


def        lowerCAmelCase_  (      __lowerCamelCase							,			__lowerCamelCase			):
							if number < 0 or shift_amount < 0:
														raise ValueError("both inputs must be positive integers"			)

							__snake_case	:    Dict											=     str(bin(__lowerCamelCase			)			)
							binary_number += "0" * shift_amount
							return binary_number







def        lowerCAmelCase_  (      __lowerCamelCase							,			__lowerCamelCase			):
							if number < 0 or shift_amount < 0:
														raise ValueError("both inputs must be positive integers"			)

							__snake_case	:    List[str]											=     str(bin(__lowerCamelCase			)			)[2:]
							if shift_amount >= len(__lowerCamelCase			):
														return "0b0"
							__snake_case	:    Union[str, Any]											=     binary_number[: len(__lowerCamelCase			) - shift_amount]
							return "0b" + shifted_binary_number







def        lowerCAmelCase_  (      __lowerCamelCase							,			__lowerCamelCase			):
							if number >= 0:  # Get binary representation of positive number
														__snake_case	:    Optional[int]											=     "0" + str(bin(__lowerCamelCase			)			).strip("-"			)[2:]
							else:  # Get binary (2's complement) representation of negative number
														__snake_case	:    Tuple											=     len(bin(__lowerCamelCase			)[3:]			)  # Find 2's complement of number
														__snake_case	:    Any											=     bin(abs(__lowerCamelCase			) - (1 << binary_number_length)			)[3:]
														__snake_case	:    Optional[Any]											=     (
														    "1" + "0" * (binary_number_length - len(__lowerCamelCase			)) + binary_number
														)

							if shift_amount >= len(__lowerCamelCase			):
														return "0b" + binary_number[0] * len(__lowerCamelCase			)
							return (
							    "0b"
							    + binary_number[0] * shift_amount
							    + binary_number[: len(__lowerCamelCase			) - shift_amount]
							)


if __name__ == "__main__":
			import doctest

			doctest.testmod()





import unittest

import numpy as np







def        lowerCAmelCase_  (      __lowerCamelCase							,			__lowerCamelCase							,			__lowerCamelCase							,			__lowerCamelCase = None							,			):
							__snake_case	:    List[str]											=     np.shape(__lowerCamelCase			)
							__snake_case	:    Optional[Any]											=     np.shape(__lowerCamelCase			)
							__snake_case	:    List[str]											=     np.shape(__lowerCamelCase			)

							if shape_a[0] != shape_b[0]:
														__snake_case	:    Any											=     (
														    "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(__lowerCamelCase			)

							if shape_b[1] != shape_c[1]:
														__snake_case	:    int											=     (
														    "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(__lowerCamelCase			)

							__snake_case	:    str											=     pseudo_inv
							if a_inv is None:
														try:
																					__snake_case	:    Optional[Any]											=     np.linalg.inv(__lowerCamelCase			)
														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 					__snake_case      (							self :						Tuple       )			->					None:
														__snake_case	:    str											=     np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]       )
														__snake_case	:    str											=     np.array([[0, 3], [3, 0], [2, 3]]       )
														__snake_case	:    Dict											=     np.array([[2, 1], [6, 3]]       )

														__snake_case	:    Dict											=     schur_complement(lowerCamelCase					,			lowerCamelCase					,			lowerCamelCase       )

														__snake_case	:    int											=     np.block([[a, b], [b.T, c]]       )

														__snake_case	:    Optional[int]											=     np.linalg.det(lowerCamelCase       )
														__snake_case	:    Any											=     np.linalg.det(lowerCamelCase       )
														__snake_case	:    Tuple											=     np.linalg.det(lowerCamelCase       )

														self.assertAlmostEqual(lowerCamelCase					,			det_a * det_s       )

							def 					__snake_case      (							self :						int       )			->					None:
														__snake_case	:    str											=     np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]       )
														__snake_case	:    Dict											=     np.array([[0, 3], [3, 0], [2, 3]]       )
														__snake_case	:    Tuple											=     np.array([[2, 1], [6, 3]]       )

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






							def 					__snake_case      (							self :						List[Any]       )			->					None:
														__snake_case	:    List[Any]											=     np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]       )
														__snake_case	:    Tuple											=     np.array([[0, 3], [3, 0], [2, 3]]       )
														__snake_case	:    List[Any]											=     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()

