




from typing import Any

import numpy as np






def  snake_case (							A      )							->							bool:




	'''simple docstring'''


	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_							:		List[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  snake_case (							A      )							->							Any:




	'''simple docstring'''


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






def  snake_case (							A						,       A						,       A = 1000						,       A = 1e-8						,       )							->							Any:




	'''simple docstring'''


	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_							:		List[Any]							       =							np.zeros((np.shape(A      )[0], 1)      )
	lowerCamelCase_							:		List[str]							       =							np.copy(A      )
	lowerCamelCase_							:		List[Any]							       =							np.copy(A      )

	# Set initial errors in solution guess and residual.
	lowerCamelCase_							:		Optional[int]							       =							1e9
	lowerCamelCase_							:		Optional[int]							       =							1e9
	lowerCamelCase_							:		Tuple							       =							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_							:		List[Any]							       =							np.dot(A						,       A      )

		# The main algorithm.

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

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

		# Update variables.
		lowerCamelCase_							:		Any							       =							np.copy(A      )
		lowerCamelCase_							:		Tuple							       =							np.copy(A      )
		lowerCamelCase_							:		Any							       =							np.copy(A      )

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

	return x






def  snake_case (							)							->							None:




	'''simple docstring'''


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

	# Numpy solution.
	lowerCamelCase_							:		int							       =							np.linalg.solve(A						,       A      )

	# Our implementation.
	lowerCamelCase_							:		Any							       =							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()






from typing import Callable






def  snake_case (							A      )							->							List[str]:




	'''simple docstring'''


	def wrapper(A      ):
		lowerCamelCase_							:		Dict							       =							function_with_docstring.__doc__
		return target_function

	return wrapper
