'''
 Author: P Shreyas Shetty
 Implementation of Newton-Raphson method for solving equations of kind
 f(x) = 0. It is an iterative method where solution is found by the expression
 x[n+1] = x[n] + f(x[n])/f'(x[n])
 If no solution exists, then either the solution will not be found when iteration
 limit is reached or the gradient f'(x[n]) approaches zero. In both cases, exception
 is raised. If iteration limit is reached, try increasing maxiter.
 '''

import math as m

def calc_derivative(f, a, h=0.001):
 '''
 Calculates derivative at point a for function f using finite difference
 method
 '''
 return (f(a+h)-f(a-h))/(2*h)

def newton_raphson(f, x0=0, maxiter=100, step=0.0001, maxerror=1e-6,logsteps=False):
 
 a = x0 #set the initial guess
 steps = [a]
 error = abs(f(a))
 f1 = lambda x:calc_derivative(f, x, h=step) #Derivative of f(x)
 for _ in range(maxiter):
 if f1(a) == 0:
 raise ValueError("No converging solution found")
 a = a - f(a)/f1(a) #Calculate the next estimate
 if logsteps:
 steps.append(a)
 error = abs(f(a))
 if error < maxerror:
 break
 else:
 raise ValueError("Itheration limit reached, no converging solution found")
 if logsteps:
 #If logstep is true, then log intermediate steps
 return a, error, steps
 return a, error
 
if __name__ == '__main__':
 import matplotlib.pyplot as plt
 f = lambda x:m.tanh(x)**2-m.exp(3*x)
 solution, error, steps = newton_raphson(f, x0=10, maxiter=1000, step=1e-6, logsteps=True)
 plt.plot([abs(f(x)) for x in steps])
 plt.xlabel("step")
 plt.ylabel("error")
 plt.show()
 print("solution = {%f}, error = {%f}" % (solution, error))