





from typing import Any

import numpy as np

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

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

   # Get eigenvalues and eignevectors for a symmetric matrix.
   A_					:				Optional[int]          =			np.linalg.eigh(_lowerCAmelCase	)

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

def      __snake_case						(					_lowerCAmelCase : int	) ->						Any:
   A_					:				Any          =			np.random.randn(_lowerCAmelCase				,				_lowerCAmelCase	)
   A_					:				Any          =			np.dot(_lowerCAmelCase				,				random_matrix.T	)
   assert _is_matrix_spd(_lowerCAmelCase	)
   return spd_matrix

def      __snake_case						(					_lowerCAmelCase : np.ndarray				,				_lowerCAmelCase : np.ndarray				,				_lowerCAmelCase : int = 1000				,				_lowerCAmelCase : float = 1e-8				,				) ->						Any:
   assert np.shape(_lowerCAmelCase	)[0] == np.shape(_lowerCAmelCase	)[1]
   assert np.shape(_lowerCAmelCase	)[0] == np.shape(_lowerCAmelCase	)[0]
   assert _is_matrix_spd(_lowerCAmelCase	)

   # Initialize solution guess, residual, search direction.
   A_					:				Optional[int]          =			np.zeros((np.shape(_lowerCAmelCase	)[0], 1)	)
   A_					:				List[str]          =			np.copy(_lowerCAmelCase	)
   A_					:				Dict          =			np.copy(_lowerCAmelCase	)

   # Set initial errors in solution guess and residual.
   A_					:				Dict          =			1e9
   A_					:				Any          =			1e9
   A_					:				int          =			1e9

   # Set iteration counter to threshold number of iterations.
   A_					:				str          =			0

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

      # The main algorithm.

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

      # Calculate errors.
      A_					:				Union[str, Any]          =			np.linalg.norm(r - ra	)
      A_					:				Dict          =			np.linalg.norm(x - xa	)
      A_					:				Optional[Any]          =			np.maximum(_lowerCAmelCase				,				_lowerCAmelCase	)

      # Update variables.
      A_					:				str          =			np.copy(_lowerCAmelCase	)
      A_					:				Any          =			np.copy(_lowerCAmelCase	)
      A_					:				Union[str, Any]          =			np.copy(_lowerCAmelCase	)

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

   return x

def      __snake_case						(					) ->						None:
   A_					:				List[str]          =			3
   A_					:				Tuple          =			_create_spd_matrix(_lowerCAmelCase	)
   A_					:				Optional[Any]          =			np.random.randn(_lowerCAmelCase				,				1	)
   A_					:				Tuple          =			np.dot(_lowerCAmelCase				,				_lowerCAmelCase	)

   # Numpy solution.
   A_					:				Union[str, Any]          =			np.linalg.solve(_lowerCAmelCase				,				_lowerCAmelCase	)

   # Our implementation.
   A_					:				str          =			conjugate_gradient(_lowerCAmelCase				,				_lowerCAmelCase	)

   # 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()







import torch
from torch import autograd


class 					__magic_name__  (							autograd.Function							):





     """simple docstring"""



     @staticmethod
     def 						SCREAMING_SNAKE_CASE		(							snake_case       :Any			,      snake_case       :torch.tensor			,      snake_case       :float			,      snake_case       :bool  ):





        '''simple docstring'''




        A_					:				int          =			inputs.numel()
        A_					:				str          =			int(0.005 * nb_elems  ) + 1
        if sigmoid:
           A_					:				List[Any]          =			(torch.sigmoid(snake_case  ) > threshold).type(inputs.type()  )
        else:
           A_					:				Optional[Any]          =			(inputs > threshold).type(inputs.type()  )
        if mask.sum() < nb_min:
           # We limit the pruning so that at least 0.5% (half a percent) of the weights are remaining
           A_					:				Tuple          =			inputs.flatten().kthvalue(max(nb_elems - nb_min			,      1  )  ).values
           A_					:				Optional[int]          =			(inputs > k_threshold).type(inputs.type()  )
        return mask





     @staticmethod
     def 						SCREAMING_SNAKE_CASE		(							snake_case       :Optional[int]			,      snake_case       :int  ):





        '''simple docstring'''




        return gradOutput, None, None




class 					__magic_name__  (							autograd.Function							):





     """simple docstring"""



     @staticmethod
     def 						SCREAMING_SNAKE_CASE		(							snake_case       :Tuple			,      snake_case       :torch.tensor			,      snake_case       :float  ):





        '''simple docstring'''




        A_					:				Tuple          =			inputs.clone()
        A_      ,						A_					:				Optional[int]          =			inputs.flatten().sort(descending=snake_case  )
        A_					:				Dict          =			int(threshold * inputs.numel()  )

        # flat_out and mask access the same memory.
        A_					:				str          =			mask.flatten()
        A_					:				List[Any]          =			0
        A_					:				List[str]          =			1
        return mask





     @staticmethod
     def 						SCREAMING_SNAKE_CASE		(							snake_case       :List[Any]			,      snake_case       :Optional[Any]  ):





        '''simple docstring'''




        return gradOutput, None




class 					__magic_name__  (							lowerCamelCase__							):





     """simple docstring"""



     @staticmethod
     def 						SCREAMING_SNAKE_CASE		(							snake_case       :torch.tensor			,      snake_case       :float  ):





        '''simple docstring'''




        A_					:				str          =			inputs.clone()
        A_      ,						A_					:				Optional[int]          =			inputs.abs().flatten().sort(descending=snake_case  )
        A_					:				int          =			int(threshold * inputs.numel()  )

        # flat_out and mask access the same memory.
        A_					:				List[str]          =			mask.flatten()
        A_					:				List[str]          =			0
        A_					:				Union[str, Any]          =			1
        return mask
