

import numpy as np
def   a   (    A__			:     Optional[Any]   ,     A__			:     Tuple   ,     A__			:     Union[str, Any] = 1e-12   ,     A__			:     Tuple = 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  )
      _lowercase			 =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.

      _lowercase			 =False
      _lowercase			 =0
      _lowercase			 =0
      _lowercase			 =1e12

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

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

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

            _lowercase			 =lambda_

      if is_complex:
            _lowercase			 =np.real(lambda_  )

      return lambda_, vector
def   a   (    )					->   None:





      """simple docstring"""



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

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

            # Our implementation.
            _lowercase			 =power_iteration(_lowercase   ,     _lowercase  )

            # Numpy implementation.

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





def   a   (    A__			:     Optional[int]  )					->   Tuple:





      """simple docstring"""



      _lowercase			 =[0] * len(A__  )
      _lowercase			 =[]
      _lowercase			 =[]
      _lowercase			 =0

      for values in graph.values():
            for i in values:
                  indegree[i] += 1

      for i in range(len(A__  )  ):
            if indegree[i] == 0:
                  queue.append(A__  )

      while queue:
            _lowercase			 =queue.pop(0  )
            cnt += 1
            topo.append(A__  )
            for x in graph[vertex]:
                  indegree[x] -= 1
                  if indegree[x] == 0:
                        queue.append(A__  )

      if cnt != len(A__  ):
            print('Cycle exists'  )
      else:
            print(A__  )


# Adjacency List of Graph
lowercase_             =     {0: [1, 2], 1: [3], 2: [3], 3: [4, 5], 4: [], 5: []}







topological_sort(graph)


