






"""simple docstring"""



import numpy as np
def 		_SCREAMING_SNAKE_CASE					(       __snake_case     :    Dict       ):





  '''simple docstring'''





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

    # 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(lowercase__       ), 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()




"""simple docstring"""






from __future__ import annotations






def 		_snake_case			(      lowercase__						:    list[int | float]						)    ->							int | float:


			'''simple docstring'''





			if len(lowercase__						) == 0:
						raise ValueError("""find_min() arg is an empty sequence"""						)
			lowerCAmelCase_       :Optional[int]     	=    nums[0]
			for num in nums:
						lowerCAmelCase_       :Optional[Any]     	=    min(lowercase__ ,	lowercase__						)
			return min_num


if __name__ == "__main__":
				import doctest

				doctest.testmod(verbose=True)






