



'''simple docstring'''






import unittest

import numpy as np
def    lowerCamelCase__ (		_A				,    _A				,    _A				,    _A = None				,    ):
  a		:      Optional[int]       =  np.shape(_A       )
  a		:      Tuple       =  np.shape(_A       )
  a		:      Union[str, Any]       =  np.shape(_A       )

  if shape_a[0] != shape_b[0]:
    a		:      Tuple       =  (
        'Expected the same number of rows for A and B. '
        f"""Instead found A of size {shape_a} and B of size {shape_b}"""
    )
    raise ValueError(_A       )

  if shape_b[1] != shape_c[1]:
    a		:      Union[str, Any]       =  (
        'Expected the same number of columns for B and C. '
        f"""Instead found B of size {shape_b} and C of size {shape_c}"""
    )
    raise ValueError(_A       )

  a		:      Tuple       =  pseudo_inv
  if a_inv is None:
    try:
      a		:      Tuple       =  np.linalg.inv(_A       )
    except np.linalg.LinAlgError:
      raise ValueError(
          'Input matrix A is not invertible. Cannot compute Schur complement.'       )

  return mat_c - mat_b.T @ a_inv @ mat_b

class 							a__(      unittest.TestCase			):







    def     lowercase_ (   self				:						str			):
      a		:      str       =  np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]			)
      a		:      str       =  np.array([[0, 3], [3, 0], [2, 3]]			)
      a		:      Dict       =  np.array([[2, 1], [6, 3]]			)

      a		:      List[Any]       =  schur_complement(__snake_case		,    __snake_case		,    __snake_case			)

      a		:      Optional[int]       =  np.block([[a, b], [b.T, c]]			)

      a		:      List[Any]       =  np.linalg.det(__snake_case			)
      a		:      str       =  np.linalg.det(__snake_case			)
      a		:      Optional[Any]       =  np.linalg.det(__snake_case			)

      self.assertAlmostEqual(__snake_case		,    det_a * det_s			)







    def     lowercase_ (   self				:						Optional[int]			):
      a		:      int       =  np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]			)
      a		:      Optional[Any]       =  np.array([[0, 3], [3, 0], [2, 3]]			)
      a		:      Union[str, Any]       =  np.array([[2, 1], [6, 3]]			)

      with self.assertRaises(__snake_case			):
        schur_complement(__snake_case		,    __snake_case		,    __snake_case			)







    def     lowercase_ (   self				:						Optional[int]			):
      a		:      str       =  np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]			)
      a		:      List[str]       =  np.array([[0, 3], [3, 0], [2, 3]]			)
      a		:      Tuple       =  np.array([[2, 1, 3], [6, 3, 5]]			)

      with self.assertRaises(__snake_case			):
        schur_complement(__snake_case		,    __snake_case		,    __snake_case			)


if __name__ == "__main__":
     import doctest

     doctest.testmod()
     unittest.main()




'''simple docstring'''






def    lowerCamelCase__ (		_A = 6008_5147_5143       ):

  try:
    a		:      Optional[int]       =  int(_A       )
  except (TypeError, ValueError):
    raise TypeError('Parameter n must be int or castable to int.'       )
  if n <= 0:
    raise ValueError('Parameter n must be greater than or equal to one.'       )
  a		:      Any       =  1
  a		:      Union[str, Any]       =  2
  while i * i <= n:
    while n % i == 0:
      a		:      str       =  i
      n //= i
    i += 1
  if n > 1:
    a		:      Any       =  n
  return int(_A       )


if __name__ == "__main__":
     print(F"{solution() = }")