
"""simple docstring"""





from __future__ import annotations

from collections.abc import Iterator
from typing import Any


class A_      :




  '''simple docstring'''







  def __init__(						self							,  lowercase_    ):

     """simple docstring"""
     UpperCAmelCase_   :   Any     	=       data
     UpperCAmelCase_   :   Node | None     	=       None







  def __str__(						self    ):

     """simple docstring"""
     return F"""{self.data}"""




class A_      :




  '''simple docstring'''







  def __init__(						self    ):

     """simple docstring"""
     UpperCAmelCase_   :   Node | None     	=       None
     UpperCAmelCase_   :   Node | None     	=       None







  def __iter__(						self    ):

     """simple docstring"""
     UpperCAmelCase_   :   List[str]     	=       self.front
     while node:
        yield node.data
        UpperCAmelCase_   :   int     	=       node.next







  def __len__(						self    ):

     """simple docstring"""
     return len(tuple(iter(self    )    )    )







  def __str__(						self    ):

     """simple docstring"""
     return " <- ".join(str(lowercase_    ) for item in self    )







  def 			UpperCamelCase__   (						self    ):

     """simple docstring"""
     return len(self    ) == 0







  def 			UpperCamelCase__   (						self							,  lowercase_    ):

     """simple docstring"""
     UpperCAmelCase_   :   List[str]     	=       Node(lowercase_    )
     if self.is_empty():
        UpperCAmelCase_   :   Any     	=       node
     else:
        assert isinstance(self.rear							,  lowercase_    )
        UpperCAmelCase_   :   Union[str, Any]     	=       node
        UpperCAmelCase_   :   str     	=       node







  def 			UpperCamelCase__   (						self    ):

     """simple docstring"""
     if self.is_empty():
        raise IndexError("dequeue from empty queue"    )
     assert isinstance(self.front							,  lowercase_    )
     UpperCAmelCase_   :   int     	=       self.front
     UpperCAmelCase_   :   str     	=       self.front.next
     if self.front is None:
        UpperCAmelCase_   :   List[str]     	=       None
     return node.data







  def 			UpperCamelCase__   (						self    ):

     """simple docstring"""
     UpperCAmelCase_   :   Any     	=       None


if __name__ == "__main__":
    from doctest import testmod

    testmod()








'''simple docstring'''
import numpy as np
def 							snake_case_    (  a__						:      Tuple     ):



    """simple docstring"""




    __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()




