


"""simple docstring"""

import numpy as np


def lowerCAmelCase_      (				UpperCamelCase__	: np.ndarray				,   UpperCamelCase__	: np.ndarray				,   UpperCamelCase__	: float = 1E-12				,   UpperCamelCase__	: int = 100				,   ):






 """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__ )
 __lowercase									=	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.

 __lowercase									=	False
 __lowercase									=	0
 __lowercase									=	0
 __lowercase									=	1E12

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

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

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

  __lowercase									=	lambda_

 if is_complex:
  __lowercase									=	np.real(lambda_ )

 return lambda_, vector


def lowerCAmelCase_      (				):






 """simple docstring"""







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

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

  # Our implementation.
  __lowercase							,		__lowercase									=	power_iteration(UpperCamelCase__				,   UpperCamelCase__ )

  # Numpy implementation.

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




"""simple docstring"""

def lowerCAmelCase_      (				UpperCamelCase__	: str				,   UpperCamelCase__	: str ):






 """simple docstring"""







 def get_matched_characters(UpperCamelCase__	: str				,   UpperCamelCase__	: str ) -> str:
  __lowercase									=	[]
  __lowercase									=	min(len(_stra )				,   len(_stra ) ) // 2
  for i, l in enumerate(_stra ):
   __lowercase									=	int(max(0				,   i - limit ) )
   __lowercase									=	int(min(i + limit + 1				,   len(_stra ) ) )
   if l in _stra[left:right]:
    matched.append(UpperCamelCase__ )
    __lowercase									=	f'''{_stra[0:_stra.index(UpperCamelCase__ )]} {_stra[_stra.index(UpperCamelCase__ ) + 1:]}'''

  return "".join(UpperCamelCase__ )

 # matching characters
 __lowercase									=	get_matched_characters(UpperCamelCase__				,   UpperCamelCase__ )
 __lowercase									=	get_matched_characters(UpperCamelCase__				,   UpperCamelCase__ )
 __lowercase									=	len(UpperCamelCase__ )

 # transposition
 __lowercase									=	(
     len([(ca, ca) for ca, ca in zip(UpperCamelCase__				,   UpperCamelCase__ ) if ca != ca] ) // 2
 )

 if not match_count:
  __lowercase									=	0.0
 else:
  __lowercase									=	(
      1
      / 3
      * (
          match_count / len(UpperCamelCase__ )
          + match_count / len(UpperCamelCase__ )
          + (match_count - transpositions) / match_count
      )
  )

 # common prefix up to 4 characters
 __lowercase									=	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"))
