




'''simple docstring'''







class 	lowerCamelCase_				:




					'''simple docstring'''






					def __init__(       self,    __UpperCamelCase					):
							_snake_case      			=			data
							_snake_case      			=			42
							_snake_case      			=			42






def 						_lowercase						(      lowercase_    ):



		'''simple docstring'''







		_snake_case      			=			0
		_snake_case      			=			x






def 						_lowercase						(      lowercase_		,				lowercase_    ):



		'''simple docstring'''







		_snake_case			,       _snake_case      			=			find_set(lowercase_    ), find_set(lowercase_    )
		if x == y:
				return

		elif x.rank > y.rank:
				_snake_case      			=			x
		else:
				_snake_case      			=			y
				if x.rank == y.rank:
						y.rank += 1






def 						_lowercase						(      lowercase_    ):



		'''simple docstring'''







		if x != x.parent:
				_snake_case      			=			find_set(x.parent    )
		return x.parent






def 						_lowercase						(      lowercase_    ):



		'''simple docstring'''







		_snake_case      			=			({0, 1, 2}, {3, 4, 5})
		for s in sets:
				if node.data in s:
						return s
		_snake_case      			=			f'''{node.data} is not in {sets}'''
		raise ValueError(lowercase_    )






def 						_lowercase						(      ):



		'''simple docstring'''







		_snake_case      			=			[Node(lowercase_    ) for i in range(6    )]
		for v in vertex:
				make_set(lowercase_    )

		union_set(vertex[0]		,				vertex[1]    )
		union_set(vertex[1]		,				vertex[2]    )
		union_set(vertex[3]		,				vertex[4]    )
		union_set(vertex[3]		,				vertex[5]    )

		for nodea in vertex:
				for nodea in vertex:
						if find_python_set(lowercase_    ).isdisjoint(find_python_set(lowercase_    )    ):
								assert find_set(lowercase_    ) != find_set(lowercase_    )
						else:
								assert find_set(lowercase_    ) == find_set(lowercase_    )


if __name__ == "__main__":
						test_disjoint_set()











'''simple docstring'''







import numpy as np






def 						_lowercase						(      lowercase_    ):



		'''simple docstring'''







		_snake_case			,       _snake_case      			=			a.shape
		_snake_case      			=			min(lowercase_		,				lowercase_    )
		_snake_case      			=			np.eye(lowercase_    )
		_snake_case      			=			a.copy()

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

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

				_snake_case      			=			q @ q_k.T
				_snake_case      			=			q_k @ r

		return q, r


if __name__ == "__main__":
						import doctest

						doctest.testmod()





