
'''simple docstring'''





from typing import Any

import numpy as np
def 	snake_case_							(				_a  :				np.ndarray      ):
      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.
      snake_case_							,						snake_case_    							=				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 	snake_case_							(				_a  :				int      ):
      snake_case_    							=				np.random.randn(_a			,       _a      )
      snake_case_    							=				np.dot(_a			,       random_matrix.T      )
      assert _is_matrix_spd(_a      )
      return spd_matrix
def 	snake_case_							(				_a  :				np.ndarray			,       _a  :				np.ndarray			,       _a  :				int = 1_000			,       _a  :				float = 1E-8			,       ):
      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.
      snake_case_    							=				np.zeros((np.shape(_a      )[0], 1)      )
      snake_case_    							=				np.copy(_a      )
      snake_case_    							=				np.copy(_a      )

      # Set initial errors in solution guess and residual.
      snake_case_    							=				1E9
      snake_case_    							=				1E9
      snake_case_    							=				1E9

      # Set iteration counter to threshold number of iterations.
      snake_case_    							=				0

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

            # The main algorithm.

            # Update search direction magnitude.
            snake_case_    							=				np.dot(ra.T			,       _a      ) / np.dot(pa.T			,       _a      )
            # Update solution guess.
            snake_case_    							=				xa + alpha * pa
            # Calculate new residual.
            snake_case_    							=				ra - alpha * w
            # Calculate new Krylov subspace scale.
            snake_case_    							=				np.dot(r.T			,       _a      ) / np.dot(ra.T			,       _a      )
            # Calculate new A conjuage search direction.
            snake_case_    							=				r + beta * pa

            # Calculate errors.
            snake_case_    							=				np.linalg.norm(r - ra      )
            snake_case_    							=				np.linalg.norm(x - xa      )
            snake_case_    							=				np.maximum(_a			,       _a      )

            # Update variables.
            snake_case_    							=				np.copy(_a      )
            snake_case_    							=				np.copy(_a      )
            snake_case_    							=				np.copy(_a      )

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

      return x
def 	snake_case_							(				):
      snake_case_    							=				3
      snake_case_    							=				_create_spd_matrix(_a      )
      snake_case_    							=				np.random.randn(_a			,       1      )
      snake_case_    							=				np.dot(_a			,       _a      )

      # Numpy solution.
      snake_case_    							=				np.linalg.solve(_a			,       _a      )

      # Our implementation.
      snake_case_    							=				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()


"""simple docstring"""







_a	:	Optional[Any]=					{
    "+": lambda x, y: x + y,
    "-": lambda x, y: x - y,
    "*": lambda x, y: x * y,
    "/": lambda x, y: x / y,
}
def     __UpperCAmelCase							(       UpperCAmelCase_       :							List[Any]    )				->     List[Any]:

	'''simple docstring'''





	return c.isdigit()
def     __UpperCAmelCase							(       UpperCAmelCase_       :							int    )				->     List[Any]:

	'''simple docstring'''





	__snake_case     :				Dict 						=					[]

	# iterate over the string in reverse order
	for c in expression.split()[::-1]:
		# push operand to stack
		if is_operand(UpperCAmelCase_    ):
			stack.append(int(UpperCAmelCase_    )    )

		else:
			# pop values from stack can calculate the result
			# push the result onto the stack again
			__snake_case     :				Optional[Any] 						=					stack.pop()
			__snake_case     :				int 						=					stack.pop()
			stack.append(calc[c](UpperCAmelCase_     ,				UpperCAmelCase_    )    )

	return stack.pop()


# Driver code
if __name__ == "__main__":
	_a	:	Union[str, Any]=					"+ 9 * 2 6"
	print(evaluate(test_expression))

	_a	:	List[Any]=					"/ * 10 2 + 4 1 "
	print(evaluate(test_expression))





