
"""simple docstring"""

import numpy as np




def       lowercase__  (   lowerCAmelCase			:   List[str]	)	-> Tuple:






     """simple docstring"""




     UpperCAmelCase     ,							UpperCAmelCase       =   a.shape
     UpperCAmelCase       =   min(lowerCAmelCase	,			lowerCAmelCase	)
     UpperCAmelCase       =   np.eye(lowerCAmelCase	)
     UpperCAmelCase       =   a.copy()

     for k in range(t - 1	):
          # select a column of modified matrix A':
          UpperCAmelCase       =   r[k:, [k]]
          # construct first basis vector
          UpperCAmelCase       =   np.zeros_like(lowerCAmelCase	)
          UpperCAmelCase       =   1.0
          # determine scaling factor
          UpperCAmelCase       =   np.linalg.norm(lowerCAmelCase	)
          # construct vector v for Householder reflection
          UpperCAmelCase       =   x + np.sign(x[0]	) * alpha * ea
          v /= np.linalg.norm(lowerCAmelCase	)

          # construct the Householder matrix
          UpperCAmelCase       =   np.eye(m - k	) - 2.0 * v @ v.T
          # pad with ones and zeros as necessary
          UpperCAmelCase       =   np.block([[np.eye(lowerCAmelCase	), np.zeros((k, m - k)	)], [np.zeros((m - k, k)	), q_k]]	)

          UpperCAmelCase       =   q @ q_k.T
          UpperCAmelCase       =   q_k @ r

     return q, r


if __name__ == "__main__":
   import doctest

   doctest.testmod()





"""simple docstring"""

SCREAMING_SNAKE_CASE_         =  {
    0: '''underwear''',
    1: '''pants''',
    2: '''belt''',
    3: '''suit''',
    4: '''shoe''',
    5: '''socks''',
    6: '''shirt''',
    7: '''tie''',
    8: '''watch''',
}

SCREAMING_SNAKE_CASE_         =  [[1, 4], [2, 4], [3], [], [], [4], [2, 7], [3], []]

SCREAMING_SNAKE_CASE_         =  [0 for x in range(len(graph))]
SCREAMING_SNAKE_CASE_         =  []




def       lowercase__  (   lowerCAmelCase			:   Optional[int]	,			lowerCAmelCase			:   Any	)	-> List[str]:






    """simple docstring"""




    UpperCAmelCase       =   1
    while stack:
        UpperCAmelCase       =   stack.pop()
        print(lowerCAmelCase	,			clothes[current_clothing]	)
        order += 1




def       lowercase__  (   lowerCAmelCase			:   List[Any]	,			lowerCAmelCase			:   Any	,			lowerCAmelCase			:   Dict	)	-> Optional[Any]:






    """simple docstring"""




    UpperCAmelCase       =   1
    for v in graph[u]:
        if not visited[v]:
            depth_first_search(lowerCAmelCase	,			lowerCAmelCase	,			lowerCAmelCase	)

    stack.append(lowerCAmelCase	)




def       lowercase__  (   lowerCAmelCase			:   Tuple	,			lowerCAmelCase			:   Dict	)	-> Optional[Any]:






    """simple docstring"""




    for v in range(len(lowerCAmelCase	)	):
        if not visited[v]:
            depth_first_search(lowerCAmelCase	,			lowerCAmelCase	,			lowerCAmelCase	)


if __name__ == "__main__":
   topological_sort(graph, visited)
   print(stack)
   print_stack(stack, clothes)



