






def      lowercase__      (     __snake_case    :							List[str]     ,       __snake_case    :							str     ,       __snake_case    :							int     ,       __snake_case    :							Union[str, Any]     ,       __snake_case    :							Tuple     ,       __snake_case    :							str  ):


					'''simple docstring'''


					if index == r:
										for j in range(__snake_case  ):
															print(data[j]     ,       end=' '  )
										print(' '  )
										return
					#  When no more elements are there to put in data[]
					if i >= n:
										return
					# current is included, put next at next location
					UpperCAmelCase_    :      Tuple            =						arr[i]
					combination_util(__snake_case     ,       __snake_case     ,       __snake_case     ,       index + 1     ,       __snake_case     ,       i + 1  )
					# current is excluded, replace it with
					# next (Note that i+1 is passed, but
					# index is not changed)
					combination_util(__snake_case     ,       __snake_case     ,       __snake_case     ,       __snake_case     ,       __snake_case     ,       i + 1  )
					# The main function that prints all combinations
					# of size r in arr[] of size n. This function
					# mainly uses combinationUtil()




def      lowercase__      (     __snake_case    :							List[str]     ,       __snake_case    :							List[str]     ,       __snake_case    :							Dict  ):


					'''simple docstring'''


					UpperCAmelCase_    :      Tuple            =						[0] * r
					# Print all combination using temporary array 'data[]'
					combination_util(__snake_case     ,       __snake_case     ,       __snake_case     ,       0     ,       __snake_case     ,       0  )


if __name__ == "__main__":
		# Driver code to check the function above
		__UpperCAmelCase								=						[10, 20, 30, 40, 50]
		print_combination(arr, len(arr), 3)
		# This code is contributed by Ambuj sahu











"""simple docstring"""
from collections.abc import Callable

import numpy as np

def       _snake_case						(						UpperCamelCase						:		Callable			,		UpperCamelCase						:		float			,		UpperCamelCase						:		float			,		UpperCamelCase						:		float			,		UpperCamelCase						:		float						):
 UpperCAmelCase	:					Any	=	int(np.ceil((x_end - xa) / step_size						)						)
 UpperCAmelCase	:					Optional[Any]	=	np.zeros((n + 1,)						)
 UpperCAmelCase	:					Optional[int]	=	ya
 UpperCAmelCase	:					int	=	xa

 for k in range(UpperCamelCase						):
  UpperCAmelCase	:					Optional[int]	=	y[k] + step_size * ode_func(UpperCamelCase			,		y[k]						)
  UpperCAmelCase	:					Optional[int]	=	y[k] + (
      (step_size / 2) * (ode_func(UpperCamelCase			,		y[k]						) + ode_func(x + step_size			,		UpperCamelCase						))
  )
  x += step_size

 return y


if __name__ == "__main__":
      import doctest

      doctest.testmod()
