
"""simple docstring"""




from math import pow




def  _lowerCAmelCase		(		_UpperCamelCase ,				_UpperCamelCase ,				_UpperCamelCase ,				_UpperCamelCase ,				_UpperCamelCase ,				):






  """simple docstring"""



  if current_sum == needed_sum:
    # If the sum of the powers is equal to needed_sum, then we have a solution.
    solutions_count += 1
    return current_sum, solutions_count

  _lowercase:			List[str]       =     int(pow(_UpperCamelCase ,				_UpperCamelCase						)						)
  if current_sum + i_to_n <= needed_sum:
    # If the sum of the powers is less than needed_sum, then continue adding powers.
    current_sum += i_to_n
    _lowercase ,				_lowercase:			Tuple       =     backtrack(
        _UpperCamelCase ,				_UpperCamelCase ,				current_number + 1 ,				_UpperCamelCase ,				_UpperCamelCase						)
    current_sum -= i_to_n
  if i_to_n < needed_sum:
    # If the power of i is less than needed_sum, then try with the next power.
    _lowercase ,				_lowercase:			Any       =     backtrack(
        _UpperCamelCase ,				_UpperCamelCase ,				current_number + 1 ,				_UpperCamelCase ,				_UpperCamelCase						)
  return current_sum, solutions_count




def  _lowerCAmelCase		(		_UpperCamelCase ,				_UpperCamelCase						):






  """simple docstring"""



  if not (1 <= needed_sum <= 1_000 and 2 <= power <= 10):
    raise ValueError(
        '''Invalid input\n'''
        '''needed_sum must be between 1 and 1000, power between 2 and 10.'''						)

  return backtrack(_UpperCamelCase ,				_UpperCamelCase ,				1 ,				0 ,				0						)[1]  # Return the solutions_count


if __name__ == "__main__":
  import doctest

  doctest.testmod()


"""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()
