



'''simple docstring'''






def 				_SCREAMING_SNAKE_CASE(					snake_case_    :     list[int]  )		->list[int]:






							'''simple docstring'''
							if len(snake_case_  ) > 1:
														_lowercase							:		Optional[Any]				=   len(snake_case_  ) // 2  # Finds the middle of the array
														_lowercase							:		Union[str, Any]				=   arr[
														    :middle_length
														]  # Creates an array of the elements in the first half.
														_lowercase							:		str				=   arr[
														    middle_length:
														]  # Creates an array of the elements in the second half.
														_lowercase							:		Union[str, Any]				=   len(snake_case_  )
														_lowercase							:		List[str]				=   len(snake_case_  )
														merge(snake_case_  )  # Starts sorting the left.
														merge(snake_case_  )  # Starts sorting the right
														_lowercase							:		Tuple				=   0  # Left Counter
														_lowercase							:		str				=   0  # Right Counter
														_lowercase							:		Dict				=   0  # Position Counter
														while (
														    left_index < left_size and right_index < right_size
														):  # Runs until the lowers size of the left and right are sorted.
																					if left_array[left_index] < right_array[right_index]:
																												_lowercase							:		List[Any]				=   left_array[left_index]
																												left_index += 1
																					else:
																												_lowercase							:		List[Any]				=   right_array[right_index]
																												right_index += 1
																					index += 1
														while (
														    left_index < left_size
														):  # Adds the left over elements in the left half of the array
																					_lowercase							:		int				=   left_array[left_index]
																					left_index += 1
																					index += 1
														while (
														    right_index < right_size
														):  # Adds the left over elements in the right half of the array
																					_lowercase							:		str				=   right_array[right_index]
																					right_index += 1
																					index += 1
							return arr


if __name__ == "__main__":
						import doctest

						doctest.testmod()








'''simple docstring'''






import math as m





def 				_SCREAMING_SNAKE_CASE(					snake_case_    :     Union[str, Any] ,       snake_case_    :     int ,       snake_case_    :     Optional[Any]=0.0_0_1  )		->Tuple:






							'''simple docstring'''
							return (f(a + h  ) - f(a - h  )) / (2 * h)





def 				_SCREAMING_SNAKE_CASE(					snake_case_    :     Dict ,       snake_case_    :     List[Any]=0 ,       snake_case_    :     Dict=1_00 ,       snake_case_    :     List[str]=0.0_0_0_1 ,       snake_case_    :     str=1E-6 ,       snake_case_    :     Tuple=False  )		->List[Any]:






							'''simple docstring'''
							_lowercase							:		Tuple				=   xa  # set the initial guess
							_lowercase							:		List[str]				=   [a]
							_lowercase							:		str				=   abs(f(snake_case_  )  )
							_lowercase							:		Dict				=   lambda snake_case_  : calc_derivative(snake_case_ ,       snake_case_ ,       h=snake_case_  )  # noqa: E731  Derivative of f(x)
							for _ in range(snake_case_  ):
														if fa(snake_case_  ) == 0:
																					raise ValueError('''No converging solution found'''  )
														_lowercase							:		Tuple				=   a - f(snake_case_  ) / fa(snake_case_  )  # Calculate the next estimate
														if logsteps:
																					steps.append(snake_case_  )
														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__	  =		lambda x: m.tanh(x) ** 2 - m.exp(3 * x)  # noqa: E731
						lowerCamelCase__  ,      lowerCamelCase__  ,      lowerCamelCase__	  =		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}}}''')



