






from typing import Any

import numpy as np

def        _lowerCamelCase		(  __A      :							np.ndarray	) ->   bool:
    assert np.shape(__A	)[0] == np.shape(__A	)[1]

    # If matrix not symmetric, exit right away.
    if np.allclose(__A						,						matrix.T	) is False:
        return False

    # Get eigenvalues and eignevectors for a symmetric matrix.
    _UpperCAmelCase					,	_UpperCAmelCase			: Optional[Any]     					=			np.linalg.eigh(__A	)

    # Check sign of all eigenvalues.
    # np.all returns a value of type np.bool_
    return bool(np.all(eigen_values > 0	)	)

def        _lowerCamelCase		(  __A      :							int	) ->   Any:
    _UpperCAmelCase			: str     					=			np.random.randn(__A						,						__A	)
    _UpperCAmelCase			: Any     					=			np.dot(__A						,						random_matrix.T	)
    assert _is_matrix_spd(__A	)
    return spd_matrix

def        _lowerCamelCase		(  __A      :							np.ndarray						,						__A      :							np.ndarray						,						__A      :							int = 1_000						,						__A      :							float = 1e-8						,						) ->   Any:
    assert np.shape(__A	)[0] == np.shape(__A	)[1]
    assert np.shape(__A	)[0] == np.shape(__A	)[0]
    assert _is_matrix_spd(__A	)

    # Initialize solution guess, residual, search direction.
    _UpperCAmelCase			: Dict     					=			np.zeros((np.shape(__A	)[0], 1)	)
    _UpperCAmelCase			: Union[str, Any]     					=			np.copy(__A	)
    _UpperCAmelCase			: Optional[int]     					=			np.copy(__A	)

    # Set initial errors in solution guess and residual.
    _UpperCAmelCase			: Union[str, Any]     					=			1e9
    _UpperCAmelCase			: List[Any]     					=			1e9
    _UpperCAmelCase			: Tuple     					=			1e9

    # Set iteration counter to threshold number of iterations.
    _UpperCAmelCase			: List[Any]     					=			0

    while error > tol:
        # Save this value so we only calculate the matrix-vector product once.
        _UpperCAmelCase			: List[Any]     					=			np.dot(__A						,						__A	)

        # The main algorithm.

        # Update search direction magnitude.
        _UpperCAmelCase			: List[Any]     					=			np.dot(ra.T						,						__A	) / np.dot(pa.T						,						__A	)
        # Update solution guess.
        _UpperCAmelCase			: str     					=			xa + alpha * pa
        # Calculate new residual.
        _UpperCAmelCase			: Dict     					=			ra - alpha * w
        # Calculate new Krylov subspace scale.
        _UpperCAmelCase			: List[str]     					=			np.dot(r.T						,						__A	) / np.dot(ra.T						,						__A	)
        # Calculate new A conjuage search direction.
        _UpperCAmelCase			: Optional[Any]     					=			r + beta * pa

        # Calculate errors.
        _UpperCAmelCase			: List[Any]     					=			np.linalg.norm(r - ra	)
        _UpperCAmelCase			: List[str]     					=			np.linalg.norm(x - xa	)
        _UpperCAmelCase			: str     					=			np.maximum(__A						,						__A	)

        # Update variables.
        _UpperCAmelCase			: Optional[int]     					=			np.copy(__A	)
        _UpperCAmelCase			: Tuple     					=			np.copy(__A	)
        _UpperCAmelCase			: str     					=			np.copy(__A	)

        # Update number of iterations.
        iterations += 1
        if iterations > max_iterations:
            break

    return x

def        _lowerCamelCase		(  ) ->   None:
    _UpperCAmelCase			: Any     					=			3
    _UpperCAmelCase			: List[str]     					=			_create_spd_matrix(__A	)
    _UpperCAmelCase			: Dict     					=			np.random.randn(__A						,						1	)
    _UpperCAmelCase			: Optional[Any]     					=			np.dot(__A						,						__A	)

    # Numpy solution.
    _UpperCAmelCase			: List[str]     					=			np.linalg.solve(__A						,						__A	)

    # Our implementation.
    _UpperCAmelCase			: List[str]     					=			conjugate_gradient(__A						,						__A	)

    # Ensure both solutions are close to x_true (and therefore one another).
    assert np.linalg.norm(x_numpy - x_true	) <= 1e-6
    assert np.linalg.norm(x_conjugate_gradient - x_true	) <= 1e-6


if __name__ == "__main__":
  import doctest

  doctest.testmod()
  test_conjugate_gradient()












def        _lowerCamelCase		(  __A      :							dict						,						__A      :							int	) ->   str:
    if not root or root not in binary_tree_mirror_dictionary:
        return
    _UpperCAmelCase					,	_UpperCAmelCase			: int     					=			binary_tree_mirror_dictionary[root][:2]
    _UpperCAmelCase			: int     					=			[right_child, left_child]
    binary_tree_mirror_dict(__A						,						__A	)
    binary_tree_mirror_dict(__A						,						__A	)

def        _lowerCamelCase		(  __A      :							dict						,						__A      :							int = 1	) ->   dict:
    if not binary_tree:
        raise ValueError('''binary tree cannot be empty'''	)
    if root not in binary_tree:
        _UpperCAmelCase			: Dict     					=			f'''root {root} is not present in the binary_tree'''
        raise ValueError(__A	)
    _UpperCAmelCase			: int     					=			dict(__A	)
    binary_tree_mirror_dict(__A						,						__A	)
    return binary_tree_mirror_dictionary


if __name__ == "__main__":
  SCREAMING_SNAKE_CASE								=  {1: [2, 3], 2: [4, 5], 3: [6, 7], 7: [8, 9]}
  print(F'Binary tree: {binary_tree}')
  SCREAMING_SNAKE_CASE								=  binary_tree_mirror(binary_tree, 5)
  print(F'Binary tree mirror: {binary_tree_mirror_dictionary}')




