
'''simple docstring'''



import numpy as np
def  lowerCAmelCase_						(					SCREAMING_SNAKE_CASE_     ,   SCREAMING_SNAKE_CASE_     ,   SCREAMING_SNAKE_CASE_ = 1e-12     ,   SCREAMING_SNAKE_CASE_ = 1_00     ,   )  ->  tuple[float, np.ndarray]:




   """simple docstring"""
   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__    )
   _SCREAMING_SNAKE_CASE  			=					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.

   _SCREAMING_SNAKE_CASE  			=					False
   _SCREAMING_SNAKE_CASE  			=					0
   _SCREAMING_SNAKE_CASE  			=					0
   _SCREAMING_SNAKE_CASE  			=					1e12

   while not convergence:
      # Multiple matrix by the vector.
      _SCREAMING_SNAKE_CASE  			=					np.dot(UpperCamelCase__     ,   UpperCamelCase__    )
      # Normalize the resulting output vector.
      _SCREAMING_SNAKE_CASE  			=					w / np.linalg.norm(UpperCamelCase__    )
      # 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(UpperCamelCase__     ,   np.dot(UpperCamelCase__     ,   UpperCamelCase__    )    )

      # 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  lowerCAmelCase_						(					)  ->  None:




   """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(UpperCamelCase__     ,   UpperCamelCase__    )

      # 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(UpperCamelCase__    )
      # 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(UpperCamelCase__    ) - np.abs(UpperCamelCase__    )    ) <= 1e-6


if __name__ == "__main__":
     import doctest

     doctest.testmod()
     test_power_iteration()












from string import ascii_uppercase

__A								=       {str(ord(c) - 55): c for c in ascii_uppercase}







def      lowerCamelCase_						(			UpperCamelCase__							: int					,	UpperCamelCase__							: int )	->			str:





				"""simple docstring"""


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

				return new_value[::-1]


if __name__ == "__main__":
					import doctest

					doctest.testmod()

					for base in range(2, 37):
										for num in range(10_00):
															assert int(decimal_to_any(num, base), base) == num, (
															    num,
															    base,
															    decimal_to_any(num, base),
															    int(decimal_to_any(num, base), base),
															)


