





"""simple docstring"""




def 						UpperCAmelCase__		(					_UpperCAmelCase ) ->    Tuple:






  """simple docstring"""





  A_  :	str    						=  len(_UpperCAmelCase )

  # Set the gap value to be decreased by a factor of 1.3
  # after each iteration
  A_  :	Dict    						=  1.3

  # Continue sorting until the gap is 1
  while gap > 1:
    # Decrease the gap value
    A_  :	int    						=  int(gap / shrink )

    # Sort the elements using insertion sort
    for i in range(_UpperCAmelCase			, len(_UpperCAmelCase ) ):
      A_  :	Tuple    						=  collection[i]
      A_  :	str    						=  i
      while j >= gap and collection[j - gap] > temp:
        A_  :	Dict    						=  collection[j - gap]
        j -= gap
      A_  :	List[str]    						=  temp

  return collection


if __name__ == "__main__":
   import doctest

   doctest.testmod()






"""simple docstring"""




import math as m





def 						UpperCAmelCase__		(					_UpperCAmelCase			, _UpperCAmelCase			, _UpperCAmelCase=0.001 ):






  """simple docstring"""





  return (f(a + h ) - f(a - h )) / (2 * h)





def 						UpperCAmelCase__		(					_UpperCAmelCase			, _UpperCAmelCase=0			, _UpperCAmelCase=100			, _UpperCAmelCase=0.0_001			, _UpperCAmelCase=1E-6			, _UpperCAmelCase=False ):






  """simple docstring"""





  A_  :	Optional[int]    						=  xa  # set the initial guess
  A_  :	Any    						=  [a]
  A_  :	str    						=  abs(f(_UpperCAmelCase ) )
  A_  :	str    						=  lambda _UpperCAmelCase : calc_derivative(_UpperCAmelCase			, _UpperCAmelCase			, h=_UpperCAmelCase )  # noqa: E731  Derivative of f(x)
  for _ in range(_UpperCAmelCase ):
    if fa(_UpperCAmelCase ) == 0:
      raise ValueError('No converging solution found' )
    A_  :	Union[str, Any]    						=  a - f(_UpperCAmelCase ) / fa(_UpperCAmelCase )  # Calculate the next estimate
    if logsteps:
      steps.append(_UpperCAmelCase )
    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__":
   from matplotlib import pyplot as plt

   lowerCamelCase_					:						str      = lambda x: m.tanh(x) ** 2 - m.exp(3 * x)  # noqa: E731
   lowerCamelCase_ , lowerCamelCase_ , lowerCamelCase_					:						int      = newton_raphson(
       f, xa=10, maxiter=10_00, step=1E-6, logsteps=True
   )
   plt.plot([abs(f(x)) for x in steps])
   plt.xlabel('step')
   plt.ylabel('error')
   plt.show()
   print(F"solution = {{{solution:f}}}, error = {{{error:f}}}")