
"""simple docstring"""




from __future__ import annotations

from typing import TypedDict


class 			__magic_name__       (       SCREAMING_SNAKE_CASE__ ):
       UpperCamelCase_   =		42
       UpperCamelCase_   =		42




def  _lowerCAmelCase		(		_UpperCamelCase						):






  """simple docstring"""



  if not isinstance(_UpperCamelCase ,				_UpperCamelCase						):
    raise TypeError('''The parameter s type must be str.'''						)

  return [s[i:] + s[:i] for i in range(len(_UpperCamelCase						)						)]




def  _lowerCAmelCase		(		_UpperCamelCase						):






  """simple docstring"""



  if not isinstance(_UpperCamelCase ,				_UpperCamelCase						):
    raise TypeError('''The parameter s type must be str.'''						)
  if not s:
    raise ValueError('''The parameter s must not be empty.'''						)

  _lowercase:			Dict       =     all_rotations(_UpperCamelCase						)
  rotations.sort()  # sort the list of rotations in alphabetically order
  # make a string composed of the last char of each rotation
  _lowercase:			BWTTransformDict       =     {
      "bwt_string": "".join([word[-1] for word in rotations]						),
      "idx_original_string": rotations.index(_UpperCamelCase						),
  }
  return response




def  _lowerCAmelCase		(		_UpperCamelCase ,				_UpperCamelCase						):






  """simple docstring"""



  if not isinstance(_UpperCamelCase ,				_UpperCamelCase						):
    raise TypeError('''The parameter bwt_string type must be str.'''						)
  if not bwt_string:
    raise ValueError('''The parameter bwt_string must not be empty.'''						)
  try:
    _lowercase:			Tuple       =     int(_UpperCamelCase						)
  except ValueError:
    raise TypeError(
        '''The parameter idx_original_string type must be int or passive'''
        ''' of cast to int.'''						)
  if idx_original_string < 0:
    raise ValueError('''The parameter idx_original_string must not be lower than 0.'''						)
  if idx_original_string >= len(_UpperCamelCase						):
    raise ValueError(
        '''The parameter idx_original_string must be lower than''' ''' len(bwt_string).'''						)

  _lowercase:			Dict       =     [''''''] * len(_UpperCamelCase						)
  for _ in range(len(_UpperCamelCase						)						):
    for i in range(len(_UpperCamelCase						)						):
      _lowercase:			Optional[Any]       =     bwt_string[i] + ordered_rotations[i]
    ordered_rotations.sort()
  return ordered_rotations[idx_original_string]


if __name__ == "__main__":
  A__   :					Any							=   'Provide a string that I will generate its BWT transform: '
  A__   :					Optional[Any]							=   input(entry_msg).strip()
  A__   :					str							=   bwt_transform(s)
  print(
      f"""Burrows Wheeler transform for string '{s}' results """
      f"""in '{result["bwt_string"]}'"""
  )
  A__   :					Optional[int]							=   reverse_bwt(result['bwt_string'], result['idx_original_string'])
  print(
      f"""Reversing Burrows Wheeler transform for entry '{result["bwt_string"]}' """
      f"""we get original string '{original_string}'"""
  )


"""simple docstring"""




from typing import Any

import numpy as np




def  _lowerCAmelCase		(		_UpperCamelCase						):






  """simple docstring"""



  return np.array_equal(_UpperCamelCase ,				matrix.conjugate().T						)




def  _lowerCAmelCase		(		_UpperCamelCase ,				_UpperCamelCase						):






  """simple docstring"""



  _lowercase:			str       =     v.conjugate().T
  _lowercase:			List[Any]       =     v_star.dot(_UpperCamelCase						)
  assert isinstance(_UpperCamelCase ,				np.ndarray						)
  return (v_star_dot.dot(_UpperCamelCase						)) / (v_star.dot(_UpperCamelCase						))




def  _lowerCAmelCase		(		):






  """simple docstring"""



  _lowercase:			int       =     np.array([[2, 2 + 1j, 4], [2 - 1j, 3, 1j], [4, -1j, 1]]						)
  _lowercase:			str       =     np.array([[1], [2], [3]]						)
  assert is_hermitian(_UpperCamelCase						), f'''{a} is not hermitian.'''
  print(rayleigh_quotient(_UpperCamelCase ,				_UpperCamelCase						)						)

  _lowercase:			Any       =     np.array([[1, 2, 4], [2, 3, -1], [4, -1, 1]]						)
  assert is_hermitian(_UpperCamelCase						), f'''{a} is not hermitian.'''
  assert rayleigh_quotient(_UpperCamelCase ,				_UpperCamelCase						) == float(3						)


if __name__ == "__main__":
  import doctest

  doctest.testmod()
  tests()
