96 lines
2.7 KiB
Python
96 lines
2.7 KiB
Python
import matplotlib.pyplot as plt
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def euler(f, n: int, h: float, x_0: float, y_0: float) -> tuple[list[float], list[float]]:
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"""
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:param f: la fonction f (ou y)
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:param n: jusqu'où va le calcul (le nombre d'itérations)
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:param h: le pas de chaque itération
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:param x_0: dans la condition initiale, c'est la valeur passée en entrée de f
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:param y_0: dans la condition initiale, c'est la valeur de sortie de f.
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:return:
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"""
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results_x = [x_0]
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results_y = [y_0]
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# h c'est delta_x
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x = x_0
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y = y_0
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for i in range(n):
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if i < 4:
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x = x + h
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y = y + h * f(x, y)
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else:
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h = .3
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x = x + h
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y = y + h * f(x, y)
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results_x.append(x)
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results_y.append(y)
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return results_x, results_y
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def runge_kutta(f, n: int, h: float, x_0: float, y_0: float) -> tuple[list[float], list[float]]:
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results_x = [x_0]
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results_y = [y_0]
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x = x_0
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y = y_0
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for i in range(n):
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if i < 4:
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k_1 = f(x, y)
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k_2 = f(x + (h / 2), y + (h / 2) * k_1)
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x = x + h
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y = y + h * k_2
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else:
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h = .3
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k_1 = f(x, y)
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k_2 = f(x + (h / 2), y + (h / 2) * k_1)
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x = x + h
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y = y + h * k_2
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results_x.append(x)
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results_y.append(y)
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return results_x, results_y
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def taylor(f, f_derive_x, f_derive_y, n: int, h: float, x_0: float, y_0: float) -> tuple[list[float], list[float]]:
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results_x = [x_0]
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results_y = [y_0]
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x = x_0
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y = y_0
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for i in range(n):
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if i < 4:
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x = x + h
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y = y + h * f(x, y) + (h ** 2) / 2 * (f_derive_x(x, y) + f_derive_y(x, y) * f(x, y))
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else:
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h = .3
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x = x + h
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y = y + h * f(x, y) + (h ** 2) / 2 * (f_derive_x(x, y) + f_derive_y(x, y) * f(x, y))
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results_x.append(x)
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results_y.append(y)
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return results_x, results_y
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def exercice1() -> None:
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f = lambda x, y: -2 * x * y + 1
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f_derive_x = lambda x, y: -2 * y + 1
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f_derive_y = lambda x, y: -2 * x + 1
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results_x_euler, results_y_euler = euler(f, n=5, x_0=0, y_0=1, h=.2)
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results_x_runge, results_y_runge = runge_kutta(f, n=5, x_0=0, y_0=1, h=.2)
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results_x_taylor, results_y_taylor = taylor(f, f_derive_x, f_derive_y, 5, x_0=0, y_0=1, h=.2)
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plt.plot(results_x_euler, results_y_euler, color='green')
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plt.plot(results_x_runge, results_y_runge, color='red')
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plt.plot(results_x_taylor, results_y_taylor, color='blue')
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plt.title('La giga fonction pour les equations différentielles. (mieux que le sex garanti 3 ans)')
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plt.show()
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if __name__ == '__main__':
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exercice1()
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