


"""simple docstring"""






import numpy as np





def 							A_					(   snake_case__		)				->     List[str]:
			_UpperCamelCase    ,			_UpperCamelCase				:str				    =  a.shape
			_UpperCamelCase				:Tuple				    =  min(snake_case__					,				snake_case__		)
			_UpperCamelCase				:List[str]				    =  np.eye(snake_case__		)
			_UpperCamelCase				:Tuple				    =  a.copy()

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

						# construct the Householder matrix
						_UpperCamelCase				:Any				    =  np.eye(m - k		) - 2.0 * v @ v.T
						# pad with ones and zeros as necessary
						_UpperCamelCase				:Optional[int]				    =  np.block([[np.eye(snake_case__		), np.zeros((k, m - k)		)], [np.zeros((m - k, k)		), q_k]]		)

						_UpperCamelCase				:Optional[Any]				    =  q @ q_k.T
						_UpperCamelCase				:List[str]				    =  q_k @ r

			return q, r


if __name__ == "__main__":
					import doctest

					doctest.testmod()





"""simple docstring"""






class     A:

		"""simple docstring"""





		def __init__(   self      )			->  List[str]:

					"""simple docstring"""



					_UpperCamelCase				:List[Any]				    =  []
					_UpperCamelCase				:Dict				    =  0
					_UpperCamelCase				:List[Any]				    =  0
		def __str__(   self      )			->  Optional[int]:

					"""simple docstring"""



					_UpperCamelCase				:Optional[Any]				    =  '''<''' + str(self.entries      )[1:-1] + '''>'''
					return printed
		def      _UpperCamelCase(   self				,							SCREAMING_SNAKE_CASE__      )			->  Tuple:

					"""simple docstring"""



					self.entries.append(SCREAMING_SNAKE_CASE__      )
					_UpperCamelCase				:Union[str, Any]				    =  self.length + 1
		def      _UpperCamelCase(   self      )			->  Dict:

					"""simple docstring"""



					_UpperCamelCase				:str				    =  self.length - 1
					_UpperCamelCase				:Optional[int]				    =  self.entries[self.front]
					# self.front-=1
					# self.entries = self.entries[self.front:]
					_UpperCamelCase				:Tuple				    =  self.entries[1:]
					return dequeued
		def      _UpperCamelCase(   self				,							SCREAMING_SNAKE_CASE__      )			->  str:

					"""simple docstring"""



					for _ in range(SCREAMING_SNAKE_CASE__      ):
								self.put(self.get()      )
		def      _UpperCamelCase(   self      )			->  str:

					"""simple docstring"""



					return self.entries[0]
		def      _UpperCamelCase(   self      )			->  Union[str, Any]:

					"""simple docstring"""



					return self.length

