




from __future__ import annotations




class 		UpperCamelCase__     :

    '''simple docstring'''



    def __init__(   self					,  UpperCamelCase__					)					->				None:
           lowerCamelCase       :				List[str]      			= value
           lowerCamelCase       :				Node | None      			= None
           lowerCamelCase       :				Node | None      			= None




class 		UpperCamelCase__     :

    '''simple docstring'''



    lowerCamelCase_				:     int







    def __init__(   self					)					->				None:
           lowerCamelCase       :				Any      			= 0







    def    _lowercase					(   self					,  UpperCamelCase__					,  UpperCamelCase__					)					->				None:
           if node is None:
                  return

           if path_sum == self.target:
                  self.paths += 1

           if node.left:
                  self.depth_first_search(node.left					,  path_sum + node.left.value					)
           if node.right:
                  self.depth_first_search(node.right					,  path_sum + node.right.value					)







    def    _lowercase					(   self					,  UpperCamelCase__					,  UpperCamelCase__ = None					)					->				int:
           if node is None:
                  return 0
           if target is not None:
                  lowerCamelCase       :				List[Any]      			= target

           self.depth_first_search(UpperCamelCase__					,  node.value					)
           self.path_sum(node.left					)
           self.path_sum(node.right					)

           return self.paths


if __name__ == "__main__":
       import doctest

       doctest.testmod()












import numpy as np

def 					A				(				_SCREAMING_SNAKE_CASE   )     ->			str:
       lowerCamelCase			,				lowerCamelCase       :				Dict      			= a.shape
       lowerCamelCase       :				Union[str, Any]      			= min(_SCREAMING_SNAKE_CASE			,_SCREAMING_SNAKE_CASE   )
       lowerCamelCase       :				Union[str, Any]      			= np.eye(_SCREAMING_SNAKE_CASE   )
       lowerCamelCase       :				List[Any]      			= a.copy()

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

              # construct the Householder matrix
              lowerCamelCase       :				List[Any]      			= np.eye(m - k   ) - 2.0 * v @ v.T
              # pad with ones and zeros as necessary
              lowerCamelCase       :				Union[str, Any]      			= np.block([[np.eye(_SCREAMING_SNAKE_CASE   ), np.zeros((k, m - k)   )], [np.zeros((m - k, k)   ), q_k]]   )

              lowerCamelCase       :				Any      			= q @ q_k.T
              lowerCamelCase       :				List[Any]      			= q_k @ r

       return q, r


if __name__ == "__main__":
       import doctest

       doctest.testmod()






