


"""simple docstring"""



import unittest

import numpy as np






def       lowerCamelCase					(a_						:np.ndarray				,				a_						:np.ndarray				,				a_						:np.ndarray				,				a_						:np.ndarray | None = None				,				)     ->							np.ndarray:
							lowercase		:str       							=				np.shape(a_)
							lowercase		:Dict       							=				np.shape(a_)
							lowercase		:Optional[int]       							=				np.shape(a_)

							if shape_a[0] != shape_b[0]:
														lowercase		: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]:
														lowercase		: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_)

							lowercase		:Optional[Any]       							=				pseudo_inv
							if a_inv is None:
														try:
																					lowercase		:int       							=				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 					__magic_name__		(			unittest.TestCase     ):






							def  __snake_case						(   self  :    int				):

														'''simple docstring'''






														lowercase		:Dict       							=				np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]				)
														lowercase		:List[str]       							=				np.array([[0, 3], [3, 0], [2, 3]]				)
														lowercase		:List[str]       							=				np.array([[2, 1], [6, 3]]				)

														lowercase		:Any       							=				schur_complement(snake_case__  ,     snake_case__  ,     snake_case__				)

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

														lowercase		:Tuple       							=				np.linalg.det(snake_case__				)
														lowercase		:Any       							=				np.linalg.det(snake_case__				)
														lowercase		:List[Any]       							=				np.linalg.det(snake_case__				)

														self.assertAlmostEqual(snake_case__  ,     det_a * det_s				)






							def  __snake_case						(   self  :    Optional[int]				):

														'''simple docstring'''






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

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







							def  __snake_case						(   self  :    List[Any]				):

														'''simple docstring'''






														lowercase		:List[str]       							=				np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]				)
														lowercase		:int       							=				np.array([[0, 3], [3, 0], [2, 3]]				)
														lowercase		:List[Any]       							=				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"""



from __future__ import annotations

from decimal import Decimal

from numpy import array






def       lowerCamelCase					(a_						:list[list[float]])     ->							list[list[float]]:
       lowercase		:List[str]       							=				Decimal

       # Check if the provided matrix has 2 rows and 2 columns
       # since this implementation only works for 2x2 matrices
       if len(a_) == 2 and len(matrix[0]) == 2 and len(matrix[1]) == 2:
              # Calculate the determinant of the matrix
              lowercase		:List[str]       							=				float(
                  d(matrix[0][0]) * d(matrix[1][1]) - d(matrix[1][0]) * d(matrix[0][1]))
              if determinant == 0:
                     raise ValueError('''This matrix has no inverse.''')

              # Creates a copy of the matrix with swapped positions of the elements
              lowercase		:Optional[Any]       							=				[[0.0, 0.0], [0.0, 0.0]]
              lowercase							,  lowercase		:List[Any]       							=				matrix[1][1], matrix[0][0]
              lowercase							,  lowercase		:Union[str, Any]       							=				-matrix[1][0], -matrix[0][1]

              # Calculate the inverse of the matrix
              return [
                  [(float(d(a_)) / determinant) or 0.0 for n in row] for row in swapped_matrix
              ]
       elif (
           len(a_) == 3
           and len(matrix[0]) == 3
           and len(matrix[1]) == 3
           and len(matrix[2]) == 3
       ):
              # Calculate the determinant of the matrix using Sarrus rule
              lowercase		:Any       							=				float(
                  (
                      (d(matrix[0][0]) * d(matrix[1][1]) * d(matrix[2][2]))
                      + (d(matrix[0][1]) * d(matrix[1][2]) * d(matrix[2][0]))
                      + (d(matrix[0][2]) * d(matrix[1][0]) * d(matrix[2][1]))
                  )
                  - (
                      (d(matrix[0][2]) * d(matrix[1][1]) * d(matrix[2][0]))
                      + (d(matrix[0][1]) * d(matrix[1][0]) * d(matrix[2][2]))
                      + (d(matrix[0][0]) * d(matrix[1][2]) * d(matrix[2][1]))
                  ))
              if determinant == 0:
                     raise ValueError('''This matrix has no inverse.''')

              # Creating cofactor matrix
              lowercase		:Optional[Any]       							=				[
                  [d(0.0), d(0.0), d(0.0)],
                  [d(0.0), d(0.0), d(0.0)],
                  [d(0.0), d(0.0), d(0.0)],
              ]
              lowercase		:Optional[int]       							=				(d(matrix[1][1]) * d(matrix[2][2])) - (
                  d(matrix[1][2]) * d(matrix[2][1])
              )
              lowercase		:str       							=				-(
                  (d(matrix[1][0]) * d(matrix[2][2])) - (d(matrix[1][2]) * d(matrix[2][0]))
              )
              lowercase		:Union[str, Any]       							=				(d(matrix[1][0]) * d(matrix[2][1])) - (
                  d(matrix[1][1]) * d(matrix[2][0])
              )
              lowercase		:Union[str, Any]       							=				-(
                  (d(matrix[0][1]) * d(matrix[2][2])) - (d(matrix[0][2]) * d(matrix[2][1]))
              )
              lowercase		:Union[str, Any]       							=				(d(matrix[0][0]) * d(matrix[2][2])) - (
                  d(matrix[0][2]) * d(matrix[2][0])
              )
              lowercase		:List[str]       							=				-(
                  (d(matrix[0][0]) * d(matrix[2][1])) - (d(matrix[0][1]) * d(matrix[2][0]))
              )
              lowercase		:Union[str, Any]       							=				(d(matrix[0][1]) * d(matrix[1][2])) - (
                  d(matrix[0][2]) * d(matrix[1][1])
              )
              lowercase		:Union[str, Any]       							=				-(
                  (d(matrix[0][0]) * d(matrix[1][2])) - (d(matrix[0][2]) * d(matrix[1][0]))
              )
              lowercase		:Any       							=				(d(matrix[0][0]) * d(matrix[1][1])) - (
                  d(matrix[0][1]) * d(matrix[1][0])
              )

              # Transpose the cofactor matrix (Adjoint matrix)
              lowercase		:Union[str, Any]       							=				array(a_)
              for i in range(3):
                     for j in range(3):
                            lowercase		:Any       							=				cofactor_matrix[j][i]

        # Inverse of the matrix using the formula (1/determinant) * adjoint matrix
              lowercase		:str       							=				array(a_)
              for i in range(3):
                     for j in range(3):
                            inverse_matrix[i][j] /= d(a_)

        # Calculate the inverse of the matrix
              return [[float(d(a_)) or 0.0 for n in row] for row in inverse_matrix]
       raise ValueError('''Please provide a matrix of size 2x2 or 3x3.''')

