

from __future__ import annotations



def      snake_case__      (   SCREAMING_SNAKE_CASE_		:   Dict     ,					SCREAMING_SNAKE_CASE_		:   Optional[int] = None     ,					SCREAMING_SNAKE_CASE_		:   Optional[int] = None      ):



     '''simple docstring'''





     if start is None:
          lowercase__   :       Union[str, Any]				=	0

     if end is None:
          lowercase__   :       List[str]				=	len(lowercase_      ) - 1

     if start >= end:
          return

     lowercase__   :       Optional[Any]				=	(start + end) // 2

     slowsort(lowercase_     ,					lowercase_     ,					lowercase_      )
     slowsort(lowercase_     ,					mid + 1     ,					lowercase_      )

     if sequence[end] < sequence[mid]:
          lowercase__				,			lowercase__   :       Any				=	sequence[mid], sequence[end]

     slowsort(lowercase_     ,					lowercase_     ,					end - 1      )


if __name__ == "__main__":
       from doctest import testmod

       testmod()




import numpy as np

def      snake_case__      (   SCREAMING_SNAKE_CASE_		:   str     ,					SCREAMING_SNAKE_CASE_		:   Optional[int]     ,					SCREAMING_SNAKE_CASE_		:   Optional[int]     ,					SCREAMING_SNAKE_CASE_		:   Union[str, Any]     ,					SCREAMING_SNAKE_CASE_		:   List[Any]      ):



     '''simple docstring'''





     lowercase__   :       List[Any]				=	int(np.ceil((x_end - xa) / h      )      )
     lowercase__   :       Dict				=	np.zeros((n + 1,)      )
     lowercase__   :       Dict				=	ya
     lowercase__   :       Union[str, Any]				=	xa

     for k in range(SCREAMING_SNAKE_CASE_      ):
          lowercase__   :       Optional[int]				=	f(SCREAMING_SNAKE_CASE_     ,					y[k]      )
          lowercase__   :       Dict				=	f(x + 0.5 * h     ,					y[k] + 0.5 * h * ka      )
          lowercase__   :       Any				=	f(x + 0.5 * h     ,					y[k] + 0.5 * h * ka      )
          lowercase__   :       List[Any]				=	f(x + h     ,					y[k] + h * ka      )
          lowercase__   :       Tuple				=	y[k] + (1 / 6) * h * (ka + 2 * ka + 2 * ka + ka)
          x += h

     return y


if __name__ == "__main__":
       import doctest

       doctest.testmod()

