
"""simple docstring"""

import numpy as np




def 			lowercase__  (    lowerCAmelCase :    Dict       )       ->					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"""

class    _UpperCAmelCase							:




      def __init__(							self      ,				lowercase_=None							)  ->    Union[str, Any]:
          UpperCAmelCase  	=    data
          UpperCAmelCase  	=    None




      def __repr__(							self							)  ->    Tuple:
          UpperCAmelCase  	=    ''
          UpperCAmelCase  	=    self
          while temp:
              string_rep += F"<{temp.data}> ---> "
              UpperCAmelCase  	=    temp.next
          string_rep += "<END>"
          return string_rep




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






    """simple docstring"""




    if not elements_list:
        raise Exception('The Elements List is empty'       )

    # Set first element as Head
    UpperCAmelCase  	=    Node(elements_list[0]       )
    UpperCAmelCase  	=    head
    # Loop through elements from position 1
    for data in elements_list[1:]:
        UpperCAmelCase  	=    Node(lowerCAmelCase       )
        UpperCAmelCase  	=    current.next
    return head


SCREAMING_SNAKE_CASE_			    =  [1, 3, 5, 32, 44, 12, 43]
print(F'List: {list_data}')
print('''Creating Linked List from List.''')
SCREAMING_SNAKE_CASE_			    =  make_linked_list(list_data)
print('''Linked List:''')







print(linked_list)



