


def  UpperCAmelCase__    (    lowercase__						,       lowercase__						,       lowercase__   ) ->      List[str]:
						if n == 0:
												return 1

						elif n % 2 == 1:
												return (binary_exponentiation(lowercase__						,       n - 1						,       lowercase__   ) * a) % mod

						else:
												__lowercase					=				binary_exponentiation(lowercase__						,       n / 2						,       lowercase__   )
												return (b * b) % mod


# a prime number
UpperCamelCase__ =						7_01

UpperCamelCase__ =						10_00_00_00_00
UpperCamelCase__ =						10

# using binary exponentiation function, O(log(p)):
print((a / b) % p == (a * binary_exponentiation(b, p - 2, p)) % p)







print((a / b) % p == (a * b ** (p - 2)) % p)










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()





