





"""simple docstring"""
import numpy as np

def 		_SCREAMING_SNAKE_CASE			(_UpperCAmelCase					:    np.ndarray  ,     _UpperCAmelCase					:    np.ndarray  ,     _UpperCAmelCase					:    float = 1e-12  ,     _UpperCAmelCase					:    int = 100  ,     ):
 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			)
 lowerCAmelCase    				= 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.

 lowerCAmelCase    				= False
 lowerCAmelCase    				= 0
 lowerCAmelCase    				= 0
 lowerCAmelCase    				= 1e12

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

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

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

  lowerCAmelCase    				= lambda_

 if is_complex:
  lowerCAmelCase    				= np.real(lambda_			)

 return lambda_, vector

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

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

  # Our implementation.
  lowerCAmelCase				,lowerCAmelCase    				= power_iteration(_UpperCAmelCase  ,     _UpperCAmelCase			)

  # Numpy implementation.

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









def 						a__			(				A__,      A__				):

			def get_matched_characters(A__,      A__				) -> str:
						SCREAMING_SNAKE_CASE_		:     Dict					=					[]
						SCREAMING_SNAKE_CASE_		:     Any					=					min(len(_stra				),      len(_stra				)				) // 2
						for i, l in enumerate(_stra				):
									SCREAMING_SNAKE_CASE_		:     Union[str, Any]					=					int(max(0,      i - limit				)				)
									SCREAMING_SNAKE_CASE_		:     Union[str, Any]					=					int(min(i + limit + 1,      len(_stra				)				)				)
									if l in _stra[left:right]:
												matched.append(A__				)
												SCREAMING_SNAKE_CASE_		:     List[str]					=					F'''{_stra[0:_stra.index(A__				)]} {_stra[_stra.index(A__				) + 1:]}'''

						return "".join(A__				)

			# matching characters
			SCREAMING_SNAKE_CASE_		:     Union[str, Any]					=					get_matched_characters(A__,      A__				)
			SCREAMING_SNAKE_CASE_		:     int					=					get_matched_characters(A__,      A__				)
			SCREAMING_SNAKE_CASE_		:     Any					=					len(A__				)

			# transposition
			SCREAMING_SNAKE_CASE_		:     Optional[int]					=					(
			    len([(ca, ca) for ca, ca in zip(A__,      A__				) if ca != ca]				) // 2
			)

			if not match_count:
						SCREAMING_SNAKE_CASE_		:     Dict					=					0.0
			else:
						SCREAMING_SNAKE_CASE_		:     Optional[Any]					=					(
						    1
						    / 3
						    * (
						        match_count / len(A__				)
						        + match_count / len(A__				)
						        + (match_count - transpositions) / match_count
						    )
						)

			# common prefix up to 4 characters
			SCREAMING_SNAKE_CASE_		:     List[Any]					=					0
			for ca, ca in zip(stra[:4],      stra[:4]				):
						if ca == ca:
									prefix_len += 1
						else:
									break

			return jaro + 0.1 * prefix_len * (1 - jaro)


if __name__ == "__main__":
				import doctest

				doctest.testmod()
				print(jaro_winkler('hello', 'world'))





