor_points[1], SD.Taylor_derivatives_values[SD.Taylor_points[1]])
+
+    y2 = compute_taylor_series(
+        t, SD.Taylor_points[2], SD.Taylor_derivatives_values[SD.Taylor_points[2]])
+
+    fig, axs = plt.subplots(2, 2, figsize=(18, 12))
+
+    axs[0, 0].plot(t, SD.f(t), color='black')
+    axs[0, 0].set_title('Fonction $f$ originale')
+    axs[0, 0].grid()
+
+    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].set_title(
+        f'Développement de $T_f$ en $a = {SD.Taylor_points[0]}$')
+    axs[0, 1].set_ylim([-1.2, 1.2])
+    axs[0, 1].grid()
+    axs[0, 1].legend()
+
+    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].set_title(
+        f'Développement de $T_f$ en $a = {SD.Taylor_points[1]}$')
+    axs[1, 0].set_ylim([-1.2, 1.2])
+    axs[1, 0].grid()
+    axs[1, 0].legend()
+
+    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].set_title(
+        f'Développement de $T_f$ en $a = {SD.Taylor_points[2]}$')
+    axs[1, 1].set_ylim([-1.2, 1.2])
+    axs[1, 1].grid()
+    axs[1, 1].legend()
+
+    fig.tight_layout()
+    plt.show()
+
+
+ex2()
+
+# Graphique des polynômes de Taylor
+# fig, axes = plt.subplots(1, 3)
+# for i, pt in enumerate(SD.Taylor_points):
+#     ax = axes[i]
+#     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()

report.qmd

+---
+title: |
+    Contrôle 4
+subtitle: Mathématiques en technologies de l'information 4 -- ISC_421
+author:
+    - Iliya Saroukhanian
+lang: fr-CH
+date: today
+date-format: long
+format:
+    pdf:
+        documentclass: scrreprt
+        papersize: a4
+        highlight-style: github
+        toc: true
+        lof: true
+        number-sections: true
+        colorlinks: true
+---
+
+# Graphe de divers polynômes de Taylor
+
+![Graphiques de la fonction $f$ et de ses développements de Taylor en divers points](./figs/ex2_taylor.png)
+
+# Exercice 2
+
+# Exercice 3
+
+# Exercice 4
+
+# Conclusion