
from __future__ import annotations

from collections import deque
from queue import Queue
from timeit import timeit

lowercase__							:							int													=     {
    "A": ["B", "C"],
    "B": ["A", "D", "E"],
    "C": ["A", "F"],
    "D": ["B"],
    "E": ["B", "F"],
    "F": ["C", "E"],
}




def 		A_	(							snake_case							:				dict	, snake_case							:				str	) ->      list[str]:



						'''simple docstring'''

						__UpperCamelCase       					=     {start}
						__UpperCamelCase       					=     [start]
						__UpperCamelCase       					=     Queue()
						queue.put(snake_case	)
						while not queue.empty():
												__UpperCamelCase       					=     queue.get()
												for w in graph[v]:
																		if w not in explored:
																								explored.add(snake_case	)
																								result.append(snake_case	)
																								queue.put(snake_case	)
						return result




def 		A_	(							snake_case							:				dict	, snake_case							:				str	) ->      list[str]:



						'''simple docstring'''

						__UpperCamelCase       					=     {start}
						__UpperCamelCase       					=     [start]
						__UpperCamelCase       					=     deque([start]	)
						while queue:
												__UpperCamelCase       					=     queue.popleft()
												for child in graph[v]:
																		if child not in visited:
																								visited.add(snake_case	)
																								result.append(snake_case	)
																								queue.append(snake_case	)
						return result




def 		A_	(							snake_case							:				str	) ->      None:



						'''simple docstring'''

						__UpperCamelCase       					=     f"from __main__ import G, {name}"
						__UpperCamelCase       					=     10000
						__UpperCamelCase       					=     timeit(f"{name}(G, 'A')"	, setup=snake_case	, number=snake_case	)
						print(f"{name:<35} finished {number} runs in {res:.5f} seconds"	)


if __name__ == "__main__":
				import doctest

				doctest.testmod()

				benchmark_function("breadth_first_search")
				benchmark_function("breadth_first_search_with_deque")
				# breadth_first_search                finished 10000 runs in 0.20999 seconds
				# breadth_first_search_with_deque     finished 10000 runs in 0.01421 seconds






from typing import Any

import numpy as np




def 		A_	(							snake_case							:				np.ndarray	) ->      bool:



						'''simple docstring'''

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

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

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

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




def 		A_	(							snake_case							:				int	) ->      Any:



						'''simple docstring'''

						__UpperCamelCase       					=     np.random.randn(snake_case	, snake_case	)
						__UpperCamelCase       					=     np.dot(snake_case	, random_matrix.T	)
						assert _is_matrix_spd(snake_case	)
						return spd_matrix




def 		A_	(							snake_case							:				np.ndarray	, snake_case							:				np.ndarray	, snake_case							:				int = 1000	, snake_case							:				float = 1e-8	, ) ->      Any:



						'''simple docstring'''

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

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

						# Set initial errors in solution guess and residual.
						__UpperCamelCase       					=     1e9
						__UpperCamelCase       					=     1e9
						__UpperCamelCase       					=     1e9

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

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

												# The main algorithm.

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

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

												# Update variables.
												__UpperCamelCase       					=     np.copy(snake_case	)
												__UpperCamelCase       					=     np.copy(snake_case	)
												__UpperCamelCase       					=     np.copy(snake_case	)

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

						return x




def 		A_	(							) ->      None:



						'''simple docstring'''

						__UpperCamelCase       					=     3
						__UpperCamelCase       					=     _create_spd_matrix(snake_case	)
						__UpperCamelCase       					=     np.random.randn(snake_case	, 1	)
						__UpperCamelCase       					=     np.dot(snake_case	, snake_case	)

						# Numpy solution.
						__UpperCamelCase       					=     np.linalg.solve(snake_case	, snake_case	)

						# Our implementation.
						__UpperCamelCase       					=     conjugate_gradient(snake_case	, snake_case	)

						# 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()




