





'''simple docstring'''




import unittest

import numpy as np






def        _lowerCAmelCase			(  __snake_case  : np.ndarray ,							__snake_case  : np.ndarray ,							__snake_case  : np.ndarray ,							__snake_case  : np.ndarray | None = None ,							)     ->     np.ndarray:
     __A						: Tuple      =       np.shape(__snake_case			)
     __A						: Dict      =       np.shape(__snake_case			)
     __A						: Any      =       np.shape(__snake_case			)

     if shape_a[0] != shape_b[0]:
          __A						: Optional[int]      =       (
              '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(__snake_case			)

     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(__snake_case			)

     __A						: List[Any]      =       pseudo_inv
     if a_inv is None:
          try:
               __A						: Tuple      =       np.linalg.inv(__snake_case			)
          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  SCREAMING_SNAKE_CASE    (unittest.TestCase							):


     def 							SCREAMING_SNAKE_CASE  (						self):

          '''simple docstring'''






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

          __A						: str      =       schur_complement(_UpperCAmelCase						,	_UpperCAmelCase						,	_UpperCAmelCase)

          __A						: List[str]      =       np.block([[a, b], [b.T, c]])

          __A						: Any      =       np.linalg.det(_UpperCAmelCase)
          __A						: List[Any]      =       np.linalg.det(_UpperCAmelCase)
          __A						: List[Any]      =       np.linalg.det(_UpperCAmelCase)

          self.assertAlmostEqual(_UpperCAmelCase						,	det_a * det_s)


     def 							SCREAMING_SNAKE_CASE  (						self):

          '''simple docstring'''






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

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


     def 							SCREAMING_SNAKE_CASE  (						self):

          '''simple docstring'''






          __A						: Tuple      =       np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]])
          __A						: Any      =       np.array([[0, 3], [3, 0], [2, 3]])
          __A						: str      =       np.array([[2, 1, 3], [6, 3, 5]])

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


if __name__ == "__main__":
  import doctest

  doctest.testmod()
  unittest.main()






'''simple docstring'''




import random






def        _lowerCAmelCase			(  __snake_case  : int ,							__snake_case  : float ,							__snake_case  : bool = False			)     ->     dict:
     __A						: dict      =       {i: [] for i in range(__snake_case			)}

     # if probability is greater or equal than 1, then generate a complete graph
     if probability >= 1:
          return complete_graph(__snake_case			)
     # if probability is lower or equal than 0, then return a graph without edges
     if probability <= 0:
          return graph

     # for each couple of nodes, add an edge from u to v
     # if the number randomly generated is greater than probability probability
     for i in range(__snake_case			):
          for j in range(i + 1 ,							__snake_case			):
               if random.random() < probability:
                    graph[i].append(__snake_case			)
                    if not directed:
                         # if the graph is undirected, add an edge in from j to i, either
                         graph[j].append(__snake_case			)
     return graph






def        _lowerCAmelCase			(  __snake_case  : int			)     ->     dict:
     return {
         i: [j for j in range(__snake_case			) if i != j] for i in range(__snake_case			)
     }


if __name__ == "__main__":
  import doctest

  doctest.testmod()