
"""simple docstring"""







import numpy as np




def       A_      (       lowercase    ,			lowercase    ,			lowercase = 1E-12    ,			lowercase = 100    ,			)  ->				tuple[float, np.ndarray]:


     """simple docstring"""







     assert np.shape(lowercase   )[0] == np.shape(lowercase   )[1]
     # Ensure proper dimensionality.
     assert np.shape(lowercase   )[0] == np.shape(lowercase   )[0]
     # Ensure inputs are either both complex or both real
     assert np.iscomplexobj(lowercase   ) == np.iscomplexobj(lowercase   )
     UpperCAmelCase_    :     Tuple            =	np.iscomplexobj(lowercase   )
     if is_complex:
          # Ensure complex input_matrix is Hermitian
          assert np.array_equal(lowercase    ,			input_matrix.conj().T   )

     # Set convergence to False. Will define convergence when we exceed max_iterations
     # or when we have small changes from one iteration to next.

     UpperCAmelCase_    :     Union[str, Any]            =	False
     UpperCAmelCase_    :     Tuple            =	0
     UpperCAmelCase_    :     Optional[Any]            =	0
     UpperCAmelCase_    :     Any            =	1E12

     while not convergence:
          # Multiple matrix by the vector.
          UpperCAmelCase_    :     str            =	np.dot(lowercase    ,			lowercase   )
          # Normalize the resulting output vector.
          UpperCAmelCase_    :     Tuple            =	w / np.linalg.norm(lowercase   )
          # Find rayleigh quotient
          # (faster than usual b/c we know vector is normalized already)
          UpperCAmelCase_    :     Dict            =	vector.conj().T if is_complex else vector.T
          UpperCAmelCase_    :     int            =	np.dot(lowercase    ,			np.dot(lowercase    ,			lowercase   )   )

          # Check convergence.
          UpperCAmelCase_    :     str            =	np.abs(lambda_ - lambda_previous   ) / lambda_
          iterations += 1

          if error <= error_tol or iterations >= max_iterations:
               UpperCAmelCase_    :     str            =	True

          UpperCAmelCase_    :     Any            =	lambda_

     if is_complex:
          UpperCAmelCase_    :     str            =	np.real(lambda_   )

     return lambda_, vector




def       A_      (       )  ->				None:


     """simple docstring"""







     UpperCAmelCase_    :     Optional[int]            =	np.array([[41, 4, 20], [4, 26, 30], [20, 30, 50]]   )
     UpperCAmelCase_    :     Union[str, Any]            =	np.array([41, 4, 20]   )
     UpperCAmelCase_    :     Optional[Any]            =	real_input_matrix.astype(np.complexaaa   )
     UpperCAmelCase_    :     Tuple            =	np.triu(1J * complex_input_matrix    ,			1   )
     complex_input_matrix += imag_matrix
     complex_input_matrix += -1 * imag_matrix.T
     UpperCAmelCase_    :     Optional[Any]            =	np.array([41, 4, 20]   ).astype(np.complexaaa   )

     for problem_type in ["real", "complex"]:
          if problem_type == "real":
               UpperCAmelCase_    :     Optional[int]            =	real_input_matrix
               UpperCAmelCase_    :     List[str]            =	real_vector
          elif problem_type == "complex":
               UpperCAmelCase_    :     int            =	complex_input_matrix
               UpperCAmelCase_    :     Dict            =	complex_vector

          # Our implementation.
          UpperCAmelCase_       ,UpperCAmelCase_    :     Optional[int]            =	power_iteration(lowercase    ,			lowercase   )

          # Numpy implementation.

          # Get eigenvalues and eigenvectors using built-in numpy
          # eigh (eigh used for symmetric or hermetian matrices).
          UpperCAmelCase_       ,UpperCAmelCase_    :     Union[str, Any]            =	np.linalg.eigh(lowercase   )
          # Last eigenvalue is the maximum one.
          UpperCAmelCase_    :     List[Any]            =	eigen_values[-1]
          # Last column in this matrix is eigenvector corresponding to largest eigenvalue.
          UpperCAmelCase_    :     int            =	eigen_vectors[:, -1]

          # Check our implementation and numpy gives close answers.
          assert np.abs(eigen_value - eigen_value_max   ) <= 1E-6
          # Take absolute values element wise of each eigenvector.
          # as they are only unique to a minus sign.
          assert np.linalg.norm(np.abs(lowercase   ) - np.abs(lowercase   )   ) <= 1E-6


if __name__ == "__main__":
       import doctest

       doctest.testmod()
       test_power_iteration()



"""simple docstring"""







class 		UpperCAmelCase_						:
 """simple docstring"""




 def __init__(			self							:			Optional[Any]  , a_							:			int			)->       str:



      """simple docstring"""

      UpperCAmelCase_    :     Any            =	n
      UpperCAmelCase_    :     str            =	[None] * self.n
      UpperCAmelCase_    :     List[Any]            =	0  # index of the first element
      UpperCAmelCase_    :     List[Any]            =	0
      UpperCAmelCase_    :     Dict            =	0




 def __len__(			self							:			Union[str, Any]			)->       int:



      """simple docstring"""

      return self.size




 def 	a   (			self							:			Dict			)->       bool:



      """simple docstring"""

      return self.size == 0




 def 	a   (			self							:			List[Any]			)->       Optional[int]:



      """simple docstring"""

      return False if self.is_empty() else self.array[self.front]




 def 	a   (			self							:			Dict  , a_							:			int			)->       Optional[int]:



      """simple docstring"""

      if self.size >= self.n:
           raise Exception("""QUEUE IS FULL"""			)

      UpperCAmelCase_    :     List[str]            =	data
      UpperCAmelCase_    :     Optional[int]            =	(self.rear + 1) % self.n
      self.size += 1
      return self






 def 	a   (			self							:			Union[str, Any]			)->       List[Any]:



      """simple docstring"""

      if self.size == 0:
           raise Exception("""UNDERFLOW"""			)

      UpperCAmelCase_    :     Tuple            =	self.array[self.front]
      UpperCAmelCase_    :     List[str]            =	None
      UpperCAmelCase_    :     Dict            =	(self.front + 1) % self.n
      self.size -= 1
      return temp

