




'''simple docstring'''




from typing import Any

import numpy as np




def 						snake_case__    ( lowerCamelCase__				:   np.ndarray			)   ->   bool:
					assert np.shape(lowerCamelCase__			)[0] == np.shape(lowerCamelCase__			)[1]

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

					# Get eigenvalues and eignevectors for a symmetric matrix.
					A_ ,A_ :     str =							np.linalg.eigh(lowerCamelCase__			)

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




def 						snake_case__    ( lowerCamelCase__				:   int			)   ->   Any:
					A_ :     Optional[int] =							np.random.randn(lowerCamelCase__			,		lowerCamelCase__			)
					A_ :     List[Any] =							np.dot(lowerCamelCase__			,		random_matrix.T			)
					assert _is_matrix_spd(lowerCamelCase__			)
					return spd_matrix




def 						snake_case__    ( lowerCamelCase__				:   np.ndarray			,		lowerCamelCase__				:   np.ndarray			,		lowerCamelCase__				:   int = 1_0_0_0			,		lowerCamelCase__				:   float = 1e-8			,		)   ->   Any:
					assert np.shape(lowerCamelCase__			)[0] == np.shape(lowerCamelCase__			)[1]
					assert np.shape(lowerCamelCase__			)[0] == np.shape(lowerCamelCase__			)[0]
					assert _is_matrix_spd(lowerCamelCase__			)

					# Initialize solution guess, residual, search direction.
					A_ :     Optional[Any] =							np.zeros((np.shape(lowerCamelCase__			)[0], 1)			)
					A_ :     int =							np.copy(lowerCamelCase__			)
					A_ :     Any =							np.copy(lowerCamelCase__			)

					# Set initial errors in solution guess and residual.
					A_ :     str =							1e9
					A_ :     List[Any] =							1e9
					A_ :     int =							1e9

					# Set iteration counter to threshold number of iterations.
					A_ :     Any =							0

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

										# The main algorithm.

										# Update search direction magnitude.
										A_ :     List[str] =							np.dot(ra.T			,		lowerCamelCase__			) / np.dot(pa.T			,		lowerCamelCase__			)
										# Update solution guess.
										A_ :     Optional[Any] =							xa + alpha * pa
										# Calculate new residual.
										A_ :     List[str] =							ra - alpha * w
										# Calculate new Krylov subspace scale.
										A_ :     str =							np.dot(r.T			,		lowerCamelCase__			) / np.dot(ra.T			,		lowerCamelCase__			)
										# Calculate new A conjuage search direction.
										A_ :     Any =							r + beta * pa

										# Calculate errors.
										A_ :     Any =							np.linalg.norm(r - ra			)
										A_ :     str =							np.linalg.norm(x - xa			)
										A_ :     Optional[int] =							np.maximum(lowerCamelCase__			,		lowerCamelCase__			)

										# Update variables.
										A_ :     Optional[int] =							np.copy(lowerCamelCase__			)
										A_ :     List[Any] =							np.copy(lowerCamelCase__			)
										A_ :     str =							np.copy(lowerCamelCase__			)

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

					return x




def 						snake_case__    ( )   ->   None:
					A_ :     Optional[int] =							3
					A_ :     str =							_create_spd_matrix(lowerCamelCase__			)
					A_ :     int =							np.random.randn(lowerCamelCase__			,		1			)
					A_ :     Optional[Any] =							np.dot(lowerCamelCase__			,		lowerCamelCase__			)

					# Numpy solution.
					A_ :     str =							np.linalg.solve(lowerCamelCase__			,		lowerCamelCase__			)

					# Our implementation.
					A_ :     Any =							conjugate_gradient(lowerCamelCase__			,		lowerCamelCase__			)

					# 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'''




def 						snake_case__    ( lowerCamelCase__				:   int			)   ->   bool:
					if not isinstance(lowerCamelCase__			,		lowerCamelCase__			):
										A_ :     int =							f'Input value of [number={number}] must be an integer'
										raise TypeError(lowerCamelCase__			)
					if number < 0 or number % 2 == 1:
										return False
					A_ :     Dict =							int(number**0.5			)
					return number == number_sqrt * (number_sqrt + 1)


if __name__ == "__main__":
							import doctest

							doctest.testmod()



