



'''simple docstring'''





import numpy as np





def 						__snake_case(				_lowerCAmelCase     ,   _lowerCAmelCase     ,   _lowerCAmelCase = 1e-12     ,   _lowerCAmelCase = 100     ,   )		-> Optional[int]:
      assert np.shape(SCREAMING_SNAKE_CASE_		)[0] == np.shape(SCREAMING_SNAKE_CASE_		)[1]
      # Ensure proper dimensionality.
      assert np.shape(SCREAMING_SNAKE_CASE_		)[0] == np.shape(SCREAMING_SNAKE_CASE_		)[0]
      # Ensure inputs are either both complex or both real
      assert np.iscomplexobj(SCREAMING_SNAKE_CASE_		) == np.iscomplexobj(SCREAMING_SNAKE_CASE_		)
      snake_case__  :    Union[str, Any]    						=      np.iscomplexobj(SCREAMING_SNAKE_CASE_		)
      if is_complex:
            # Ensure complex input_matrix is Hermitian
            assert np.array_equal(SCREAMING_SNAKE_CASE_     ,   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.

      snake_case__  :    int    						=      False
      snake_case__  :    Optional[int]    						=      0
      snake_case__  :    Optional[int]    						=      0
      snake_case__  :    Optional[Any]    						=      1e12

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

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

            if error <= error_tol or iterations >= max_iterations:
                  snake_case__  :    Union[str, Any]    						=      True

            snake_case__  :    Union[str, Any]    						=      lambda_

      if is_complex:
            snake_case__  :    Any    						=      np.real(lambda_		)

      return lambda_, vector





def 						__snake_case(				)		-> Tuple:
      snake_case__  :    List[Any]    						=      np.array([[41, 4, 20], [4, 26, 30], [20, 30, 50]]		)
      snake_case__  :    Optional[int]    						=      np.array([41, 4, 20]		)
      snake_case__  :    List[Any]    						=      real_input_matrix.astype(np.complexaaa		)
      snake_case__  :    Tuple    						=      np.triu(1J * complex_input_matrix     ,   1		)
      complex_input_matrix += imag_matrix
      complex_input_matrix += -1 * imag_matrix.T
      snake_case__  :    Tuple    						=      np.array([41, 4, 20]		).astype(np.complexaaa		)

      for problem_type in ["real", "complex"]:
            if problem_type == "real":
                  snake_case__  :    Any    						=      real_input_matrix
                  snake_case__  :    List[str]    						=      real_vector
            elif problem_type == "complex":
                  snake_case__  :    Tuple    						=      complex_input_matrix
                  snake_case__  :    int    						=      complex_vector

            # Our implementation.
            snake_case__						,				snake_case__  :    Optional[int]    						=      power_iteration(SCREAMING_SNAKE_CASE_     ,   SCREAMING_SNAKE_CASE_		)

            # Numpy implementation.

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


if __name__ == "__main__":
       import doctest

       doctest.testmod()
       test_power_iteration()






class 			a__    (					snake_case__      ):
						pass






class 			a__    (					snake_case__      ):
						pass






class 			a__    :






						def __init__(   self			):




											"""simple docstring"""



											__lowerCAmelCase				   =       [
											    [],
											    [],
											    [],
											]






						def 				__SCREAMING_SNAKE_CASE(   self		,			_A		,			_A			):




											"""simple docstring"""



											try:
																if len(self.queues[priority]			) >= 1_0_0:
																					raise OverflowError("Maximum queue size is 100"			)
																self.queues[priority].append(_A			)
											except IndexError:
																raise ValueError("Valid priorities are 0, 1, and 2"			)






						def 				__SCREAMING_SNAKE_CASE(   self			):




											"""simple docstring"""



											for queue in self.queues:
																if queue:
																					return queue.pop(0			)
											raise UnderFlowError("All queues are empty"			)






						def __str__(   self			):




											"""simple docstring"""



											return "\n".join(f"""Priority {i}: {q}""" for i, q in enumerate(self.queues			)			)






class 			a__    :






						def __init__(   self			):




											"""simple docstring"""



											__lowerCAmelCase				   =       []






						def 				__SCREAMING_SNAKE_CASE(   self		,			_A			):




											"""simple docstring"""



											if len(self.queue			) == 1_0_0:
																raise OverFlowError("Maximum queue size is 100"			)
											self.queue.append(_A			)






						def 				__SCREAMING_SNAKE_CASE(   self			):




											"""simple docstring"""



											if not self.queue:
																raise UnderFlowError("The queue is empty"			)
											else:
																__lowerCAmelCase				   =       min(self.queue			)
																self.queue.remove(_A			)
																return data






						def __str__(   self			):




											"""simple docstring"""



											return str(self.queue			)







def 	_a      (						):
					__lowerCAmelCase				   =       FixedPriorityQueue()
					fpq.enqueue(0      ,      10     )
					fpq.enqueue(1      ,      70     )
					fpq.enqueue(0      ,      1_00     )
					fpq.enqueue(2      ,      1     )
					fpq.enqueue(2      ,      5     )
					fpq.enqueue(1      ,      7     )
					fpq.enqueue(2      ,      4     )
					fpq.enqueue(1      ,      64     )
					fpq.enqueue(0      ,      1_28     )
					print(SCREAMING_SNAKE_CASE_     )
					print(fpq.dequeue()     )
					print(fpq.dequeue()     )
					print(fpq.dequeue()     )
					print(fpq.dequeue()     )
					print(fpq.dequeue()     )
					print(SCREAMING_SNAKE_CASE_     )
					print(fpq.dequeue()     )
					print(fpq.dequeue()     )
					print(fpq.dequeue()     )
					print(fpq.dequeue()     )
					print(fpq.dequeue()     )







def 	_a      (						):
					__lowerCAmelCase				   =       ElementPriorityQueue()
					epq.enqueue(10     )
					epq.enqueue(70     )
					epq.enqueue(1_00     )
					epq.enqueue(1     )
					epq.enqueue(5     )
					epq.enqueue(7     )
					epq.enqueue(4     )
					epq.enqueue(64     )
					epq.enqueue(1_28     )
					print(SCREAMING_SNAKE_CASE_     )
					print(epq.dequeue()     )
					print(epq.dequeue()     )
					print(epq.dequeue()     )
					print(epq.dequeue()     )
					print(epq.dequeue()     )
					print(SCREAMING_SNAKE_CASE_     )
					print(epq.dequeue()     )
					print(epq.dequeue()     )
					print(epq.dequeue()     )
					print(epq.dequeue()     )
					print(epq.dequeue()     )


if __name__ == "__main__":
					fixed_priority_queue()
					element_priority_queue()




