





import unittest

import numpy as np



def 				lowerCAmelCase_ (							A_    ,A_    ,A_    ,A_ = None    ,):
    UpperCamelCase__:	Any					=     np.shape(A_)
    UpperCamelCase__:	Tuple					=     np.shape(A_)
    UpperCamelCase__:	List[Any]					=     np.shape(A_)

    if shape_a[0] != shape_b[0]:
        UpperCamelCase__:	List[str]					=     (
            "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]:
        UpperCamelCase__:	List[str]					=     (
            "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_)

    UpperCamelCase__:	str					=     pseudo_inv
    if a_inv is None:
        try:
            UpperCamelCase__:	Any					=     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):

      """simple docstring"""


      def 			UpperCAmelCase_       (		self:			List[Any]					):
          '''simple docstring'''


          UpperCamelCase__:	Union[str, Any]					=     np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]					)
          UpperCamelCase__:	Any					=     np.array([[0, 3], [3, 0], [2, 3]]					)
          UpperCamelCase__:	Dict					=     np.array([[2, 1], [6, 3]]					)

          UpperCamelCase__:	int					=     schur_complement(__lowerCamelCase	,    __lowerCamelCase	,    __lowerCamelCase					)

          UpperCamelCase__:	List[Any]					=     np.block([[a, b], [b.T, c]]					)

          UpperCamelCase__:	Tuple					=     np.linalg.det(__lowerCamelCase					)
          UpperCamelCase__:	Optional[int]					=     np.linalg.det(__lowerCamelCase					)
          UpperCamelCase__:	List[str]					=     np.linalg.det(__lowerCamelCase					)

          self.assertAlmostEqual(__lowerCamelCase	,    det_a * det_s					)






      def 			UpperCAmelCase_       (		self:			Dict					):
          '''simple docstring'''


          UpperCamelCase__:	Any					=     np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]					)
          UpperCamelCase__:	List[Any]					=     np.array([[0, 3], [3, 0], [2, 3]]					)
          UpperCamelCase__:	List[Any]					=     np.array([[2, 1], [6, 3]]					)

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






      def 			UpperCAmelCase_       (		self:			List[str]					):
          '''simple docstring'''


          UpperCamelCase__:	Union[str, Any]					=     np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]					)
          UpperCamelCase__:	List[str]					=     np.array([[0, 3], [3, 0], [2, 3]]					)
          UpperCamelCase__:	List[str]					=     np.array([[2, 1, 3], [6, 3, 5]]					)

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


if __name__ == "__main__":
   import doctest

   doctest.testmod()
   unittest.main()








def 				lowerCAmelCase_ (							A_    ,A_):
    UpperCamelCase__:	str					=     [1]
    for i in range(2    ,A_):
        factorials.append(factorials[-1] * i)
    assert 0 <= k < factorials[-1] * n, "k out of bounds"

    UpperCamelCase__:	Optional[int]					=     []
    UpperCamelCase__:	Optional[Any]					=     list(range(A_))

    # Find permutation
    while factorials:
        UpperCamelCase__:	Dict					=     factorials.pop()
        UpperCamelCase__				, UpperCamelCase__:	Optional[int]					=     divmod(A_    ,A_)
        permutation.append(elements[number])
        elements.remove(elements[number])
    permutation.append(elements[0])

    return permutation


if __name__ == "__main__":
   import doctest

   doctest.testmod()

