





import unittest

import numpy as np





def   lowerCamelCase__       (       UpperCamelCase__     :    np.ndarray    ,      UpperCamelCase__     :    np.ndarray    ,      UpperCamelCase__     :    np.ndarray    ,      UpperCamelCase__     :    np.ndarray | None = None    ,      )     ->	np.ndarray:



					'''simple docstring'''




					_snake_case													=			np.shape(UpperCamelCase__  )
					_snake_case													=			np.shape(UpperCamelCase__  )
					_snake_case													=			np.shape(UpperCamelCase__  )

					if shape_a[0] != shape_b[0]:
										_snake_case													=			(
										    '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]:
										_snake_case													=			(
										    '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__  )

					_snake_case													=			pseudo_inv
					if a_inv is None:
										try:
															_snake_case													=			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      UpperCamelCase_						(					unittest.TestCase   ):




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

										_snake_case													=			schur_complement(lowerCAmelCase_			,  lowerCAmelCase_			,  lowerCAmelCase_					)

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

										_snake_case													=			np.linalg.det(lowerCAmelCase_					)
										_snake_case													=			np.linalg.det(lowerCAmelCase_					)
										_snake_case													=			np.linalg.det(lowerCAmelCase_					)

										self.assertAlmostEqual(lowerCAmelCase_			,  det_a * det_s					)




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

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






					def 		lowerCAmelCase     (						self					)						->							None:
										_snake_case													=			np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]					)
										_snake_case													=			np.array([[0, 3], [3, 0], [2, 3]]					)
										_snake_case													=			np.array([[2, 1, 3], [6, 3, 5]]					)

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


if __name__ == "__main__":
					import doctest

					doctest.testmod()
					unittest.main()










def   lowerCamelCase__       (       UpperCamelCase__     :    Tuple    ,      UpperCamelCase__     :    Optional[int]  )     ->	Tuple:



					'''simple docstring'''




					_snake_case													=			[0 for i in range(r + 1  )]
					# nc0 = 1
					_snake_case													=			1
					for i in range(1    ,      n + 1  ):
										# to compute current row from previous row.
										_snake_case													=			min(UpperCamelCase__    ,      UpperCamelCase__  )
										while j > 0:
															c[j] += c[j - 1]
															j -= 1
					return c[r]




print(binomial_coefficient(n=10, r=5))



