
import unittest

import numpy as np

def a__    (       A_,     A_,     A_,     A_ = None,     ):






  '''simple docstring'''





  __magic_name__				     =       np.shape(A_					)
  __magic_name__				     =       np.shape(A_					)
  __magic_name__				     =       np.shape(A_					)

  if shape_a[0] != shape_b[0]:
    __magic_name__				     =       (
        """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]:
    __magic_name__				     =       (
        """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_					)

  __magic_name__				     =       pseudo_inv
  if a_inv is None:
    try:
      __magic_name__				     =       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        UpperCAmelCase_			( unittest.TestCase     ):


  '''simple docstring'''







  def 							_lowercase							( self			: int    )			->	None:







    """simple docstring"""






    __magic_name__				     =       np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]    )
    __magic_name__				     =       np.array([[0, 3], [3, 0], [2, 3]]    )
    __magic_name__				     =       np.array([[2, 1], [6, 3]]    )

    __magic_name__				     =       schur_complement(UpperCamelCase__			,							UpperCamelCase__			,							UpperCamelCase__    )

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

    __magic_name__				     =       np.linalg.det(UpperCamelCase__    )
    __magic_name__				     =       np.linalg.det(UpperCamelCase__    )
    __magic_name__				     =       np.linalg.det(UpperCamelCase__    )

    self.assertAlmostEqual(UpperCamelCase__			,							det_a * det_s    )
  def 							_lowercase							( self			: List[Any]    )			->	None:







    """simple docstring"""






    __magic_name__				     =       np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]    )
    __magic_name__				     =       np.array([[0, 3], [3, 0], [2, 3]]    )
    __magic_name__				     =       np.array([[2, 1], [6, 3]]    )

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





  def 							_lowercase							( self			: int    )			->	None:







    """simple docstring"""






    __magic_name__				     =       np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]    )
    __magic_name__				     =       np.array([[0, 3], [3, 0], [2, 3]]    )
    __magic_name__				     =       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


def  snake_case_     (					_lowerCAmelCase  :						list[int | float]   ,      _lowerCAmelCase  :						int   ,      _lowerCAmelCase  :						int  )      ->   int | float:
							if len(_lowerCAmelCase  ) == 0:
														raise ValueError('''find_max() arg is an empty sequence'''  )
							if (
							    left >= len(_lowerCAmelCase  )
							    or left < -len(_lowerCAmelCase  )
							    or right >= len(_lowerCAmelCase  )
							    or right < -len(_lowerCAmelCase  )
							):
														raise IndexError('''list index out of range'''  )
							if left == right:
														return nums[left]
							UpperCAmelCase :       List[Any] 							=		(left + right) >> 1  # the middle
							UpperCAmelCase :       Optional[Any] 							=		find_max(_lowerCAmelCase   ,      _lowerCAmelCase   ,      _lowerCAmelCase  )  # find max in range[left, mid]
							UpperCAmelCase :       Dict 							=		find_max(_lowerCAmelCase   ,      mid + 1   ,      _lowerCAmelCase  )  # find max in range[mid + 1, right]

							return left_max if left_max >= right_max else right_max


if __name__ == "__main__":
						import doctest

						doctest.testmod(verbose=True)





