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'''

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)

if error < maxerror:

break

else:

raise ValueError("Iteration 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))