



from __future__ import annotations

UpperCamelCase__ =						10






def  UpperCAmelCase__    (    lowercase__   ) ->      list[int]:
						__lowercase					=				1
						__lowercase					=				max(lowercase__   )
						while placement <= max_digit:
												# declare and initialize empty buckets
												__lowercase					=				[[] for _ in range(lowercase__   )]
												# split list_of_ints between the buckets
												for i in list_of_ints:
																		__lowercase					=				int((i / placement) % RADIX   )
																		buckets[tmp].append(lowercase__   )
												# put each buckets' contents into list_of_ints
												__lowercase					=				0
												for b in range(lowercase__   ):
																		for i in buckets[b]:
																								__lowercase					=				i
																								a += 1
        # move to next
												placement *= RADIX
						return list_of_ints


if __name__ == "__main__":
			import doctest

			doctest.testmod()










import unittest

import numpy as np






def  UpperCAmelCase__    (    lowercase__						,       lowercase__						,       lowercase__						,       lowercase__ = None						,       ) ->      np.ndarray:
						__lowercase					=				np.shape(lowercase__   )
						__lowercase					=				np.shape(lowercase__   )
						__lowercase					=				np.shape(lowercase__   )

						if shape_a[0] != shape_b[0]:
												__lowercase					=				(
												    """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]:
												__lowercase					=				(
												    """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__   )

						__lowercase					=				pseudo_inv
						if a_inv is None:
												try:
																		__lowercase					=				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 			_lowerCAmelCase   ( unittest.TestCase			):






			"""simple docstring"""





			def 		snake_case__			(				self	:       Dict   )						->					None:
									"""simple docstring"""







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

									__lowercase					=				schur_complement(lowercase			,     lowercase			,     lowercase   )

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

									__lowercase					=				np.linalg.det(lowercase   )
									__lowercase					=				np.linalg.det(lowercase   )
									__lowercase					=				np.linalg.det(lowercase   )

									self.assertAlmostEqual(lowercase			,     det_a * det_s   )





			def 		snake_case__			(				self	:       Tuple   )						->					None:
									"""simple docstring"""







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

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







			def 		snake_case__			(				self	:       Tuple   )						->					None:
									"""simple docstring"""







									__lowercase					=				np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]   )
									__lowercase					=				np.array([[0, 3], [3, 0], [2, 3]]   )
									__lowercase					=				np.array([[2, 1, 3], [6, 3, 5]]   )

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


if __name__ == "__main__":
			import doctest

			doctest.testmod()
			unittest.main()





