



'''simple docstring'''





from typing import Any

import numpy as np





def 						__snake_case(				_lowerCAmelCase		)		-> bool:
      return np.array_equal(_lowerCAmelCase     ,   matrix.conjugate().T		)





def 						__snake_case(				_lowerCAmelCase     ,   _lowerCAmelCase		)		-> Any:
      snake_case__  :    int    						=      v.conjugate().T
      snake_case__  :    Optional[Any]    						=      v_star.dot(_lowerCAmelCase		)
      assert isinstance(_lowerCAmelCase     ,   np.ndarray		)
      return (v_star_dot.dot(_lowerCAmelCase		)) / (v_star.dot(_lowerCAmelCase		))





def 						__snake_case(				)		-> None:
      snake_case__  :    int    						=      np.array([[2, 2 + 1J, 4], [2 - 1J, 3, 1J], [4, -1J, 1]]		)
      snake_case__  :    List[str]    						=      np.array([[1], [2], [3]]		)
      assert is_hermitian(_lowerCAmelCase		), f"{a} is not hermitian."
      print(rayleigh_quotient(_lowerCAmelCase     ,   _lowerCAmelCase		)		)

      snake_case__  :    Optional[int]    						=      np.array([[1, 2, 4], [2, 3, -1], [4, -1, 1]]		)
      assert is_hermitian(_lowerCAmelCase		), f"{a} is not hermitian."
      assert rayleigh_quotient(_lowerCAmelCase     ,   _lowerCAmelCase		) == float(3		)


if __name__ == "__main__":
       import doctest

       doctest.testmod()
       tests()

"""simple docstring"""


from __future__ import annotations

from collections import Counter
from random import random


class 				a :


    def __init__(    self		:     Union[str, Any]    ):
      _UpperCAmelCase         =					{}


    def    lowerCAmelCase_     (    self		:     Optional[int]      , __lowerCAmelCase		:     str    ):
      _UpperCAmelCase         =					{}


    def    lowerCAmelCase_     (    self		:     str      , __lowerCAmelCase		:     str      , __lowerCAmelCase		:     str      , __lowerCAmelCase		:     float    ):
      if nodea not in self.connections:
        self.add_node(__lowerCAmelCase    )
      if nodea not in self.connections:
        self.add_node(__lowerCAmelCase    )
      _UpperCAmelCase         =					probability


    def    lowerCAmelCase_     (    self		:     Optional[Any]    ):
      return list(self.connections    )


    def    lowerCAmelCase_     (    self		:     List[Any]      , __lowerCAmelCase		:     str    ):
      _UpperCAmelCase         =					0
      _UpperCAmelCase         =					random()

      for dest in self.connections[node]:
        current_probability += self.connections[node][dest]
        if current_probability > random_value:
          return dest
      return ""






def 			__UpperCAmelCase    (	lowercase      ,lowercase      ,lowercase						):







  """simple docstring"""
  _UpperCAmelCase         =					MarkovChainGraphUndirectedUnweighted()

  for nodea, nodea, probability in transitions:
    graph.add_transition_probability(lowercase      ,lowercase      ,lowercase						)

  _UpperCAmelCase         =					Counter(graph.get_nodes()						)
  _UpperCAmelCase         =					start

  for _ in range(lowercase						):
    _UpperCAmelCase         =					graph.transition(lowercase						)
    visited[node] += 1

  return visited


if __name__ == "__main__":
    import doctest

    doctest.testmod()
