code: from typing import Any
  
  import numpy as np
  
  
  def SCREAMING_SNAKE_CASE				(     lowercase_:	np.ndarray				):
         assert np.shape(lowercase_				)[0] == np.shape(lowercase_				)[1]
  
         # If matrix not symmetric, exit right away.
         if np.allclose(lowercase_		,						matrix.T				) is False:
                return False
  
         # Get eigenvalues and eignevectors for a symmetric matrix.
         _UpperCamelCase						,      _UpperCamelCase     :       int  						=  np.linalg.eigh(lowercase_				)
  
         # Check sign of all eigenvalues.
         # np.all returns a value of type np.bool_
         return bool(np.all(eigen_values > 0				)				)
  
  
  def SCREAMING_SNAKE_CASE				(     lowercase_:	int				):
         _UpperCamelCase     :       Optional[int]  						=  np.random.randn(lowercase_		,						lowercase_				)
         _UpperCamelCase     :       int  						=  np.dot(lowercase_		,						random_matrix.T				)
         assert _is_matrix_spd(lowercase_				)
         return spd_matrix
  
  
  def SCREAMING_SNAKE_CASE				(     lowercase_:	np.ndarray		,						lowercase_:	np.ndarray		,						lowercase_:	int = 1000		,						lowercase_:	float = 1e-8		,						):
         assert np.shape(lowercase_				)[0] == np.shape(lowercase_				)[1]
         assert np.shape(lowercase_				)[0] == np.shape(lowercase_				)[0]
         assert _is_matrix_spd(lowercase_				)
  
         # Initialize solution guess, residual, search direction.
         _UpperCamelCase     :       int  						=  np.zeros((np.shape(lowercase_				)[0], 1)				)
         _UpperCamelCase     :       Union[str, Any]  						=  np.copy(lowercase_				)
         _UpperCamelCase     :       str  						=  np.copy(lowercase_				)
  
         # Set initial errors in solution guess and residual.
         _UpperCamelCase     :       List[Any]  						=  1e9
         _UpperCamelCase     :       int  						=  1e9
         _UpperCamelCase     :       List[Any]  						=  1e9
  
         # Set iteration counter to threshold number of iterations.
         _UpperCamelCase     :       str  						=  0
  
         while error > tol:
                # Save this value so we only calculate the matrix-vector product once.
                _UpperCamelCase     :       Optional[int]  						=  np.dot(lowercase_		,						lowercase_				)
  
                # The main algorithm.
  
                # Update search direction magnitude.
                _UpperCamelCase     :       Any  						=  np.dot(ra.T		,						lowercase_				) / np.dot(pa.T		,						lowercase_				)
                # Update solution guess.
                _UpperCamelCase     :       Any  						=  xa + alpha * pa
                # Calculate new residual.
                _UpperCamelCase     :       Dict  						=  ra - alpha * w
                # Calculate new Krylov subspace scale.
                _UpperCamelCase     :       Dict  						=  np.dot(r.T		,						lowercase_				) / np.dot(ra.T		,						lowercase_				)
                # Calculate new A conjuage search direction.
                _UpperCamelCase     :       Dict  						=  r + beta * pa
  
                # Calculate errors.
                _UpperCamelCase     :       Dict  						=  np.linalg.norm(r - ra				)
                _UpperCamelCase     :       Any  						=  np.linalg.norm(x - xa				)
                _UpperCamelCase     :       Any  						=  np.maximum(lowercase_		,						lowercase_				)
  
                # Update variables.
                _UpperCamelCase     :       Union[str, Any]  						=  np.copy(lowercase_				)
                _UpperCamelCase     :       Tuple  						=  np.copy(lowercase_				)
                _UpperCamelCase     :       str  						=  np.copy(lowercase_				)
  
                # Update number of iterations.
                iterations += 1
                if iterations > max_iterations:
                       break
  
         return x
  
  
  def SCREAMING_SNAKE_CASE				(     ):
         _UpperCamelCase     :       Optional[int]  						=  3
         _UpperCamelCase     :       Union[str, Any]  						=  _create_spd_matrix(lowercase_				)
         _UpperCamelCase     :       str  						=  np.random.randn(lowercase_		,						1				)
         _UpperCamelCase     :       List[Any]  						=  np.dot(lowercase_		,						lowercase_				)
  
         # Numpy solution.
         _UpperCamelCase     :       List[str]  						=  np.linalg.solve(lowercase_		,						lowercase_				)
  
         # Our implementation.
         _UpperCamelCase     :       List[Any]  						=  conjugate_gradient(lowercase_		,						lowercase_				)
  
         # 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()
  
  
  
  
style_context: from .chunk_utils import chunk_layer
  from .data_transforms import make_atomaa_masks
  from .feats import atomaa_to_atomaa, frames_and_literature_positions_to_atomaa_pos, torsion_angles_to_frames
  from .loss import compute_predicted_aligned_error, compute_tm
  from .protein import Protein as OFProtein
  from .protein import to_pdb
  from .rigid_utils import Rigid, Rotation
  from .tensor_utils import dict_multimap, flatten_final_dims, permute_final_dims
  
  
  
  