old_contents: '''
      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))
new_contents: '''
      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("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))
  
subject: Update newton_raphson.py (#891)