
import numpy as np






def 							UpperCAmelCase_		(							__SCREAMING_SNAKE_CASE							):
	lowercase  , lowercase					  =   a.shape
	lowercase					  =   min(__SCREAMING_SNAKE_CASE			,							__SCREAMING_SNAKE_CASE							)
	lowercase					  =   np.eye(__SCREAMING_SNAKE_CASE							)
	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(__SCREAMING_SNAKE_CASE							)
		lowercase					  =   1.0
		# determine scaling factor
		lowercase					  =   np.linalg.norm(__SCREAMING_SNAKE_CASE							)
		# construct vector v for Householder reflection
		lowercase					  =   x + np.sign(x[0]							) * alpha * ea
		v /= np.linalg.norm(__SCREAMING_SNAKE_CASE							)

		# 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(__SCREAMING_SNAKE_CASE							), 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()



import unittest
from timeit import timeit






def 							UpperCAmelCase_		(							__SCREAMING_SNAKE_CASE			,							__SCREAMING_SNAKE_CASE							):
	lowercase					  =   first_num if first_num >= second_num else second_num
	lowercase					  =   max_num
	while (common_mult % first_num > 0) or (common_mult % second_num > 0):
		common_mult += max_num
	return common_mult






def 							UpperCAmelCase_		(							__SCREAMING_SNAKE_CASE			,							__SCREAMING_SNAKE_CASE							):
	return b if a == 0 else greatest_common_divisor(b % a			,							__SCREAMING_SNAKE_CASE							)






def 							UpperCAmelCase_		(							__SCREAMING_SNAKE_CASE			,							__SCREAMING_SNAKE_CASE							):
	return first_num // greatest_common_divisor(__SCREAMING_SNAKE_CASE			,							__SCREAMING_SNAKE_CASE							) * second_num






def 							UpperCAmelCase_		(							):
	lowercase					  =   (
	    'from __main__ import least_common_multiple_slow, least_common_multiple_fast'
	)
	print(
	    'least_common_multiple_slow():'			,							timeit('least_common_multiple_slow(1000, 999)'			,							setup=__SCREAMING_SNAKE_CASE							)			,							)
	print(
	    'least_common_multiple_fast():'			,							timeit('least_common_multiple_fast(1000, 999)'			,							setup=__SCREAMING_SNAKE_CASE							)			,							)







class 	A_       (      unittest.TestCase       ):




	'''simple docstring'''


	_UpperCamelCase					:					Any										=	(
	    (10, 20),
	    (13, 15),
	    (4, 31),
	    (10, 42),
	    (43, 34),
	    (5, 12),
	    (12, 25),
	    (10, 25),
	    (6, 9),
	)
	_UpperCamelCase					:					int										=	(20, 195, 124, 210, 1462, 60, 300, 50, 18)





	def      SCREAMING_SNAKE_CASE__						(	self	):
		for i, (first_num, second_num) in enumerate(self.test_inputs	):
			lowercase					  =   least_common_multiple_slow(snake_case		,   snake_case	)
			lowercase					  =   least_common_multiple_fast(snake_case		,   snake_case	)
			with self.subTest(i=snake_case	):
				self.assertEqual(snake_case		,   self.expected_results[i]	)
				self.assertEqual(snake_case		,   self.expected_results[i]	)


if __name__ == "__main__":
	benchmark()
	unittest.main()

