# %%
import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

# %%
# "Einfacher" Newton Verfahren
# Parameter
N = 200
x = sp.Symbol("x")

# Funktion
f_sym = sp.sin(3 * x) + 0.02 * x**2

# Ableitungen
f = sp.lambdify(x, f_sym, "numpy")
f_prime = sp.lambdify(x, sp.diff(f_sym, x), "numpy")
f_second = sp.lambdify(x, sp.diff(f_sym, x, 2), "numpy")

# Daten
x_data = np.linspace(0, 10, N)
f_data = f(x_data)

# Plot
plt.figure(1)
plt.plot(x_data, f_data)
plt.xlabel("x")
plt.ylabel("y")
plt.grid("on")
plt.axis("image")

# Newton Verfahren
# Startwert
startwerte = [6.6, 3.4, 8.38]

# Iterationsformel
x_n = lambda x: x - f_prime(x) / f_second(x)

# Iteration
for x_0 in startwerte:
    n = 0
    x_i = x_0
    f_x_prime = f_prime(x_0)
    print(f"x_{n}: {x_0}\nf'(x_{n}): {f_x_prime}\n")

    limit = 4

    while np.abs(f_x_prime) > 1e-16 and limit > 0:
        n += 1
        x_i = x_n(x_i)
        f_x_prime = f_prime(x_i)
        print(f"x_{n}: {x_i}\nf'(x_{n}): {f_x_prime}\n")
        limit -= 1

# %%
# "Modifizierter" Newton Verfahren
# Parameter
N = 200
x = sp.Symbol("x")

# Funktion
f_sym = sp.sin(3 * x) + 0.02 * x**2

# Ableitungen
f = sp.lambdify(x, f_sym, "numpy")
f_prime = sp.lambdify(x, sp.diff(f_sym, x), "numpy")
f_second = sp.lambdify(x, sp.diff(f_sym, x, 2), "numpy")

# Daten
x_data = np.linspace(0, 10, N)
f_data = f(x_data)

# Plot
plt.figure(1)
plt.plot(x_data, f_data)
plt.xlabel("x")
plt.ylabel("y")
plt.grid("on")
plt.axis("image")

# Newton Verfahren
# Startwert
startwerte = [6.6, 3.4, 8.38]

# Iterationsformel
x_n = lambda x: x - f_prime(x) / np.abs(f_second(x))

# Iteration
for x_0 in startwerte:
    n = 0
    x_i = x_0
    f_x_prime = f_prime(x_0)
    print(f"x_{n}: {x_0}\nf'(x_{n}): {f_x_prime}\n")

    limit = 1000

    while np.abs(f_x_prime) > 1e-16 and limit > 0:
        n += 1
        x_i = x_n(x_i)
        f_x_prime = f_prime(x_i)
        print(f"x_{n}: {x_i}\nf'(x_{n}): {f_x_prime}\n")
        limit -= 1
# %%