Feat: Add dichotomie and Newton-Raphson

ex5.py

@@ -1,56 +1,80 @@
+import matplotlib.pyplot as plt
+import numpy as np
 
-class Point:
-    def __init__(self, x: float, y: float):
-        self.x = x
-        self.y = y
+dich_steps = []
+new_raph_steps = []
+
+def f(x) -> float:
+    return np.exp(x) - 2 * np.cos(x)
 
 
-def interpolation_lagrange(points: list[Point], x: float) -> float:
-    interpolation: float = 0.
-    n = len(points)
-    for i in range(n):
-        term: float = 1. # 0.0 will always give 0, it's a product ! So lets put 1
-        # Lagrange L(x)
-        for k in range(n):
-            if k != i:
-                term *= ((x - points[k].x) / (points[i].x - points[k].x))
-        # interpolation with a piece of Lagrange (y_i)
-        interpolation += term * points[i].y
+def dichotomie(x_0, a, b, epsilon, n=0) -> float:
+    delta = f(a) * f(x_0)
+    x_1 = (a + b) / 2
 
-    return interpolation
+    dich_steps.append(x_1)
+
+    if np.abs(b - a) <= epsilon:
+        return x_1
+
+    if delta < 0:
+        return dichotomie(x_1, a, x_0, epsilon, n+1)
+    elif delta == 0:
+        return x_1
+    else:
+        return dichotomie(x_1, x_0, b, epsilon, n+1)
 
 
-def interpolation_newton(points: list[Point], x: float) -> float:
-    interpolation = 0. # alpha sum
-    n = len(points)
-    for i in range(n):
-        a_i = 1.
-        for k in range(n-1):
-            a_i *= (points[k+1].y - points[k].y) / (points[k+1].x - points[k].x)
+def f_prime(x) -> float:
+    return 1 + np.sin(x)
 
-    return interpolation
+
+def newton_raphson(x_0, epsilon) -> float:
+    x_1 = x_0 - (f(x_0) / f_prime(x_0))
+    new_raph_steps.append(x_1)
+
+    if np.abs(x_1 - x_0) < epsilon:
+        return x_1
+    elif np.abs(f(x_0)) < epsilon:
+        return x_1
+    return newton_raphson(x_1, epsilon)
 
 
 def ex1() -> None:
-    p1 = Point(-1, 0)
-    p2 = Point(0, -1)
-    p3 = Point(1, 0)
-    p4 = Point(3, 70)
+    x = np.arange(0, 1, 0.001)
+    f = [np.exp(i) - 2 * np.cos(i) for i in x]
+    f1 = [np.exp(i) for i in x]
+    f2 = [2 * np.cos(i) for i in x]
 
-    points = []
-    points.append(p1)
-    points.append(p2)
-    points.append(p3)
-    points.append(p4)
+    for i in range(len(x)):
+        if f1[i] == f2[i]:
+            print(f1[i])
+
+    plt.plot(x, f)
+    plt.plot(x, f1)
+    plt.plot(x, f2)
+    plt.grid()
+    plt.show()
+
+    dichotomie_res = dichotomie(0, 0, 1, 10e-6)
+    print(dichotomie_res)
+
+    plt.plot(dich_steps)
+    plt.title('Convergence de la dichotomie')
+    plt.show()
+
+    newton_raphson_res = newton_raphson(0.1, 10e-6)
+    print(newton_raphson_res)
+
+    plt.plot(new_raph_steps)
+    plt.title('Convergence de Newton-Raphson')
+    plt.show()
 
     """
-    
+    La dichotomie à moins d'étape que NR, mais NR à l'air plus rapide,
+    comme Flash McQueen
     """
 
-    racine_x_2 = interpolation_lagrange(points, 2.0)
-    racine_x_3 = interpolation_lagrange(points, 3.0)
-    print(f'racine X = 2 = {racine_x_2}, racine X = 3 = {racine_x_3}')
-
 
 if __name__ == '__main__':
-    ex1()
+    ex1()