70 lines
1.6 KiB
Python
70 lines
1.6 KiB
Python
import numpy as np
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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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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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# 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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if __name__ == '__main__':
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exercice1()
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