





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.
     lowerCAmelCase_ ,	lowerCAmelCase_       :							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:
     lowerCAmelCase_       :							Any			 =		np.random.randn(_a       , _a      )
     lowerCAmelCase_       :							Optional[int]			 =		np.dot(_a       , random_matrix.T      )
     assert _is_matrix_spd(_a      )
     return spd_matrix






def 				_lowerCAmelCase			(	_a :       np.ndarray       , _a :       np.ndarray       , _a :       int = 10_00       , _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.
     lowerCAmelCase_       :							Optional[int]			 =		np.zeros((np.shape(_a      )[0], 1)      )
     lowerCAmelCase_       :							Any			 =		np.copy(_a      )
     lowerCAmelCase_       :							Any			 =		np.copy(_a      )

     # Set initial errors in solution guess and residual.
     lowerCAmelCase_       :							str			 =		1E9
     lowerCAmelCase_       :							List[Any]			 =		1E9
     lowerCAmelCase_       :							Union[str, Any]			 =		1E9

     # Set iteration counter to threshold number of iterations.
     lowerCAmelCase_       :							Any			 =		0

     while error > tol:
          # Save this value so we only calculate the matrix-vector product once.
          lowerCAmelCase_       :							Any			 =		np.dot(_a       , _a      )

          # The main algorithm.

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

          # Calculate errors.
          lowerCAmelCase_       :							List[str]			 =		np.linalg.norm(r - ra      )
          lowerCAmelCase_       :							str			 =		np.linalg.norm(x - xa      )
          lowerCAmelCase_       :							Dict			 =		np.maximum(_a       , _a      )

          # Update variables.
          lowerCAmelCase_       :							Dict			 =		np.copy(_a      )
          lowerCAmelCase_       :							Tuple			 =		np.copy(_a      )
          lowerCAmelCase_       :							Union[str, Any]			 =		np.copy(_a      )

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

     return x






def 				_lowerCAmelCase			(	)						->      None:
     lowerCAmelCase_       :							str			 =		3
     lowerCAmelCase_       :							str			 =		_create_spd_matrix(_a      )
     lowerCAmelCase_       :							List[str]			 =		np.random.randn(_a       , 1      )
     lowerCAmelCase_       :							str			 =		np.dot(_a       , _a      )

     # Numpy solution.
     lowerCAmelCase_       :							Dict			 =		np.linalg.solve(_a       , _a      )

     # Our implementation.
     lowerCAmelCase_       :							Tuple			 =		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 :       str      )						->      int:
     lowerCAmelCase_       :							List[str]			 =		53_81
     for x in s:
          lowerCAmelCase_       :							Dict			 =		((hash_value << 5) + hash_value) + ord(_a      )
     return hash_value & 0XFFFFFFFF
