151 lines
3.7 KiB
Python
151 lines
3.7 KiB
Python
import random
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import numpy as np
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import matplotlib.pyplot as plt
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def trapeze_formule(f, a: float, b: float, n: int) -> float:
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"""
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:param f: la fonction f (bien passer une fonction ou une lambda)
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:param a: la petite borne
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:param b: la grande borne
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:param n: nombre de "tranche" de calcul
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:return:
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"""
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# dx = (b-a) / n
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dx = (b - a) / n
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# somme_inferieure = sum de i = 1 jusqu'a n - 1 faire f(xi)
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somme_inferieure = 0.
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for i in range(1, n):
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xi = a + i * dx
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somme_inferieure += f(xi)
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# I = (delta X / 2) * [f(a) + f(b) + 2 * somme_inferieure ]
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I = (dx / 2) * (f(a) + f(b) + 2 * somme_inferieure)
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return I
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def simpson(f, a: float, b: float, n: int) -> float:
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if n % 2 != 0:
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raise ValueError(f'n must be even, got {n}')
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dx = (b - a) / n
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somme_paire = 0.
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somme_impaire = 0.
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for i in range(1, n):
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xi = a + i * dx
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if i % 2 == 0:
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somme_paire += f(xi)
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else:
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somme_impaire += f(xi)
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I = (dx / 3) * (f(a) + f(b) + 4 * somme_impaire + 2 * somme_paire)
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return I
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# Les b possibles dans Newton Cotes
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POIDS = {
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2: [1/2,1/2],
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3: [1/6, 4/6, 1/6],
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4: [1/8, 3/8, 3/8, 1/8],
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5: [
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7 / 90,
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32 / 90,
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12 / 90,
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32 / 90,
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7 / 90,
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],
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6: [19/288, 75/288, 50/288, 50/288, 75/255, 19/255],
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7: [41/840, 216/840, 27/840, 272/840, 27/840, 216/840, 41/840]
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}
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def newton_cotes(f, a: float, b: float, s: int, n: int) -> float:
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weights = POIDS[s+1] # Poids pour l'ordre s
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h = (b - a) / n
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Aj = 0.
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for j in range(n):
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somme = 0.
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x_j = a + j * h
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for i in range(s):
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x = x_j + (i / (s - 1)) * h # points à l'intérieur du sous-intervalle
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somme += weights[i] * f(x)
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somme += h / (s-1) # facteur h/(s-1) selon Newton-Cotes
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Aj += somme * h
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return Aj
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def exercice1():
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f = lambda x: x ** 2
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approx = trapeze_formule(f, a=0, b=1, n=10)
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print(approx)
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g = lambda x : np.sin(x)
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trapeze1 = trapeze_formule(g, a=0, b=np.pi, n=100)
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trapeze2 = trapeze_formule(g, a=0, b=np.pi, n=200)
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simpson1 = simpson(g, a=0, b=np.pi, n=100)
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simpson2 = simpson(g, a=0, b=np.pi, n=200)
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newton1 = newton_cotes(g, a=0, b=np.pi, n=100, s=4)
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newton2 = newton_cotes(g, a=0, b=np.pi, n=200, s=4)
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# le vrai résultat est 2.
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print(f'{trapeze1}, {trapeze2} erreur: {2-trapeze1} {2-trapeze2}')
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print(f'{simpson1}, {simpson2} erreur: {2 - simpson1} {2 - simpson2}')
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print(f'{newton1}, {newton2} erreur: {2 - newton1} {2 - newton2}')
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def monte_carlo_2d(f, a: float, b: float, c: float, d: float, n: int) -> float:
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"""
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:param f: la fonction
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:param a: borne a (pour x)
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:param b: borne b (pour x)
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:param c: borne c (pour y)
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:param d: borne d (pour y)
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:param n: nombre de tirages aléatoire (N grand → epsilon petit)
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:return: I
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"""
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sum = 0.
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points_x = []
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points_y = []
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for i in range(1, n):
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x = a + (b - a) * random.random()
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y = c + (d - c) * random.random()
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if a < x < b and c < y < d:
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sum += f(x, y)
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points_x.append(x)
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points_y.append(y)
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I = (sum / n) * (b - a) * (d - c)
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epsilon = np.abs(.58 - I)
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print(f'Err absolue : {epsilon}')
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plt.scatter(points_x, points_y, color='blue')
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plt.title('Surface fonction')
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plt.show()
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return I
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def exercice2():
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g = lambda x, y: 1
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g2 = lambda x, y: np.exp(-x ** 2 - y ** 2)
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ig = monte_carlo_2d(g, a=-1, b=1, c=-1, d=1, n=10_000)
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ig2 = monte_carlo_2d(g2, a=-1, b=1, c=-1, d=1, n=10_000)
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print(f'I (g) = {ig}, I (g2) = {ig2}')
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if __name__ == '__main__':
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#exercice1()
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exercice2()
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