104 lines
1.9 KiB
Python
104 lines
1.9 KiB
Python
# %%
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import numpy as np
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import sympy as sp
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import matplotlib.pyplot as plt
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# %%
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# "Einfacher" Newton Verfahren
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# Parameter
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N = 200
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x = sp.Symbol("x")
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# Funktion
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f_sym = sp.sin(3 * x) + 0.02 * x**2
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# Ableitungen
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f = sp.lambdify(x, f_sym, "numpy")
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f_prime = sp.lambdify(x, sp.diff(f_sym, x), "numpy")
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f_second = sp.lambdify(x, sp.diff(f_sym, x, 2), "numpy")
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# Daten
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x_data = np.linspace(0, 10, N)
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f_data = f(x_data)
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# Plot
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plt.figure(1)
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plt.plot(x_data, f_data)
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plt.xlabel("x")
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plt.ylabel("y")
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plt.grid("on")
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plt.axis("image")
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# Newton Verfahren
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# Startwert
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startwerte = [6.6, 3.4, 8.38]
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# Iterationsformel
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x_n = lambda x: x - f_prime(x) / f_second(x)
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# Iteration
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for x_0 in startwerte:
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n = 0
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x_i = x_0
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f_x_prime = f_prime(x_0)
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print(f"x_{n}: {x_0}\nf'(x_{n}): {f_x_prime}\n")
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limit = 4
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while np.abs(f_x_prime) > 1e-16 and limit > 0:
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n += 1
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x_i = x_n(x_i)
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f_x_prime = f_prime(x_i)
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print(f"x_{n}: {x_i}\nf'(x_{n}): {f_x_prime}\n")
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limit -= 1
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# %%
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# "Modifizierter" Newton Verfahren
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# Parameter
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N = 200
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x = sp.Symbol("x")
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# Funktion
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f_sym = sp.sin(3 * x) + 0.02 * x**2
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# Ableitungen
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f = sp.lambdify(x, f_sym, "numpy")
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f_prime = sp.lambdify(x, sp.diff(f_sym, x), "numpy")
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f_second = sp.lambdify(x, sp.diff(f_sym, x, 2), "numpy")
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# Daten
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x_data = np.linspace(0, 10, N)
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f_data = f(x_data)
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# Plot
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plt.figure(1)
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plt.plot(x_data, f_data)
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plt.xlabel("x")
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plt.ylabel("y")
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plt.grid("on")
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plt.axis("image")
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# Newton Verfahren
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# Startwert
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startwerte = [6.6, 3.4, 8.38]
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# Iterationsformel
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x_n = lambda x: x - f_prime(x) / np.abs(f_second(x))
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# Iteration
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for x_0 in startwerte:
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n = 0
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x_i = x_0
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f_x_prime = f_prime(x_0)
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print(f"x_{n}: {x_0}\nf'(x_{n}): {f_x_prime}\n")
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limit = 1000
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while np.abs(f_x_prime) > 1e-16 and limit > 0:
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n += 1
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x_i = x_n(x_i)
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f_x_prime = f_prime(x_i)
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print(f"x_{n}: {x_i}\nf'(x_{n}): {f_x_prime}\n")
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limit -= 1
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# %%
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