

'''simple docstring'''







def    UpperCAmelCase__     (   UpperCAmelCase__					)      ->      bool:
       A_						       =   0
       for ch in input_str:
              A_						       =   ord(UpperCAmelCase__					)
              A_						       =   pow(2,							UpperCAmelCase__					)

              # If we already turned on bit for current character's unicode
              if bitmap >> ch_unicode & 1 == 1:
                     return False
              bitmap |= ch_bit_index_on
       return True


if __name__ == "__main__":
      import doctest

      doctest.testmod()



'''simple docstring'''







import numpy as np
def    UpperCAmelCase__     (   UpperCAmelCase__,							UpperCAmelCase__,							UpperCAmelCase__ = 1e-12,							UpperCAmelCase__ = 1_00,							)      ->      tuple[float, np.ndarray]:
       assert np.shape(UpperCAmelCase__					)[0] == np.shape(UpperCAmelCase__					)[1]
       # Ensure proper dimensionality.
       assert np.shape(UpperCAmelCase__					)[0] == np.shape(UpperCAmelCase__					)[0]
       # Ensure inputs are either both complex or both real
       assert np.iscomplexobj(UpperCAmelCase__					) == np.iscomplexobj(UpperCAmelCase__					)
       A_						       =   np.iscomplexobj(UpperCAmelCase__					)
       if is_complex:
              # Ensure complex input_matrix is Hermitian
              assert np.array_equal(UpperCAmelCase__,							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.

       A_						       =   False
       A_						       =   0
       A_						       =   0
       A_						       =   1e12

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

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

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

              A_						       =   lambda_

       if is_complex:
              A_						       =   np.real(lambda_					)

       return lambda_, vector
def    UpperCAmelCase__     (   )      ->      None:
       A_						       =   np.array([[41, 4, 20], [4, 26, 30], [20, 30, 50]]					)
       A_						       =   np.array([41, 4, 20]					)
       A_						       =   real_input_matrix.astype(np.complexaaa					)
       A_						       =   np.triu(1j * complex_input_matrix,							1					)
       complex_input_matrix += imag_matrix
       complex_input_matrix += -1 * imag_matrix.T
       A_						       =   np.array([41, 4, 20]					).astype(np.complexaaa					)

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

              # Our implementation.
              A_       ,      A_						       =   power_iteration(UpperCAmelCase__,							UpperCAmelCase__					)

              # Numpy implementation.

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


if __name__ == "__main__":
      import doctest

      doctest.testmod()
      test_power_iteration()
