Newton's Method

# Newton's Method (Numerical Optimization)
# Yong-Jun Shin (2019)

import numpy as np
import matplotlib.pyplot as plt

def f(x):
  return x**2 + 4*x + 5          # x_opt = -1 => f(-2) = 1

N = 5                            # total number of iterations
iteration = np.arange(0, N, 1)   # iteration vector [0,..., N-1]
x_opt = np.empty(N)              # optimal x (x_opt) candidate (vector)
x_opt[0] = 0                     # initial x_opt value = 0
dx = 0.001                       # delta x
c = 1                            # constant c

for n in range (0, N-1):                       # n: iteration index
  slope = (f(x_opt[n]) - f(x_opt[n]-dx))/dx    # slope (first derivative) at x_opt(n)
  hess = (f(x_opt[n]+dx) - 2*f(x_opt[n]) + f(x_opt[n]-dx))/dx**2   # second derivative at x_opt(n)
  x_opt[n+1] = x_opt[n] - (c/hess) * slope     # Newton's method (step size = c/hess)

plt.plot(iteration,x_opt)
plt.xlabel('iteration (n)')
plt.ylabel('optimal x (opt_x)')
plt.title('Newton\'s Method')
plt.grid(True)
plt.show()