Python/maths/newton_raphson.py

'''
    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))