


"""simple docstring"""





from collections.abc import Callable

__UpperCAmelCase    				=      Callable[[float], float]  # type alias for a real -> real function





def     lowercase__	(  lowerCamelCase     :  RealFunc				,	lowerCamelCase     :  RealFunc				,	lowerCamelCase     :  int				,	)					->     float:
  lowerCAmelCase__    :    List[str]								=					float(lowerCamelCase				)
  while True:
    try:
      lowerCAmelCase__    :    Any								=					prev_guess - function(lowerCamelCase				) / derivative(lowerCamelCase				)
    except ZeroDivisionError:
      raise ZeroDivisionError("Could not find root"				) from None
    if abs(prev_guess - next_guess				) < 1_0**-5:
      return next_guess
    lowerCAmelCase__    :    str								=					next_guess





def     lowercase__	(  lowerCamelCase     :  float				)					->     float:
  return (x**3) - (2 * x) - 5





def     lowercase__	(  lowerCamelCase     :  float				)					->     float:
  return 3 * (x**2) - 2


if __name__ == "__main__":
     print(newton(f, fa, 3))




"""simple docstring"""





from __future__ import annotations





def     lowercase__	(  lowerCamelCase     :  list				,	lowerCamelCase     :  list				)					->     list:
  lowerCAmelCase__    :    Dict								=					[None] * (len(lowerCamelCase				) + len(lowerCamelCase				))

  lowerCAmelCase__    :    Optional[Any]								=					0  # pointer to current index for left Half
  lowerCAmelCase__    :    Any								=					0  # pointer to current index for the right Half
  lowerCAmelCase__    :    Dict								=					0  # pointer to current index for the sorted array Half

  while pointera < len(lowerCamelCase				) and pointera < len(lowerCamelCase				):
    if left_half[pointera] < right_half[pointera]:
      lowerCAmelCase__    :    Tuple								=					left_half[pointera]
      pointera += 1
      index += 1
    else:
      lowerCAmelCase__    :    int								=					right_half[pointera]
      pointera += 1
      index += 1
  while pointera < len(lowerCamelCase				):
    lowerCAmelCase__    :    Tuple								=					left_half[pointera]
    pointera += 1
    index += 1

  while pointera < len(lowerCamelCase				):
    lowerCAmelCase__    :    Optional[int]								=					right_half[pointera]
    pointera += 1
    index += 1

  return sorted_array





def     lowercase__	(  lowerCamelCase     :  list				)					->     list:
  if len(lowerCamelCase				) <= 1:
    return array
  # the actual formula to calculate the middle element = left + (right - left) // 2
  # this avoids integer overflow in case of large N
  lowerCAmelCase__    :    Optional[int]								=					0 + (len(lowerCamelCase				) - 0) // 2

  # Split the array into halves till the array length becomes equal to One
  # merge the arrays of single length returned by mergeSort function and
  # pass them into the merge arrays function which merges the array
  lowerCAmelCase__    :    Union[str, Any]								=					array[:middle]
  lowerCAmelCase__    :    List[str]								=					array[middle:]

  return merge(merge_sort(lowerCamelCase				)				,	merge_sort(lowerCamelCase				)				)


if __name__ == "__main__":
   import doctest

   doctest.testmod()
