




import unittest

import numpy as np







def 				lowerCAmelCase_	(     snake_case_,snake_case_,snake_case_,snake_case_ = None,):
      _A							:     Tuple			=						np.shape(UpperCamelCase_     )
      _A							:     List[Any]			=						np.shape(UpperCamelCase_     )
      _A							:     int			=						np.shape(UpperCamelCase_     )

      if shape_a[0] != shape_b[0]:
            _A							:     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(UpperCamelCase_     )

      if shape_b[1] != shape_c[1]:
            _A							:     int			=						(
                """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(UpperCamelCase_     )

      _A							:     int			=						pseudo_inv
      if a_inv is None:
            try:
                  _A							:     List[Any]			=						np.linalg.inv(UpperCamelCase_     )
            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 						lowercase							(   unittest.TestCase		):

     def 					a__	(  self				)						->							Any:
           _A							:     Optional[int]			=						np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]				)
           _A							:     int			=						np.array([[0, 3], [3, 0], [2, 3]]				)
           _A							:     Union[str, Any]			=						np.array([[2, 1], [6, 3]]				)

           _A							:     Optional[Any]			=						schur_complement(UpperCAmelCase__		,     UpperCAmelCase__		,     UpperCAmelCase__				)

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

           _A							:     List[str]			=						np.linalg.det(UpperCAmelCase__				)
           _A							:     Optional[Any]			=						np.linalg.det(UpperCAmelCase__				)
           _A							:     int			=						np.linalg.det(UpperCAmelCase__				)

           self.assertAlmostEqual(UpperCAmelCase__		,     det_a * det_s				)

     def 					a__	(  self				)						->							Any:
           _A							:     Optional[int]			=						np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]				)
           _A							:     List[str]			=						np.array([[0, 3], [3, 0], [2, 3]]				)
           _A							:     Any			=						np.array([[2, 1], [6, 3]]				)

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



     def 					a__	(  self				)						->							str:
           _A							:     Optional[Any]			=						np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]				)
           _A							:     Any			=						np.array([[0, 3], [3, 0], [2, 3]]				)
           _A							:     Optional[Any]			=						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"""





from __future__ import annotations

from typing import Any




class 					_lowerCAmelCase  :







 """simple docstring"""







 def __init__(			self							:			Any,					UpperCAmelCase__							:			int	):
    __lowercase								=						num_of_nodes
    __lowercase								=						[]
    __lowercase								=						{}


 def 					_lowercase (			self							:			Optional[Any],					UpperCAmelCase__							:			int,					UpperCAmelCase__							:			int,					UpperCAmelCase__							:			int	):
    self.m_edges.append([u_node, v_node, weight]	)


 def 					_lowercase (			self							:			Union[str, Any],					UpperCAmelCase__							:			int	):

    if self.m_component[u_node] == u_node:
       return u_node
    return self.find_component(self.m_component[u_node]	)


 def 					_lowercase (			self							:			List[Any],					UpperCAmelCase__							:			int	):

    if self.m_component[u_node] != u_node:
       for k in self.m_component:
          __lowercase								=						self.find_component(UpperCAmelCase__	)


 def 					_lowercase (			self							:			Union[str, Any],					UpperCAmelCase__							:			list[int],					UpperCAmelCase__							:			int,					UpperCAmelCase__							:			int	):

    if component_size[u_node] <= component_size[v_node]:
       __lowercase								=						v_node
       component_size[v_node] += component_size[u_node]
       self.set_component(UpperCAmelCase__	)

    elif component_size[u_node] >= component_size[v_node]:
       __lowercase								=						self.find_component(UpperCAmelCase__	)
       component_size[u_node] += component_size[v_node]
       self.set_component(UpperCAmelCase__	)



 def 					_lowercase (			self							:			Any	):
    __lowercase								=						[]
    __lowercase								=						0

    __lowercase								=						[-1] * self.m_num_of_nodes

    # A list of components (initialized to all of the nodes)
    for node in range(self.m_num_of_nodes	):
       self.m_component.update({node: node}	)
       component_size.append(1	)

    __lowercase								=						self.m_num_of_nodes

    while num_of_components > 1:
       for edge in self.m_edges:
          __lowercase		,__lowercase		,__lowercase								=						edge

          __lowercase								=						self.m_component[u]
          __lowercase								=						self.m_component[v]

          if u_component != v_component:

             for component in (u_component, v_component):
                if (
                    minimum_weight_edge[component] == -1
                    or minimum_weight_edge[component][2] > w
                ):
                   __lowercase								=						[u, v, w]

       for edge in minimum_weight_edge:
          if isinstance(UpperCAmelCase__,					UpperCAmelCase__	):
             __lowercase		,__lowercase		,__lowercase								=						edge

             __lowercase								=						self.m_component[u]
             __lowercase								=						self.m_component[v]

             if u_component != v_component:
                mst_weight += w
                self.union(UpperCAmelCase__,					UpperCAmelCase__,					UpperCAmelCase__	)
                print(F"""Added edge [{u} - {v}]\nAdded weight: {w}\n"""	)
                num_of_components -= 1

       __lowercase								=						[-1] * self.m_num_of_nodes
    print(F"""The total weight of the minimal spanning tree is: {mst_weight}"""	)







def   _A  (  )	->						None:
   '''simple docstring'''




if __name__ == "__main__":
   import doctest

   doctest.testmod()




