





from __future__ import annotations





def     lowerCamelCase__  (  UpperCamelCase__      :							int    )      ->			list[int]:



					'''simple docstring'''




					assert (
					    isinstance(UpperCamelCase__							,     UpperCamelCase__    ) and mask > 0
					), F'''mask needs to be positive integer, your input {mask}'''
					_snake_case						=					[]
					_snake_case						=					mask

					while submask:
										all_submasks.append(UpperCamelCase__    )
										_snake_case						=					(submask - 1) & mask

					return all_submasks


if __name__ == "__main__":
					import doctest

					doctest.testmod()










from typing import Any

import numpy as np





def     lowerCamelCase__  (  UpperCamelCase__      :							np.ndarray    )      ->			bool:



					'''simple docstring'''




					assert np.shape(UpperCamelCase__    )[0] == np.shape(UpperCamelCase__    )[1]

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

					# Get eigenvalues and eignevectors for a symmetric matrix.
					_snake_case				,  _snake_case						=					np.linalg.eigh(UpperCamelCase__    )

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





def     lowerCamelCase__  (  UpperCamelCase__      :							int    )      ->			Any:



					'''simple docstring'''




					_snake_case						=					np.random.randn(UpperCamelCase__							,     UpperCamelCase__    )
					_snake_case						=					np.dot(UpperCamelCase__							,     random_matrix.T    )
					assert _is_matrix_spd(UpperCamelCase__    )
					return spd_matrix





def     lowerCamelCase__  (  UpperCamelCase__      :							np.ndarray							,     UpperCamelCase__      :							np.ndarray							,     UpperCamelCase__      :							int = 1_000							,     UpperCamelCase__      :							float = 1e-8							,     )      ->			Any:



					'''simple docstring'''




					assert np.shape(UpperCamelCase__    )[0] == np.shape(UpperCamelCase__    )[1]
					assert np.shape(UpperCamelCase__    )[0] == np.shape(UpperCamelCase__    )[0]
					assert _is_matrix_spd(UpperCamelCase__    )

					# Initialize solution guess, residual, search direction.
					_snake_case						=					np.zeros((np.shape(UpperCamelCase__    )[0], 1)    )
					_snake_case						=					np.copy(UpperCamelCase__    )
					_snake_case						=					np.copy(UpperCamelCase__    )

					# 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(UpperCamelCase__							,     UpperCamelCase__    )

										# The main algorithm.

										# Update search direction magnitude.
										_snake_case						=					np.dot(ra.T							,     UpperCamelCase__    ) / np.dot(pa.T							,     UpperCamelCase__    )
										# 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							,     UpperCamelCase__    ) / np.dot(ra.T							,     UpperCamelCase__    )
										# 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(UpperCamelCase__							,     UpperCamelCase__    )

										# Update variables.
										_snake_case						=					np.copy(UpperCamelCase__    )
										_snake_case						=					np.copy(UpperCamelCase__    )
										_snake_case						=					np.copy(UpperCamelCase__    )

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

					return x





def     lowerCamelCase__  (  )      ->			None:



					'''simple docstring'''




					_snake_case						=					3
					_snake_case						=					_create_spd_matrix(UpperCamelCase__    )
					_snake_case						=					np.random.randn(UpperCamelCase__							,     1    )
					_snake_case						=					np.dot(UpperCamelCase__							,     UpperCamelCase__    )

					# Numpy solution.
					_snake_case						=					np.linalg.solve(UpperCamelCase__							,     UpperCamelCase__    )

					# Our implementation.
					_snake_case						=					conjugate_gradient(UpperCamelCase__							,     UpperCamelCase__    )

					# 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()



