
import numpy as np



def      lowerCAmelCase	(						lowerCAmelCase_	)->	Optional[Any]:
				lowerCAmelCase_	: int      			=			a.shape
				lowerCAmelCase_	: Optional[int]      			=			min(lowerCAmelCase_	,				lowerCAmelCase_	)
				lowerCAmelCase_	: Dict      			=			np.eye(lowerCAmelCase_	)
				lowerCAmelCase_	: Dict      			=			a.copy()

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

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

								lowerCAmelCase_	: int      			=			q @ q_k.T
								lowerCAmelCase_	: Optional[Any]      			=			q_k @ r

				return q, r


if __name__ == "__main__":
						import doctest

						doctest.testmod()



import math







class 			snake_case__:







						'''simple docstring'''




						def __init__(			self				,  __lowercase				,  __lowercase				,  __lowercase							)   ->      Tuple:
										lowerCAmelCase_	: List[str]      			=			x
										lowerCAmelCase_	: Dict      			=			y
										lowerCAmelCase_	: List[Any]      			=			z




						def __repr__(			self							)   ->      str:
										return f"""Point({self.x}, {self.y}, {self.z})"""






def      lowerCAmelCase	(						lowerCAmelCase_	,				lowerCAmelCase_	)->	float:
				return math.sqrt(abs((b.x - a.x) ** 2 + (b.y - a.y) ** 2 + (b.z - a.z) ** 2	)	)


if __name__ == "__main__":
						import doctest

						doctest.testmod()