

from __future__ import annotations
def 							__SCREAMING_SNAKE_CASE     (   a__    :   List[str]			,a__    :   Dict			,a__    :   Union[str, Any]			,a__    :   Any       )					->				Optional[int]:  # noqa: E741
	while r - l > 1:
		__A :			Any		    =			(l + r) // 2
		if v[m] >= key:
			__A :			Optional[int]		    =			m
		else:
			__A :			List[Any]		    =			m  # noqa: E741
	return r
def 							__SCREAMING_SNAKE_CASE     (   a__    :   list[int]       )					->				int:
	if len(a__       ) == 0:
		return 0

	__A :			str		    =			[0] * len(a__       )
	__A :			List[str]		    =			1

	__A :			List[Any]		    =			v[0]

	for i in range(1			,len(a__       )       ):
		if v[i] < tail[0]:
			__A :			int		    =			v[i]
		elif v[i] > tail[length - 1]:
			__A :			Union[str, Any]		    =			v[i]
			length += 1
		else:
			__A :			Any		    =			v[i]

	return length


if __name__ == "__main__":
			import doctest

			doctest.testmod()











import unittest

import numpy as np







def lowerCamelCase__  (     _lowercase					,		_lowercase					,		_lowercase					,		_lowercase = None					,		):

     '''simple docstring'''


     UpperCAmelCase_     :				Dict  =     np.shape(_lowercase )
     UpperCAmelCase_     :				Optional[Any]  =     np.shape(_lowercase )
     UpperCAmelCase_     :				Tuple  =     np.shape(_lowercase )

     if shape_a[0] != shape_b[0]:
          UpperCAmelCase_     :				Tuple  =     (
              '''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(_lowercase )

     if shape_b[1] != shape_c[1]:
          UpperCAmelCase_     :				List[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(_lowercase )

     UpperCAmelCase_     :				Dict  =     pseudo_inv
     if a_inv is None:
          try:
               UpperCAmelCase_     :				Any  =     np.linalg.inv(_lowercase )
          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     __a(	unittest.TestCase  ):






    """simple docstring"""





    def 				a__ ( self    )					->						None:
         UpperCAmelCase_     :				str  =     np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]    )
         UpperCAmelCase_     :				Any  =     np.array([[0, 3], [3, 0], [2, 3]]    )
         UpperCAmelCase_     :				List[str]  =     np.array([[2, 1], [6, 3]]    )

         UpperCAmelCase_     :				Tuple  =     schur_complement(_SCREAMING_SNAKE_CASE  ,_SCREAMING_SNAKE_CASE  ,_SCREAMING_SNAKE_CASE    )

         UpperCAmelCase_     :				List[Any]  =     np.block([[a, b], [b.T, c]]    )

         UpperCAmelCase_     :				List[Any]  =     np.linalg.det(_SCREAMING_SNAKE_CASE    )
         UpperCAmelCase_     :				List[Any]  =     np.linalg.det(_SCREAMING_SNAKE_CASE    )
         UpperCAmelCase_     :				Any  =     np.linalg.det(_SCREAMING_SNAKE_CASE    )

         self.assertAlmostEqual(_SCREAMING_SNAKE_CASE  ,det_a * det_s    )





    def 				a__ ( self    )					->						None:
         UpperCAmelCase_     :				str  =     np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]    )
         UpperCAmelCase_     :				Optional[int]  =     np.array([[0, 3], [3, 0], [2, 3]]    )
         UpperCAmelCase_     :				Optional[int]  =     np.array([[2, 1], [6, 3]]    )

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





    def 				a__ ( self    )					->						None:
         UpperCAmelCase_     :				Optional[int]  =     np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]    )
         UpperCAmelCase_     :				Optional[Any]  =     np.array([[0, 3], [3, 0], [2, 3]]    )
         UpperCAmelCase_     :				int  =     np.array([[2, 1, 3], [6, 3, 5]]    )

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


if __name__ == "__main__":
     import doctest

     doctest.testmod()
     unittest.main()