

'''simple docstring'''



from string import ascii_uppercase

UpperCAmelCase :       Dict			   =     {str(ord(c) - 5_5): c for c in ascii_uppercase}







def    a__       (  a__      ,    a__						):




     """simple docstring"""


     if isinstance(a__      ,    a__						):
          raise TypeError("""int() can't convert non-string with explicit base"""						)
     if num < 0:
          raise ValueError("""parameter must be positive int"""						)
     if isinstance(a__      ,    a__						):
          raise TypeError("""'str' object cannot be interpreted as an integer"""						)
     if isinstance(a__      ,    a__						):
          raise TypeError("""'float' object cannot be interpreted as an integer"""						)
     if base in (0, 1):
          raise ValueError("""base must be >= 2"""						)
     if base > 36:
          raise ValueError("""base must be <= 36"""						)
     __SCREAMING_SNAKE_CASE     =      """"""
     __SCREAMING_SNAKE_CASE     =      0
     __SCREAMING_SNAKE_CASE     =      0
     while div != 1:
          __SCREAMING_SNAKE_CASE   ,			__SCREAMING_SNAKE_CASE     =      divmod(a__      ,    a__						)
          if base >= 11 and 9 < mod < 36:
               __SCREAMING_SNAKE_CASE     =      ALPHABET_VALUES[str(a__						)]
          else:
               __SCREAMING_SNAKE_CASE     =      str(a__						)
          new_value += actual_value
          __SCREAMING_SNAKE_CASE     =      num // base
          __SCREAMING_SNAKE_CASE     =      div
          if div == 0:
               return str(new_value[::-1]						)
          elif div == 1:
               new_value += str(a__						)
               return str(new_value[::-1]						)

     return new_value[::-1]


if __name__ == "__main__":
  import doctest

  doctest.testmod()

  for base in range(2, 3_7):
    for num in range(1_0_0_0):
      assert int(decimal_to_any(num, base), base) == num, (
          num,
          base,
          decimal_to_any(num, base),
          int(decimal_to_any(num, base), base),
      )









'''simple docstring'''



import numpy as np







def    a__       (  a__      ,    a__      ,    a__ = 1E-1_2      ,    a__ = 1_00      ,    ):




     """simple docstring"""


     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__						)
     __SCREAMING_SNAKE_CASE     =      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.

     __SCREAMING_SNAKE_CASE     =      False
     __SCREAMING_SNAKE_CASE     =      0
     __SCREAMING_SNAKE_CASE     =      0
     __SCREAMING_SNAKE_CASE     =      1E1_2

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

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

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

          __SCREAMING_SNAKE_CASE     =      lambda_

     if is_complex:
          __SCREAMING_SNAKE_CASE     =      np.real(lambda_						)

     return lambda_, vector







def    a__       (  ):




     """simple docstring"""


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

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

          # Our implementation.
          __SCREAMING_SNAKE_CASE   ,			__SCREAMING_SNAKE_CASE     =      power_iteration(a__      ,    a__						)

          # Numpy implementation.

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






