


import numpy as np






def 							_lowercase    (		SCREAMING_SNAKE_CASE_   :		np.ndarray       ,  SCREAMING_SNAKE_CASE_   :		np.ndarray       ,  SCREAMING_SNAKE_CASE_   :		float = 1e-12       ,  SCREAMING_SNAKE_CASE_   :		int = 100       ,  ):





   """simple docstring"""




   assert np.shape(SCREAMING_SNAKE_CASE_	)[0] == np.shape(SCREAMING_SNAKE_CASE_	)[1]
   # Ensure proper dimensionality.
   assert np.shape(SCREAMING_SNAKE_CASE_	)[0] == np.shape(SCREAMING_SNAKE_CASE_	)[0]
   # Ensure inputs are either both complex or both real
   assert np.iscomplexobj(SCREAMING_SNAKE_CASE_	) == np.iscomplexobj(SCREAMING_SNAKE_CASE_	)
   UpperCamelCase         =					np.iscomplexobj(SCREAMING_SNAKE_CASE_	)
   if is_complex:
      # Ensure complex input_matrix is Hermitian
      assert np.array_equal(SCREAMING_SNAKE_CASE_       ,  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         =					False
   UpperCamelCase         =					0
   UpperCamelCase         =					0
   UpperCamelCase         =					1e12

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

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

      if error <= error_tol or iterations >= max_iterations:
         UpperCamelCase         =					True

      UpperCamelCase         =					lambda_

   if is_complex:
      UpperCamelCase         =					np.real(lambda_	)

   return lambda_, vector






def 							_lowercase    (		):





   """simple docstring"""




   UpperCamelCase         =					np.array([[41, 4, 20], [4, 26, 30], [20, 30, 50]]	)
   UpperCamelCase         =					np.array([41, 4, 20]	)
   UpperCamelCase         =					real_input_matrix.astype(np.complexaaa	)
   UpperCamelCase         =					np.triu(1j * complex_input_matrix       ,  1	)
   complex_input_matrix += imag_matrix
   complex_input_matrix += -1 * imag_matrix.T
   UpperCamelCase         =					np.array([41, 4, 20]	).astype(np.complexaaa	)

   for problem_type in ["real", "complex"]:
      if problem_type == "real":
         UpperCamelCase         =					real_input_matrix
         UpperCamelCase         =					real_vector
      elif problem_type == "complex":
         UpperCamelCase         =					complex_input_matrix
         UpperCamelCase         =					complex_vector

      # Our implementation.
      UpperCamelCase  ,  UpperCamelCase         =					power_iteration(SCREAMING_SNAKE_CASE_       ,  SCREAMING_SNAKE_CASE_	)

      # Numpy implementation.

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


if __name__ == "__main__":
   import doctest

   doctest.testmod()
   test_power_iteration()








def 							_lowercase    (		SCREAMING_SNAKE_CASE_   :		int	):





			"""simple docstring"""




			UpperCamelCase         =					int(SCREAMING_SNAKE_CASE_	)
			if decimal in (0, 1):  # Exit cases for the recursion
						return str(SCREAMING_SNAKE_CASE_	)
			UpperCamelCase  ,  UpperCamelCase         =					divmod(SCREAMING_SNAKE_CASE_       ,  2	)
			return binary_recursive(SCREAMING_SNAKE_CASE_	) + str(SCREAMING_SNAKE_CASE_	)






def 							_lowercase    (		SCREAMING_SNAKE_CASE_   :		str	):





			"""simple docstring"""




			UpperCamelCase         =					str(SCREAMING_SNAKE_CASE_	).strip()
			if not number:
						raise ValueError("""No input value was provided"""	)
			UpperCamelCase         =					"""-""" if number.startswith("""-"""	) else """"""
			UpperCamelCase         =					number.lstrip("""-"""	)
			if not number.isnumeric():
						raise ValueError("""Input value is not an integer"""	)
			return f'{negative}0b{binary_recursive(int(SCREAMING_SNAKE_CASE_	)	)}'


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

			testmod()




