import numpy as np



def 							_lowerCamelCase(				__snake_case	)  ->   Union[str, Any]:
		__snake_case       ,       __snake_case		       =     a.shape
		__snake_case		       =     min(__snake_case						,						__snake_case	)
		__snake_case		       =     np.eye(__snake_case	)
		__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(__snake_case	)
				__snake_case		       =     1.0
				# determine scaling factor
				__snake_case		       =     np.linalg.norm(__snake_case	)
				# construct vector v for Householder reflection
				__snake_case		       =     x + np.sign(x[0]	) * alpha * ea
				v /= np.linalg.norm(__snake_case	)

				# 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(__snake_case	), 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()






from __future__ import annotations

from collections.abc import Iterable
from dataclasses import dataclass
from typing import Any




class 					UpperCamelCase	:
					__UpperCamelCase						=   ("""_front""", """_back""", """_len""")




					@dataclass
					class 					UpperCamelCase	:
										__UpperCamelCase						=   None
										__UpperCamelCase						=   None
										__UpperCamelCase						=   None




					class 					UpperCamelCase	:
										__UpperCamelCase						=   ("""_cur""",)




										def __init__( self :					Dict      ,_lowerCAmelCase :					Deque._Node | None						):


												"""simple docstring"""




												__snake_case		       =     cur




										def __iter__( self :					Dict						):


												"""simple docstring"""




												return self







										def 							UpperCamelCase_						( self :					Optional[Any]						):


												"""simple docstring"""




												if self._cur is None:
														# finished iterating
														raise StopIteration
												__snake_case		       =     self._cur.val
												__snake_case		       =     self._cur.next_node

												return val




					def __init__( self :					Optional[int]      ,_lowerCAmelCase :					Iterable[Any] | None = None						):


							"""simple docstring"""




							__snake_case		       =     None
							__snake_case		       =     None
							__snake_case		       =     0

							if iterable is not None:
									# append every value to the deque
									for val in iterable:
											self.append(_lowerCAmelCase						)




					def 							UpperCamelCase_						( self :					int      ,_lowerCAmelCase :					Any						):


							"""simple docstring"""




							__snake_case		       =     self._Node(_lowerCAmelCase      ,_lowerCAmelCase      ,_lowerCAmelCase						)
							if self.is_empty():
									# front = back
									__snake_case		       =     __snake_case		       =     node
									__snake_case		       =     1
							else:
									# connect nodes
									__snake_case		       =     node
									__snake_case		       =     self._back
									__snake_case		       =     node  # assign new back to the new node

									self._len += 1

									# make sure there were no errors
									assert not self.is_empty(), "Error on appending value."




					def 							UpperCamelCase_						( self :					Any      ,_lowerCAmelCase :					Any						):


							"""simple docstring"""




							__snake_case		       =     self._Node(_lowerCAmelCase      ,_lowerCAmelCase      ,_lowerCAmelCase						)
							if self.is_empty():
									# front = back
									__snake_case		       =     __snake_case		       =     node
									__snake_case		       =     1
							else:
									# connect nodes
									__snake_case		       =     self._front
									__snake_case		       =     node
									__snake_case		       =     node  # assign new front to the new node

									self._len += 1

									# make sure there were no errors
									assert not self.is_empty(), "Error on appending value."




					def 							UpperCamelCase_						( self :					int      ,_lowerCAmelCase :					Iterable[Any]						):


							"""simple docstring"""




							for val in iterable:
									self.append(_lowerCAmelCase						)




					def 							UpperCamelCase_						( self :					Optional[int]      ,_lowerCAmelCase :					Iterable[Any]						):


							"""simple docstring"""




							for val in iterable:
									self.appendleft(_lowerCAmelCase						)




					def 							UpperCamelCase_						( self :					List[Any]						):


							"""simple docstring"""




							assert not self.is_empty(), "Deque is empty."

							__snake_case		       =     self._back
							__snake_case		       =     self._back.prev_node  # set new back
							# drop the last node - python will deallocate memory automatically
							__snake_case		       =     None

							self._len -= 1

							return topop.val




					def 							UpperCamelCase_						( self :					List[str]						):


							"""simple docstring"""




							assert not self.is_empty(), "Deque is empty."

							__snake_case		       =     self._front
							__snake_case		       =     self._front.next_node  # set new front and drop the first node
							__snake_case		       =     None

							self._len -= 1

							return topop.val




					def 							UpperCamelCase_						( self :					int						):


							"""simple docstring"""




							return self._front is None




					def __len__( self :					Tuple						):


							"""simple docstring"""




							return self._len




					def __eq__( self :					Union[str, Any]      ,_lowerCAmelCase :					object						):


							"""simple docstring"""




							if not isinstance(_lowerCAmelCase      ,_lowerCAmelCase						):
									return NotImplemented

							__snake_case		       =     self._front
							__snake_case		       =     other._front

							# if the length of the dequeues are not the same, they are not equal
							if len(self						) != len(_lowerCAmelCase						):
									return False

							while me is not None and oth is not None:
									# compare every value
									if me.val != oth.val:
											return False
									__snake_case		       =     me.next_node
									__snake_case		       =     oth.next_node

							return True




					def __iter__( self :					Tuple						):


							"""simple docstring"""




							return Deque._Iterator(self._front						)







					def __repr__( self :					List[Any]						):


							"""simple docstring"""




							__snake_case		       =     []
							__snake_case		       =     self._front
							while aux is not None:
									# append the values in a list to display
									values_list.append(aux.val						)
									__snake_case		       =     aux.next_node

							return F"""[{", ".join(repr(_lowerCAmelCase						) for val in values_list						)}]"""


if __name__ == "__main__":
		import doctest

		doctest.testmod()





