
"""simple docstring"""




from __future__ import annotations

import numpy as np




def  _lowerCAmelCase		(		_UpperCamelCase						):






  """simple docstring"""



  _lowercase:			List[str]       =     np.shape(_UpperCamelCase						)
  if rows != columns:
    _lowercase:			int       =     (
        '''\'table\' has to be of square shaped array but got a '''
        f'''{rows}x{columns} array:\n{table}'''
    )
    raise ValueError(_UpperCamelCase						)

  _lowercase:			Tuple       =     np.zeros((rows, columns)						)
  _lowercase:			Tuple       =     np.zeros((rows, columns)						)
  for i in range(_UpperCamelCase						):
    for j in range(_UpperCamelCase						):
      _lowercase:			List[Any]       =     sum(lower[i][k] * upper[k][j] for k in range(_UpperCamelCase						)						)
      if upper[j][j] == 0:
        raise ArithmeticError('''No LU decomposition exists'''						)
      _lowercase:			Any       =     (table[i][j] - total) / upper[j][j]
    _lowercase:			Any       =     1
    for j in range(_UpperCamelCase ,				_UpperCamelCase						):
      _lowercase:			Optional[int]       =     sum(lower[i][k] * upper[k][j] for k in range(_UpperCamelCase						)						)
      _lowercase:			List[str]       =     table[i][j] - total
  return lower, upper


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()
