






import unittest

import numpy as np

def _UpperCAmelCase   (	a						:	List[Any]  ,					a						:	str  ,					a						:	List[str]  ,					a						:	str = None  ,					):
							snake_case__	  =       np.shape(a__ )
							snake_case__	  =       np.shape(a__ )
							snake_case__	  =       np.shape(a__ )

							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(a__ )

							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(a__ )

							snake_case__	  =       pseudo_inv
							if a_inv is None:
														try:
																					snake_case__	  =       np.linalg.inv(a__ )
														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 					__magic_name__							(      self    :    Dict):






											'''simple docstring'''







											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(lowercase_			,		lowercase_			,		lowercase_)

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

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

											self.assertAlmostEqual(lowercase_			,		det_a * det_s)



				def 					__magic_name__							(      self    :    Dict):






											'''simple docstring'''







											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(lowercase_):
																		schur_complement(lowercase_			,		lowercase_			,		lowercase_)



				def 					__magic_name__							(      self    :    int):






											'''simple docstring'''







											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(lowercase_):
																		schur_complement(lowercase_			,		lowercase_			,		lowercase_)


if __name__ == "__main__":
							import doctest

							doctest.testmod()
							unittest.main()








def _UpperCAmelCase   (	a						:	int  ,					a						:	Optional[Any] ):
							# "extended trapezoidal rule"
							# int(f) = dx/2 * (f1 + 2f2 + ... + fn)
							snake_case__	  =       (boundary[1] - boundary[0]) / steps
							snake_case__	  =       boundary[0]
							snake_case__	  =       boundary[1]
							snake_case__	  =       make_points(a  ,					a  ,					a )
							snake_case__	  =       0.0
							y += (h / 2.0) * f(a )
							for i in x_i:
														# print(i)
														y += h * f(a )
							y += (h / 2.0) * f(a )
							return y

def _UpperCAmelCase   (	a						:	Dict  ,					a						:	Union[str, Any]  ,					a						:	Union[str, Any] ):
							snake_case__	  =       a + h
							while x < (b - h):
														yield x
														snake_case__	  =       x + h

def _UpperCAmelCase   (	a						:	Any ):  # enter your function here
							snake_case__	  =       (x - 0) * (x - 0)
							return y

def _UpperCAmelCase   (	):
							snake_case__	  =       0.0  # Lower bound of integration
							snake_case__	  =       1.0  # Upper bound of integration
							snake_case__	  =       10.0  # define number of steps or resolution
							snake_case__	  =       [a, b]  # define boundary of integration
							snake_case__	  =       method_a(a  ,					a )
							print(F'''y = {y}''' )


if __name__ == "__main__":
	main()
