def 		A_    (					_lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase					)						->			int:
 if index == r:
  for j in range(_lowerCAmelCase					):
   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     : int	     =       arr[i]
 combination_util(_lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       index + 1							,       _lowerCAmelCase							,       i + 1					)
 # current is excluded, replace it with
 # next (Note that i+1 is passed, but
 # index is not changed)
 combination_util(_lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       i + 1					)
 # The main function that prints all combinations
 # of size r in arr[] of size n. This function
 # mainly uses combinationUtil()



def 		A_    (					_lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase					)						->			Optional[int]:
 # A temporary array to store all combination one by one
 UpperCamelCase     : int	     =       [0] * r
 # Print all combination using temporary array 'data[]'
 combination_util(_lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       0							,       _lowerCAmelCase							,       0					)


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







from __future__ import annotations

import numpy as np
from numpy import floataa
from numpy.typing import NDArray



def 		A_    (					_lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       _lowerCAmelCase							,       )						->			list[float]:
 UpperCamelCase     ,   UpperCamelCase     : List[Any]	     =       coefficient_matrix.shape
 UpperCamelCase     ,   UpperCamelCase     : Optional[int]	     =       constant_matrix.shape

 if rowsa != colsa:
  UpperCamelCase     : List[Any]	     =       F"""Coefficient matrix dimensions must be nxn but received {rowsa}x{colsa}"""
  raise ValueError(_lowerCAmelCase					)

 if colsa != 1:
  UpperCamelCase     : Any	     =       F"""Constant matrix must be nx1 but received {rowsa}x{colsa}"""
  raise ValueError(_lowerCAmelCase					)

 if rowsa != rowsa:
  UpperCamelCase     : Tuple	     =       (
      "Coefficient and constant matrices dimensions must be nxn and nx1 but "
      F"""received {rowsa}x{colsa} and {rowsa}x{colsa}"""
  )
  raise ValueError(_lowerCAmelCase					)

 if len(_lowerCAmelCase					) != rowsa:
  UpperCamelCase     : Any	     =       (
      "Number of initial values must be equal to number of rows in coefficient "
      F"""matrix but received {len(_lowerCAmelCase					)} and {rowsa}"""
  )
  raise ValueError(_lowerCAmelCase					)

 if iterations <= 0:
  raise ValueError("Iterations must be at least 1"					)

 UpperCamelCase     : NDArray[floataa]	     =       np.concatenate(
     (coefficient_matrix, constant_matrix)							,       axis=1					)

 UpperCamelCase     ,   UpperCamelCase     : str	     =       table.shape

 strictly_diagonally_dominant(_lowerCAmelCase					)

 # Iterates the whole matrix for given number of times
 for _ in range(_lowerCAmelCase					):
  UpperCamelCase     : Optional[Any]	     =       []
  for row in range(_lowerCAmelCase					):
   UpperCamelCase     : Optional[int]	     =       0
   for col in range(_lowerCAmelCase					):
    if col == row:
     UpperCamelCase     : Union[str, Any]	     =       table[row][col]
    elif col == cols - 1:
     UpperCamelCase     : List[Any]	     =       table[row][col]
    else:
     temp += (-1) * table[row][col] * init_val[col]
   UpperCamelCase     : Dict	     =       (temp + val) / denom
   new_val.append(_lowerCAmelCase					)
  UpperCamelCase     : List[str]	     =       new_val

 return [float(_lowerCAmelCase					) for i in new_val]



def 		A_    (					_lowerCAmelCase					)						->			bool:
 UpperCamelCase     ,   UpperCamelCase     : Dict	     =       table.shape

 UpperCamelCase     : List[Any]	     =       True

 for i in range(0							,       _lowerCAmelCase					):
  UpperCamelCase     : Union[str, Any]	     =       0
  for j in range(0							,       cols - 1					):
   if i == j:
    continue
   else:
    total += table[i][j]

  if table[i][i] <= total:
   raise ValueError("Coefficient matrix is not strictly diagonally dominant"					)

 return is_diagonally_dominant


# Test Cases
if __name__ == "__main__":
    import doctest

    doctest.testmod()
