import numpy as np

def 	__lowerCamelCase							(				__a    : Any		)      ->       Union[str, Any]:
	_lowercase   ,  _lowercase   =a.shape
	_lowercase   =min(__a      ,							__a		)
	_lowercase   =np.eye(__a		)
	_lowercase   =a.copy()

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

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

		_lowercase   =q @ q_k.T
		_lowercase   =q_k @ r

	return q, r


if __name__ == "__main__":
						import doctest

						doctest.testmod()





def 	__lowerCamelCase							(				__a    : float		)      ->       float:
	return -num if num < 0 else num

def 	__lowerCamelCase							(				__a    : list[int]		)      ->       int:
	if len(__a		) == 0:
		raise ValueError("abs_min() arg is an empty sequence"		)
	_lowercase   =x[0]
	for i in x:
		if abs_val(__a		) < abs_val(__a		):
			_lowercase   =i
	return j

def 	__lowerCamelCase							(				__a    : list[int]		)      ->       int:
	if len(__a		) == 0:
		raise ValueError("abs_max() arg is an empty sequence"		)
	_lowercase   =x[0]
	for i in x:
		if abs(__a		) > abs(__a		):
			_lowercase   =i
	return j

def 	__lowerCamelCase							(				__a    : list[int]		)      ->       int:
	if len(__a		) == 0:
		raise ValueError("abs_max_sort() arg is an empty sequence"		)
	return sorted(__a      ,							key=__a		)[-1]

def 	__lowerCamelCase							(				)      ->       Optional[Any]:
	assert abs_val(0		) == 0
	assert abs_val(34		) == 34
	assert abs_val(-100_000_000_000		) == 100_000_000_000

	_lowercase   =[-3, -1, 2, -11]
	assert abs_max(__a		) == -11
	assert abs_max_sort(__a		) == -11
	assert abs_min(__a		) == -1


if __name__ == "__main__":
						import doctest

						doctest.testmod()

						test_abs_val()
						print(abs_val(-3_4))  # --> 34




