test3.py

# -*- coding: utf-8 -*-

import numpy as np
import matplotlib.pyplot as plt
f = lambda x: 0.1*x**3 - 0.8*x**2 - 1.5*x + 5.4 
fd = lambda x: 0.3*x**2 - 1.6*x - 1.5
fdd = lambda x: 0.6*x - 1.6
viz_range = np.array([-6, 12])
max_iter = 100
min_tol = 1e-6
x_init = -2 # Try -2, 0, and 16/6 (a saddle point)
# Prepare visualization
xs = np.linspace(*viz_range, 100)
plt.plot(xs, f(xs), 'r-', label='f(x)', linewidth=2) 
plt.plot(x_init, f(x_init), 'b.', label='Each step', markersize=12) 
plt.axis((*viz_range, *f(viz_range)))
plt.legend()
x = x_init
for i in range(max_iter):
    # Run the Newton method
    xp = x
    x = x - fd(x) / abs(fdd(x)+1) # Replace the denominator as abs(fdd(x)) and (abs(fdd(x)) + 1) to resolve the maxima and saddle point problems
        # Update visualization for each iteration
    print(f'Iter: {i}, x = {xp:.3f} to {x:.3f}, f(x) = {f(xp):.3f} to {f(x):.3f} (f\'(x) = {fd(xp):.3f}, f\'\'(x) = {fdd(xp):.3f})') 
    lcolor = np.random.rand(3)
    approx = 0.5*fdd(xp)*(xs-xp)**2 + fd(xp)*(xs-xp) + f(xp)
    plt.plot(xs, approx, '-', linewidth=1, color=lcolor, alpha=0.8)
    plt.plot(x, f(x), '.', color=lcolor, markersize=12)
    # Check the terminal condition
    if abs(x - xp) < min_tol: break;
plt.show()