



import matplotlib.pyplot as plt
import numpy as np




def    UpperCAmelCase					(  lowercase__						:					np.ndarray				,					lowercase__						:					np.ndarray				,					lowercase__						:					float	):



			'''simple docstring'''





			a__					=  len(lowercase__	)  # Number of training samples
			a__					=  np.eye(lowercase__	)  # Initialize weights as identity matrix
			for j in range(lowercase__	):
						a__					=  point - x_train[j]
						a__					=  np.exp(diff @ diff.T / (-2.0 * tau**2)	)

			return weights




def    UpperCAmelCase					(  lowercase__						:					np.ndarray				,					lowercase__						:					np.ndarray				,					lowercase__						:					np.ndarray				,					lowercase__						:					float	):



			'''simple docstring'''





			a__					=  weight_matrix(lowercase__				,					lowercase__				,					lowercase__	)
			a__					=  np.linalg.inv(x_train.T @ weight_mat @ x_train	) @ (
			    x_train.T @ weight_mat @ y_train.T
			)

			return weight




def    UpperCAmelCase					(  lowercase__						:					np.ndarray				,					lowercase__						:					np.ndarray				,					lowercase__						:					float	):



			'''simple docstring'''





			a__					=  np.zeros(len(lowercase__	)	)  # Initialize array of predictions
			for i, item in enumerate(lowercase__	):
						a__					=  item @ local_weight(lowercase__				,					lowercase__				,					lowercase__				,					lowercase__	)

			return y_pred




def    UpperCAmelCase					(  lowercase__						:					str				,					lowercase__						:					str				,					lowercase__						:					str	):



			'''simple docstring'''





			import seaborn as sns

			a__					=  sns.load_dataset(lowercase__	)
			a__					=  np.array(data[x_name]	)
			a__					=  np.array(data[y_name]	)

			a__					=  np.ones(len(lowercase__	)	)

			# pairing elements of one and x_data
			a__					=  np.column_stack((one, x_data)	)

			return x_train, x_data, y_data




def    UpperCAmelCase					(  lowercase__						:					np.ndarray				,					lowercase__						:					np.ndarray				,					lowercase__						:					np.ndarray				,					lowercase__						:					np.ndarray				,					lowercase__						:					str				,					lowercase__						:					str				,					):



			'''simple docstring'''





			a__					=  np.sort(lowercase__				,					axis=0	)
			plt.scatter(lowercase__				,					lowercase__				,					color="""blue"""	)
			plt.plot(
			    x_train_sorted[:, 1]				,					preds[x_train[:, 1].argsort(0	)]				,					color="""yellow"""				,					linewidth=5				,					)
			plt.title("""Local Weighted Regression"""	)
			plt.xlabel(lowercase__	)
			plt.ylabel(lowercase__	)
			plt.show()


if __name__ == "__main__":
	import doctest

	doctest.testmod()

	# Demo with a dataset from the seaborn module
	_lowercase					,    _lowercase					,    _lowercase				:					Tuple     					=load_data("""tips""", """total_bill""", """tip""")
	_lowercase				:					Optional[int]     					=local_weight_regression(training_data_x, tip, 5)
	plot_preds(training_data_x, predictions, total_bill, tip, """total_bill""", """tip""")






from typing import Any

import numpy as np




def    UpperCAmelCase					(  lowercase__						:					np.ndarray	):



			'''simple docstring'''





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

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

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

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




def    UpperCAmelCase					(  lowercase__						:					int	):



			'''simple docstring'''





			a__					=  np.random.randn(lowercase__				,					lowercase__	)
			a__					=  np.dot(lowercase__				,					random_matrix.T	)
			assert _is_matrix_spd(lowercase__	)
			return spd_matrix




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



			'''simple docstring'''





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

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

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

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

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

						# The main algorithm.

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

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

						# Update variables.
						a__					=  np.copy(lowercase__	)
						a__					=  np.copy(lowercase__	)
						a__					=  np.copy(lowercase__	)

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

			return x




def    UpperCAmelCase					(  ):



			'''simple docstring'''





			a__					=  3
			a__					=  _create_spd_matrix(lowercase__	)
			a__					=  np.random.randn(lowercase__				,					1	)
			a__					=  np.dot(lowercase__				,					lowercase__	)

			# Numpy solution.
			a__					=  np.linalg.solve(lowercase__				,					lowercase__	)

			# Our implementation.
			a__					=  conjugate_gradient(lowercase__				,					lowercase__	)

			# 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()

