


"""simple docstring"""






import unittest

import numpy as np
def 					A_      (      _lowerCAmelCase   :    List[Any],   _lowerCAmelCase   :    List[Any],   _lowerCAmelCase   :    int,   _lowerCAmelCase   :    Union[str, Any] = None,   ):






     """simple docstring"""

     _a					= np.shape(UpperCamelCase__				)
     _a					= np.shape(UpperCamelCase__				)
     _a					= np.shape(UpperCamelCase__				)

     if shape_a[0] != shape_b[0]:
          _a					= (
              """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					= (
              """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					= pseudo_inv
     if a_inv is None:
          try:
               _a					= 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 			__lowerCamelCase     (    unittest.TestCase ):

     '''simple docstring'''


     def 							_UpperCAmelCase					(					self							)				->							None:
          _a					= np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]							)
          _a					= np.array([[0, 3], [3, 0], [2, 3]]							)
          _a					= np.array([[2, 1], [6, 3]]							)

          _a					= schur_complement(UpperCAmelCase__			, UpperCAmelCase__			, UpperCAmelCase__							)

          _a					= np.block([[a, b], [b.T, c]]							)

          _a					= np.linalg.det(UpperCAmelCase__							)
          _a					= np.linalg.det(UpperCAmelCase__							)
          _a					= np.linalg.det(UpperCAmelCase__							)

          self.assertAlmostEqual(UpperCAmelCase__			, det_a * det_s							)


     def 							_UpperCAmelCase					(					self							)				->							None:
          _a					= np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]							)
          _a					= np.array([[0, 3], [3, 0], [2, 3]]							)
          _a					= np.array([[2, 1], [6, 3]]							)

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





     def 							_UpperCAmelCase					(					self							)				->							None:
          _a					= np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]							)
          _a					= np.array([[0, 3], [3, 0], [2, 3]]							)
          _a					= 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"""




def 			lowerCAmelCase__     (     UpperCamelCase__     ,   UpperCamelCase__				):



  '''simple docstring'''

  # Check if the input is valid
  if not len(UpperCamelCase__				) == len(UpperCamelCase__				) == 3:
    raise ValueError("""Please enter a valid equation."""				)
  if equationa[0] == equationa[1] == equationa[0] == equationa[1] == 0:
    raise ValueError("""Both a & b of two equations can't be zero."""				)

  # Extract the coefficients
  _a      ,  _a      ,  _a  :       Any             =	equationa
  _a      ,  _a      ,  _a  :       Tuple             =	equationa

  # Calculate the determinants of the matrices
  _a  :       int             =	aa * ba - aa * ba
  _a  :       str             =	ca * ba - ca * ba
  _a  :       str             =	aa * ca - aa * ca

  # Check if the system of linear equations has a solution (using Cramer's rule)
  if determinant == 0:
    if determinant_x == determinant_y == 0:
      raise ValueError("""Infinite solutions. (Consistent system)"""				)
    else:
      raise ValueError("""No solution. (Inconsistent system)"""				)
  else:
    if determinant_x == determinant_y == 0:
      # Trivial solution (Inconsistent system)
      return (0.0, 0.0)
    else:
      _a  :       Dict             =	determinant_x / determinant
      _a  :       str             =	determinant_y / determinant
      # Non-Trivial Solution (Consistent system)
      return (x, y)

