
import numpy as np


def       UpperCAmelCase_					(  __lowerCAmelCase		)	->  str:
    __lowercase      , __lowercase						:     List[Any]		      =    a.shape
    __lowercase						:     Optional[Any]		      =    min(lowercase_		,   lowercase_		)
    __lowercase						:     List[Any]		      =    np.eye(lowercase_		)
    __lowercase						:     int		      =    a.copy()

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

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

        __lowercase						:     List[str]		      =    q @ q_k.T
        __lowercase						:     List[str]		      =    q_k @ r

    return q, r


if __name__ == "__main__":
 import doctest

 doctest.testmod()




from ..utils import DummyObject, requires_backends


class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     List[Any]       				=      ['''tokenizers''']




      def __init__( self						:    Any			,    *_snake_case						:    Optional[int]			,    **_snake_case						:    List[str]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Dict       				=      ['''tokenizers''']




      def __init__( self						:    Tuple			,    *_snake_case						:    List[Any]			,    **_snake_case						:    Tuple			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Optional[int]       				=      ['''tokenizers''']




      def __init__( self						:    List[Any]			,    *_snake_case						:    List[Any]			,    **_snake_case						:    List[str]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     List[str]       				=      ['''tokenizers''']




      def __init__( self						:    Optional[int]			,    *_snake_case						:    List[Any]			,    **_snake_case						:    int			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Dict       				=      ['''tokenizers''']




      def __init__( self						:    Optional[int]			,    *_snake_case						:    List[str]			,    **_snake_case						:    int			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Optional[int]       				=      ['''tokenizers''']




      def __init__( self						:    Optional[int]			,    *_snake_case						:    List[str]			,    **_snake_case						:    List[str]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Any       				=      ['''tokenizers''']




      def __init__( self						:    List[str]			,    *_snake_case						:    Union[str, Any]			,    **_snake_case						:    Optional[int]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Any       				=      ['''tokenizers''']




      def __init__( self						:    Optional[Any]			,    *_snake_case						:    Tuple			,    **_snake_case						:    List[Any]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     str       				=      ['''tokenizers''']




      def __init__( self						:    Tuple			,    *_snake_case						:    List[str]			,    **_snake_case						:    Dict			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Optional[Any]       				=      ['''tokenizers''']




      def __init__( self						:    Dict			,    *_snake_case						:    Optional[int]			,    **_snake_case						:    int			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Optional[Any]       				=      ['''tokenizers''']




      def __init__( self						:    Any			,    *_snake_case						:    Optional[int]			,    **_snake_case						:    Optional[Any]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     str       				=      ['''tokenizers''']




      def __init__( self						:    Optional[Any]			,    *_snake_case						:    Tuple			,    **_snake_case						:    str			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     int       				=      ['''tokenizers''']




      def __init__( self						:    Optional[Any]			,    *_snake_case						:    List[Any]			,    **_snake_case						:    str			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Any       				=      ['''tokenizers''']




      def __init__( self						:    int			,    *_snake_case						:    Dict			,    **_snake_case						:    Optional[Any]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     List[Any]       				=      ['''tokenizers''']




      def __init__( self						:    int			,    *_snake_case						:    Optional[Any]			,    **_snake_case						:    List[Any]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Optional[int]       				=      ['''tokenizers''']




      def __init__( self						:    Optional[Any]			,    *_snake_case						:    Dict			,    **_snake_case						:    Union[str, Any]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Optional[Any]       				=      ['''tokenizers''']




      def __init__( self						:    List[str]			,    *_snake_case						:    Union[str, Any]			,    **_snake_case						:    List[Any]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Optional[Any]       				=      ['''tokenizers''']




      def __init__( self						:    List[str]			,    *_snake_case						:    int			,    **_snake_case						:    Optional[Any]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Dict       				=      ['''tokenizers''']




      def __init__( self						:    List[Any]			,    *_snake_case						:    str			,    **_snake_case						:    Any			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Optional[int]       				=      ['''tokenizers''']




      def __init__( self						:    Any			,    *_snake_case						:    List[Any]			,    **_snake_case						:    List[str]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     int       				=      ['''tokenizers''']




      def __init__( self						:    List[str]			,    *_snake_case						:    Dict			,    **_snake_case						:    Any			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     List[Any]       				=      ['''tokenizers''']




      def __init__( self						:    Optional[Any]			,    *_snake_case						:    Union[str, Any]			,    **_snake_case						:    Dict			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     List[str]       				=      ['''tokenizers''']




      def __init__( self						:    List[Any]			,    *_snake_case						:    List[str]			,    **_snake_case						:    Optional[Any]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Any       				=      ['''tokenizers''']




      def __init__( self						:    Union[str, Any]			,    *_snake_case						:    int			,    **_snake_case						:    Any			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     List[str]       				=      ['''tokenizers''']




      def __init__( self						:    Union[str, Any]			,    *_snake_case						:    str			,    **_snake_case						:    Union[str, Any]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Tuple       				=      ['''tokenizers''']




      def __init__( self						:    Dict			,    *_snake_case						:    Any			,    **_snake_case						:    Tuple			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Dict       				=      ['''tokenizers''']




      def __init__( self						:    int			,    *_snake_case						:    List[str]			,    **_snake_case						:    List[str]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Optional[int]       				=      ['''tokenizers''']




      def __init__( self						:    List[str]			,    *_snake_case						:    Optional[Any]			,    **_snake_case						:    Tuple			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     List[str]       				=      ['''tokenizers''']




      def __init__( self						:    Dict			,    *_snake_case						:    List[str]			,    **_snake_case						:    Union[str, Any]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Union[str, Any]       				=      ['''tokenizers''']




      def __init__( self						:    Dict			,    *_snake_case						:    Union[str, Any]			,    **_snake_case						:    Any			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Dict       				=      ['''tokenizers''']




      def __init__( self						:    Any			,    *_snake_case						:    Optional[int]			,    **_snake_case						:    str			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     List[Any]       				=      ['''tokenizers''']




      def __init__( self						:    str			,    *_snake_case						:    List[str]			,    **_snake_case						:    Union[str, Any]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Union[str, Any]       				=      ['''tokenizers''']




      def __init__( self						:    Dict			,    *_snake_case						:    Dict			,    **_snake_case						:    Any			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     str       				=      ['''tokenizers''']




      def __init__( self						:    Tuple			,    *_snake_case						:    Dict			,    **_snake_case						:    Any			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Any       				=      ['''tokenizers''']




      def __init__( self						:    Optional[int]			,    *_snake_case						:    List[Any]			,    **_snake_case						:    Tuple			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Union[str, Any]       				=      ['''tokenizers''']




      def __init__( self						:    Dict			,    *_snake_case						:    str			,    **_snake_case						:    Any			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Union[str, Any]       				=      ['''tokenizers''']




      def __init__( self						:    Optional[Any]			,    *_snake_case						:    Dict			,    **_snake_case						:    List[str]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     int       				=      ['''tokenizers''']




      def __init__( self						:    Any			,    *_snake_case						:    List[str]			,    **_snake_case						:    str			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Union[str, Any]       				=      ['''tokenizers''']




      def __init__( self						:    Tuple			,    *_snake_case						:    Dict			,    **_snake_case						:    Any			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Dict       				=      ['''tokenizers''']




      def __init__( self						:    Union[str, Any]			,    *_snake_case						:    Any			,    **_snake_case						:    Optional[int]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     List[str]       				=      ['''tokenizers''']




      def __init__( self						:    Optional[int]			,    *_snake_case						:    List[Any]			,    **_snake_case						:    Any			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     List[str]       				=      ['''tokenizers''']




      def __init__( self						:    Tuple			,    *_snake_case						:    str			,    **_snake_case						:    int			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     str       				=      ['''tokenizers''']




      def __init__( self						:    Dict			,    *_snake_case						:    Tuple			,    **_snake_case						:    Dict			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     List[Any]       				=      ['''tokenizers''']




      def __init__( self						:    List[str]			,    *_snake_case						:    str			,    **_snake_case						:    List[str]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Union[str, Any]       				=      ['''tokenizers''']




      def __init__( self						:    Optional[int]			,    *_snake_case						:    List[Any]			,    **_snake_case						:    List[str]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Dict       				=      ['''tokenizers''']




      def __init__( self						:    str			,    *_snake_case						:    Tuple			,    **_snake_case						:    Optional[Any]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     int       				=      ['''tokenizers''']




      def __init__( self						:    Tuple			,    *_snake_case						:    Any			,    **_snake_case						:    List[Any]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     int       				=      ['''tokenizers''']




      def __init__( self						:    Any			,    *_snake_case						:    Optional[Any]			,    **_snake_case						:    List[Any]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Union[str, Any]       				=      ['''tokenizers''']




      def __init__( self						:    Dict			,    *_snake_case						:    str			,    **_snake_case						:    Optional[Any]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Dict       				=      ['''tokenizers''']




      def __init__( self						:    Any			,    *_snake_case						:    str			,    **_snake_case						:    Tuple			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Dict       				=      ['''tokenizers''']




      def __init__( self						:    Dict			,    *_snake_case						:    List[Any]			,    **_snake_case						:    Optional[int]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Optional[int]       				=      ['''tokenizers''']




      def __init__( self						:    Optional[Any]			,    *_snake_case						:    Dict			,    **_snake_case						:    Optional[int]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Dict       				=      ['''tokenizers''']




      def __init__( self						:    Tuple			,    *_snake_case						:    Dict			,    **_snake_case						:    int			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     str       				=      ['''tokenizers''']




      def __init__( self						:    Optional[int]			,    *_snake_case						:    List[str]			,    **_snake_case						:    str			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     List[str]       				=      ['''tokenizers''']




      def __init__( self						:    int			,    *_snake_case						:    str			,    **_snake_case						:    Union[str, Any]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Dict       				=      ['''tokenizers''']




      def __init__( self						:    Tuple			,    *_snake_case						:    Optional[Any]			,    **_snake_case						:    Optional[Any]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Dict       				=      ['''tokenizers''']




      def __init__( self						:    Dict			,    *_snake_case						:    Tuple			,    **_snake_case						:    Optional[Any]			):
          requires_backends(self			,    ['''tokenizers''']			)






class   __lowerCAmelCase		(		metaclass=lowerCAmelCase_ ):



      """simple docstring"""







      A__		:     Optional[Any]       				=      ['''tokenizers''']




      def __init__( self						:    List[str]			,    *_snake_case						:    Optional[int]			,    **_snake_case						:    List[str]			):
          requires_backends(self			,    ['''tokenizers''']			)


