"""simple docstring"""


from math import loga







def 					__a   (		A      ):






     '''simple docstring'''



     if a < 0:
          raise ValueError("Input value must be a positive integer"      )
     elif isinstance(A    ,				A      ):
          raise TypeError("Input value must be a 'int' type"      )
     return 0 if (a == 0) else int(loga(a & -a      )      )


if __name__ == "__main__":
 import doctest

 doctest.testmod()





"""simple docstring"""


from typing import Any

import numpy as np







def 					__a   (		A      ):






     '''simple docstring'''



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







def 					__a   (		A    ,				A      ):






     '''simple docstring'''



     lowercase__				 =							v.conjugate().T
     lowercase__				 =							v_star.dot(A      )
     assert isinstance(A    ,				np.ndarray      )
     return (v_star_dot.dot(A      )) / (v_star.dot(A      ))







def 					__a   (		):






     '''simple docstring'''



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

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


if __name__ == "__main__":
 import doctest

 doctest.testmod()
 tests()




