'''simple docstring'''


from typing import Any

import numpy as np


def    _lowerCamelCase       ( lowerCamelCase_   :				np.ndarray  ):



     """simple docstring"""





     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.
     UpperCAmelCase_  ,     UpperCAmelCase_				:  List[Any]  = 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    _lowerCamelCase       ( lowerCamelCase_   :				int  ):



     """simple docstring"""





     UpperCAmelCase_				:  int  = np.random.randn(lowerCamelCase_      ,	lowerCamelCase_  )
     UpperCAmelCase_				:  Union[str, Any]  = np.dot(lowerCamelCase_      ,	random_matrix.T  )
     assert _is_matrix_spd(lowerCamelCase_  )
     return spd_matrix


def    _lowerCamelCase       ( lowerCamelCase_   :				np.ndarray      ,	lowerCamelCase_   :				np.ndarray      ,	lowerCamelCase_   :				int = 1000      ,	lowerCamelCase_   :				float = 1e-8      ,	):



     """simple docstring"""





     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.
     UpperCAmelCase_				:  Union[str, Any]  = np.zeros((np.shape(lowerCamelCase_  )[0], 1)  )
     UpperCAmelCase_				:  Union[str, Any]  = np.copy(lowerCamelCase_  )
     UpperCAmelCase_				:  str  = np.copy(lowerCamelCase_  )

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

     # Set iteration counter to threshold number of iterations.
     UpperCAmelCase_				:  Union[str, Any]  = 0

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

          # The main algorithm.

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

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

          # Update variables.
          UpperCAmelCase_				:  Optional[int]  = np.copy(lowerCamelCase_  )
          UpperCAmelCase_				:  List[str]  = np.copy(lowerCamelCase_  )
          UpperCAmelCase_				:  Any  = np.copy(lowerCamelCase_  )

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

     return x


def    _lowerCamelCase       ( ):



     """simple docstring"""





     UpperCAmelCase_				:  Union[str, Any]  = 3
     UpperCAmelCase_				:  List[str]  = _create_spd_matrix(lowerCamelCase_  )
     UpperCAmelCase_				:  List[Any]  = np.random.randn(lowerCamelCase_      ,	1  )
     UpperCAmelCase_				:  int  = np.dot(lowerCamelCase_      ,	lowerCamelCase_  )

     # Numpy solution.
     UpperCAmelCase_				:  Dict  = np.linalg.solve(lowerCamelCase_      ,	lowerCamelCase_  )

     # Our implementation.
     UpperCAmelCase_				:  Union[str, Any]  = 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()








'''simple docstring'''







def 				_snake_case (		lowercase = 1_0_0_0						)       ->       int:
				__a       :     Any								=	[]
				__a       :     List[Any]								=	[]
				__a       :     Any								=	3
				__a       :     Optional[Any]								=	5
				__a       :     Optional[Any]								=	1
				while True:
								__a       :     List[str]								=	z * temp
								if result < n:
												zmulti.append(lowercase						)
												temp += 1
								else:
												__a       :     Any								=	1
												break
				while True:
								__a       :     Optional[int]								=	x * temp
								if result < n:
												xmulti.append(lowercase						)
												temp += 1
								else:
												break
				__a       :     Any								=	list(set(xmulti + zmulti						)						)
				return sum(lowercase						)


if __name__ == "__main__":
					print(f'''{solution() = }''')