
'''simple docstring'''






import numpy as np







def 		SCREAMING_SNAKE_CASE__					(							_SCREAMING_SNAKE_CASE  ):
		lowerCAmelCase_							,		lowerCAmelCase_   :       Optional[Any]    							=a.shape
		lowerCAmelCase_   :       Dict    							=min(_SCREAMING_SNAKE_CASE  ,					_SCREAMING_SNAKE_CASE  )
		lowerCAmelCase_   :       Union[str, Any]    							=np.eye(_SCREAMING_SNAKE_CASE  )
		lowerCAmelCase_   :       int    							=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[int]    							=np.zeros_like(_SCREAMING_SNAKE_CASE  )
				lowerCAmelCase_   :       Union[str, Any]    							=1.0
				# determine scaling factor
				lowerCAmelCase_   :       Dict    							=np.linalg.norm(_SCREAMING_SNAKE_CASE  )
				# construct vector v for Householder reflection
				lowerCAmelCase_   :       str    							=x + np.sign(x[0]  ) * alpha * ea
				v /= np.linalg.norm(_SCREAMING_SNAKE_CASE  )

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

				lowerCAmelCase_   :       Dict    							=q @ q_k.T
				lowerCAmelCase_   :       Any    							=q_k @ r

		return q, r


if __name__ == "__main__":
							import doctest

							doctest.testmod()








'''simple docstring'''






from __future__ import annotations







def 		SCREAMING_SNAKE_CASE__					(							_SCREAMING_SNAKE_CASE  ,					_SCREAMING_SNAKE_CASE  ,					_SCREAMING_SNAKE_CASE  ):
		if len(_SCREAMING_SNAKE_CASE  ) == 0:
				raise ValueError('''find_min() arg is an empty sequence'''  )
		if (
		    left >= len(_SCREAMING_SNAKE_CASE  )
		    or left < -len(_SCREAMING_SNAKE_CASE  )
		    or right >= len(_SCREAMING_SNAKE_CASE  )
		    or right < -len(_SCREAMING_SNAKE_CASE  )
		):
				raise IndexError('''list index out of range'''  )
		if left == right:
				return nums[left]
		lowerCAmelCase_   :       Dict    							=(left + right) >> 1  # the middle
		lowerCAmelCase_   :       Optional[int]    							=find_min(_SCREAMING_SNAKE_CASE  ,					_SCREAMING_SNAKE_CASE  ,					_SCREAMING_SNAKE_CASE  )  # find min in range[left, mid]
		lowerCAmelCase_   :       Any    							=find_min(_SCREAMING_SNAKE_CASE  ,					mid + 1  ,					_SCREAMING_SNAKE_CASE  )  # find min in range[mid + 1, right]

		return left_min if left_min <= right_min else right_min


if __name__ == "__main__":
							import doctest

							doctest.testmod(verbose=True)






