


import numpy as np


def    __lowerCamelCase   (      A__						: Optional[int]						,						A__						: List[Any]						,						A__						: List[str] = 1e-12						,						A__						: Union[str, Any] = 100						,						)     ->    Tuple:
   assert np.shape(A__					)[0] == np.shape(A__					)[1]
   # Ensure proper dimensionality.
   assert np.shape(A__					)[0] == np.shape(A__					)[0]
   # Ensure inputs are either both complex or both real
   assert np.iscomplexobj(A__					) == np.iscomplexobj(A__					)
   lowerCamelCase_						:   List[str]		= np.iscomplexobj(A__					)
   if is_complex:
      # Ensure complex input_matrix is Hermitian
      assert np.array_equal(A__						,						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.

   lowerCamelCase_						:   Any		= False
   lowerCamelCase_						:   List[Any]		= 0
   lowerCamelCase_						:   str		= 0
   lowerCamelCase_						:   Any		= 1e12

   while not convergence:
      # Multiple matrix by the vector.
      lowerCamelCase_						:   Optional[Any]		= np.dot(A__						,						A__					)
      # Normalize the resulting output vector.
      lowerCamelCase_						:   str		= w / np.linalg.norm(A__					)
      # Find rayleigh quotient
      # (faster than usual b/c we know vector is normalized already)
      lowerCamelCase_						:   str		= vector.conj().T if is_complex else vector.T
      lowerCamelCase_						:   str		= np.dot(A__						,						np.dot(A__						,						A__					)					)

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

      if error <= error_tol or iterations >= max_iterations:
         lowerCamelCase_						:   Any		= True

      lowerCamelCase_						:   Optional[int]		= lambda_

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

   return lambda_, vector


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

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

      # Our implementation.
      lowerCamelCase_						:   Optional[Any]		= power_iteration(A__						,						A__					)

      # Numpy implementation.

      # Get eigenvalues and eigenvectors using built-in numpy
      # eigh (eigh used for symmetric or hermetian matrices).
      lowerCamelCase_						:   Any		= np.linalg.eigh(A__					)
      # Last eigenvalue is the maximum one.
      lowerCamelCase_						:   Dict		= eigen_values[-1]
      # Last column in this matrix is eigenvector corresponding to largest eigenvalue.
      lowerCamelCase_						:   List[str]		= 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(A__					) - np.abs(A__					)					) <= 1e-6


if __name__ == "__main__":
      import doctest

      doctest.testmod()
      test_power_iteration()













SCREAMING_SNAKE_CASE__   :      Tuple				=		{
    """a""": """AAAAA""",
    """b""": """AAAAB""",
    """c""": """AAABA""",
    """d""": """AAABB""",
    """e""": """AABAA""",
    """f""": """AABAB""",
    """g""": """AABBA""",
    """h""": """AABBB""",
    """i""": """ABAAA""",
    """j""": """BBBAA""",
    """k""": """ABAAB""",
    """l""": """ABABA""",
    """m""": """ABABB""",
    """n""": """ABBAA""",
    """o""": """ABBAB""",
    """p""": """ABBBA""",
    """q""": """ABBBB""",
    """r""": """BAAAA""",
    """s""": """BAAAB""",
    """t""": """BAABA""",
    """u""": """BAABB""",
    """v""": """BBBAB""",
    """w""": """BABAA""",
    """x""": """BABAB""",
    """y""": """BABBA""",
    """z""": """BABBB""",
    """ """: """ """,
}


SCREAMING_SNAKE_CASE__   :      Union[str, Any]				=		{value: key for key, value in encode_dict.items()}

def   __lowercase       (			snake_case			):







				"""simple docstring"""






				__magic_name__		:Tuple   =   ''''''
				for letter in word.lower():
								if letter.isalpha() or letter == " ":
												encoded += encode_dict[letter]
								else:
												raise Exception('''encode() accepts only letters of the alphabet and spaces'''			)
				return encoded

def   __lowercase       (			snake_case			):







				"""simple docstring"""






				if set(snake_case			) - {"A", "B", " "} != set():
								raise Exception('''decode() accepts only \'A\', \'B\' and spaces'''			)
				__magic_name__		:Dict   =   ''''''
				for word in coded.split():
								while len(snake_case			) != 0:
												decoded += decode_dict[word[:5]]
												__magic_name__		:int   =   word[5:]
								decoded += " "
				return decoded.strip()


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

					testmod()






