


"""simple docstring"""



from __future__ import annotations

UpperCAmelCase      		=			[]






def       lowerCamelCase					(a_						:list[list[int]]				,				a_						:int				,				a_						:int)     ->							bool:
       for i in range(len(a_)):
              if board[row][i] == 1:
                     return False
       for i in range(len(a_)):
              if board[i][column] == 1:
                     return False
       for i, j in zip(range(a_				,				-1				,				-1)				,				range(a_				,				-1				,				-1)):
              if board[i][j] == 1:
                     return False
       for i, j in zip(range(a_				,				-1				,				-1)				,				range(a_				,				len(a_))):
              if board[i][j] == 1:
                     return False
       return True






def       lowerCamelCase					(a_						:list[list[int]]				,				a_						:int)     ->							bool:
       if row >= len(a_):
              solution.append(a_)
              printboard(a_)
              print()
              return True
       for i in range(len(a_)):
              if is_safe(a_				,				a_				,				a_):
                     lowercase		:str       							=				1
                     solve(a_				,				row + 1)
                     lowercase		:List[Any]       							=				0
       return False






def       lowerCamelCase					(a_						:list[list[int]])     ->							None:
       for i in range(len(a_)):
              for j in range(len(a_)):
                     if board[i][j] == 1:
                            print('''Q'''				,				end=''' ''')
                     else:
                            print('''.'''				,				end=''' ''')
              print()


# n=int(input("The no. of queens"))
UpperCAmelCase      		=			8
UpperCAmelCase      		=			[[0 for i in range(n)] for j in range(n)]
solve(board, 0)




print('''The total no. of solutions are :''', len(solution))





"""simple docstring"""



import unittest

import numpy as np






def       lowerCamelCase					(a_						:np.ndarray				,				a_						:np.ndarray				,				a_						:np.ndarray				,				a_						:np.ndarray | None = None				,				)     ->							np.ndarray:
       lowercase		:str       							=				np.shape(a_)
       lowercase		:Dict       							=				np.shape(a_)
       lowercase		:Optional[int]       							=				np.shape(a_)

       if shape_a[0] != shape_b[0]:
              lowercase		:List[str]       							=				(
                  '''Expected the same number of rows for A and B. '''
                  F"""Instead found A of size {shape_a} and B of size {shape_b}"""
              )
              raise ValueError(a_)

       if shape_b[1] != shape_c[1]:
              lowercase		:List[str]       							=				(
                  '''Expected the same number of columns for B and C. '''
                  F"""Instead found B of size {shape_b} and C of size {shape_c}"""
              )
              raise ValueError(a_)

       lowercase		:Optional[Any]       							=				pseudo_inv
       if a_inv is None:
              try:
                     lowercase		:int       							=				np.linalg.inv(a_)
              except np.linalg.LinAlgError:
                     raise ValueError(
                         '''Input matrix A is not invertible. Cannot compute Schur complement.''')

       return mat_c - mat_b.T @ a_inv @ mat_b
class 					__magic_name__		(			unittest.TestCase     ):






       def  __snake_case						(   self  :    int				):

              '''simple docstring'''






              lowercase		:Dict       							=				np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]				)
              lowercase		:List[str]       							=				np.array([[0, 3], [3, 0], [2, 3]]				)
              lowercase		:List[str]       							=				np.array([[2, 1], [6, 3]]				)

              lowercase		:Any       							=				schur_complement(snake_case__  ,     snake_case__  ,     snake_case__				)

              lowercase		:int       							=				np.block([[a, b], [b.T, c]]				)

              lowercase		:Tuple       							=				np.linalg.det(snake_case__				)
              lowercase		:Any       							=				np.linalg.det(snake_case__				)
              lowercase		:List[Any]       							=				np.linalg.det(snake_case__				)

              self.assertAlmostEqual(snake_case__  ,     det_a * det_s				)






       def  __snake_case						(   self  :    Optional[int]				):

              '''simple docstring'''






              lowercase		:Union[str, Any]       							=				np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]				)
              lowercase		:Union[str, Any]       							=				np.array([[0, 3], [3, 0], [2, 3]]				)
              lowercase		:List[str]       							=				np.array([[2, 1], [6, 3]]				)

              with self.assertRaises(snake_case__				):
                     schur_complement(snake_case__  ,     snake_case__  ,     snake_case__				)







       def  __snake_case						(   self  :    List[Any]				):

              '''simple docstring'''






              lowercase		:List[str]       							=				np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]				)
              lowercase		:int       							=				np.array([[0, 3], [3, 0], [2, 3]]				)
              lowercase		:List[Any]       							=				np.array([[2, 1, 3], [6, 3, 5]]				)

              with self.assertRaises(snake_case__				):
                     schur_complement(snake_case__  ,     snake_case__  ,     snake_case__				)


if __name__ == "__main__":
 import doctest

 doctest.testmod()
 unittest.main()

