





'''simple docstring'''

import unittest

import numpy as np





def 						UpperCamelCase    (							_lowerCamelCase : np.ndarray						,	_lowerCamelCase : np.ndarray						,	_lowerCamelCase : np.ndarray						,	_lowerCamelCase : np.ndarray | None = None						,	):
    A__    =      np.shape(_lowerCamelCase	)
    A__    =      np.shape(_lowerCamelCase	)
    A__    =      np.shape(_lowerCamelCase	)

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

    A__    =      pseudo_inv
    if a_inv is None:
        try:
            A__    =      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        UpperCAmelCase     (	unittest.TestCase    ):
 def 					UpperCAmelCase_							(    self				:Any							)->							None:
     A__    =      np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]							)
     A__    =      np.array([[0, 3], [3, 0], [2, 3]]							)
     A__    =      np.array([[2, 1], [6, 3]]							)

     A__    =      schur_complement(lowercase_						,	lowercase_						,	lowercase_							)

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

     A__    =      np.linalg.det(lowercase_							)
     A__    =      np.linalg.det(lowercase_							)
     A__    =      np.linalg.det(lowercase_							)

     self.assertAlmostEqual(lowercase_						,	det_a * det_s							)
 def 					UpperCAmelCase_							(    self				:Dict							)->							None:
     A__    =      np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]							)
     A__    =      np.array([[0, 3], [3, 0], [2, 3]]							)
     A__    =      np.array([[2, 1], [6, 3]]							)

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



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










'''simple docstring'''

def 						UpperCamelCase    (							_lowerCamelCase : int = 1_00_00_00	):
    A__    =      set(range(3						,	_lowerCamelCase						,	2	)	)
    primes.add(2	)
    for p in range(3						,	_lowerCamelCase						,	2	):
        if p not in primes:
            continue
        primes.difference_update(set(range(p * p						,	_lowerCamelCase						,	_lowerCamelCase	)	)	)

    A__    =      [float(_lowerCamelCase	) for n in range(limit + 1	)]

    for p in primes:
        for n in range(_lowerCamelCase						,	limit + 1						,	_lowerCamelCase	):
            phi[n] *= 1 - 1 / p

    return int(sum(phi[2:]	)	)


if __name__ == "__main__":
  print(f"""{solution() = }""")



