

from typing import Any

import numpy as np






def 					__lowercase      (     _A			)     ->						bool:
      return np.array_equal(snake_case__						,    matrix.conjugate().T			)






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






def 					__lowercase      (     )     ->						None:
      SCREAMING_SNAKE_CASE :      str       =   np.array([[2, 2 + 1J, 4], [2 - 1J, 3, 1J], [4, -1J, 1]]			)
      SCREAMING_SNAKE_CASE :      Tuple       =   np.array([[1], [2], [3]]			)
      assert is_hermitian(snake_case__			), F"{a} is not hermitian."
      print(rayleigh_quotient(snake_case__						,    snake_case__			)			)

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


if __name__ == "__main__":
   import doctest

   doctest.testmod()
   tests()







from math import ceil






def 					__lowercase      (     _A = 1001			)     ->						int:
      SCREAMING_SNAKE_CASE :      Union[str, Any]       =   1

      for i in range(1						,    int(ceil(n / 2.0			)			)			):
            SCREAMING_SNAKE_CASE :      Dict       =   2 * i + 1
            SCREAMING_SNAKE_CASE :      Optional[int]       =   2 * i
            SCREAMING_SNAKE_CASE :      str       =   total + 4 * odd**2 - 6 * even

      return total


if __name__ == "__main__":
  import sys

  if len(sys.argv) == 1:
    print(solution())
  else:
    try:
      UpperCAmelCase__ :     str				    =    int(sys.argv[1])
      print(solution(n))
    except ValueError:
      print("""Invalid entry - please enter a number""")




