Feat: Ajoute l'exercice 2 avec les polynômes de Newton

tp6.py

@@ -36,30 +36,49 @@ def interpolation_lagrange(points: list[Point], x: float, show_polynome: bool=Fa
     return interpolation
 
 
-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 calc_coef(x: list[float],y: list[float]) -> float:
+    """
+        Renvois f[x0,x1,...,xk] (diff divisé de Newton)
+        avec la relation récursive :
+        f[x0...xk] = (f[x1...xk] - f[x0..x{k-1}] / (xk - x0)
+        (calcul alpha)
+        :param x:
+        :param y:
+        :return:
+    """
+    if len(x) == 1:
+        return y[0]
+    elif len(x) == 2:
+        return (y[1]-y[0]) / (x[1] - x[0])
+    return (calc_coef(x[1:], y[1:]) - calc_coef(x[:-1], y[:-1])) / (x[-1] - x[0])
 
-    return interpolation
 
+def P_newton(x_i: list[float], y_i: list[float], x: float) -> float:
+    """
+    Calcul le polynôme de Newton
+
+    P_n(x) = y0 + sum_{k=1..n} a_k * prod_{j=0..k-1}(x - x_j),
+    avec a_k = f[x0,...,xk] calculé via calc_coef()
+    :param x_i: ensemble des x
+    :param y_i: ensemble des y
+    :param x: là ou on veut interpolé
+    :return:
+    """
+    p = y_i[0]
+    e = 1.
+
+    for k in range(1, len(x_i)):
+        e *= (x - x_i[k-1])
+        a_k = calc_coef(x_i[:k+1], y_i[:k+1])
+        p += a_k * e
+
+    return p
 
 def interpolation_moindre_carree(points: list[Point], x: float) -> float:
     interpolation = 0.
 
     pass
 
-
-def test() -> None:
-    points = [Point(0, 1), Point(1, 3), Point(2, 2)]
-    x = 1.5
-    result = interpolation_lagrange(points, x)
-    print(f"P({x}) = {result}")  # Doit afficher 2.875
-
-
 def exercice1() -> None:
     points = [Point(-1,0), Point(0,-1), Point(1,0), Point(3,70)]
 
@@ -68,16 +87,40 @@ def exercice1() -> None:
     print(f"X2={inter_x2}, X3={inter_x3}")
 
     x = np.linspace(0, 4, 300)
-    y = [interpolation_lagrange(points, x[i]) for i in range(len(x))]
+    y_lagrange = [interpolation_lagrange(points, x[i]) for i in range(len(x))]
+
+    y_newton = []
+
+    for i in range(4):
+        y_newton.append([P_newton([point.x for point in points], [point.y for point in points], i)])
+
+    print(y_lagrange)
+    print(y_newton)
 
     # extrapolation
-    plt.plot(x, y)
+    plt.plot(x, y_lagrange)
     plt.title('fonction f(x) ex 1')
     plt.show()
 
 
 def exercice2() -> None:
-    pass
+    x_i = [0, np.pi / 4, np.pi / 2,  (3 * np.pi) / 4, np.pi]
+
+    x = [i for i in range(len(x_i))]
+
+    f = lambda x : np.sin(x)
+
+    results = []
+
+    for i in x:
+        res = P_newton(x_i, [f(x_i[i]) for i in range(len(x_i))], i)
+        print(res)
+        results.append(res)
+
+    plt.plot(results)
+    plt.show()
+
 
 if __name__ == '__main__':
     exercice1()
+    #exercice2()