





'''simple docstring'''






import numpy as np
def        UpperCamelCase__				(		lowerCAmelCase							,     lowerCAmelCase							,     lowerCAmelCase = 1e-12							,     lowerCAmelCase = 1_00							,     ):





      """simple docstring"""


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

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

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

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

            _lowerCAmelCase												=							lambda_

      if is_complex:
            _lowerCAmelCase												=							np.real(lambda_  )

      return lambda_, vector
def        UpperCamelCase__				(		):





      """simple docstring"""


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

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

            # Our implementation.
            _lowerCAmelCase					,							_lowerCAmelCase												=							power_iteration(lowerCAmelCase							,     lowerCAmelCase  )

            # Numpy implementation.

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


if __name__ == "__main__":
      import doctest

      doctest.testmod()
      test_power_iteration()













'''simple docstring'''






def        UpperCamelCase__				(		lowerCAmelCase = 4_00_00_00  ):





      """simple docstring"""


      _lowerCAmelCase												=							[]
      _lowerCAmelCase					,							_lowerCAmelCase												=							0, 1
      while b <= n:
            if b % 2 == 0:
                  even_fibs.append(lowerCAmelCase  )
            _lowerCAmelCase					,							_lowerCAmelCase												=							b, a + b
      return sum(lowerCAmelCase  )


if __name__ == "__main__":
      print(F"""{solution() = }""")






