



from typing import Any

import numpy as np







def  _UpperCAmelCase  (				SCREAMING_SNAKE_CASE__  ):


						'''simple docstring'''




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

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

						# Get eigenvalues and eignevectors for a symmetric matrix.
						lowerCAmelCase     ,	lowerCAmelCase	:   Optional[Any]					     =					np.linalg.eigh(SCREAMING_SNAKE_CASE__  )

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







def  _UpperCAmelCase  (				SCREAMING_SNAKE_CASE__  ):


						'''simple docstring'''




						lowerCAmelCase	:   Tuple					     =					np.random.randn(SCREAMING_SNAKE_CASE__			,SCREAMING_SNAKE_CASE__  )
						lowerCAmelCase	:   Optional[Any]					     =					np.dot(SCREAMING_SNAKE_CASE__			,random_matrix.T  )
						assert _is_matrix_spd(SCREAMING_SNAKE_CASE__  )
						return spd_matrix







def  _UpperCAmelCase  (				SCREAMING_SNAKE_CASE__			,SCREAMING_SNAKE_CASE__			,SCREAMING_SNAKE_CASE__ = 1_0_0_0			,SCREAMING_SNAKE_CASE__ = 1e-8			,):


						'''simple docstring'''




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

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

						# Set initial errors in solution guess and residual.
						lowerCAmelCase	:   Dict					     =					1e9
						lowerCAmelCase	:   Tuple					     =					1e9
						lowerCAmelCase	:   List[str]					     =					1e9

						# Set iteration counter to threshold number of iterations.
						lowerCAmelCase	:   int					     =					0

						while error > tol:
												# Save this value so we only calculate the matrix-vector product once.
												lowerCAmelCase	:   List[str]					     =					np.dot(SCREAMING_SNAKE_CASE__			,SCREAMING_SNAKE_CASE__  )

												# The main algorithm.

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

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

												# Update variables.
												lowerCAmelCase	:   Tuple					     =					np.copy(SCREAMING_SNAKE_CASE__  )
												lowerCAmelCase	:   Dict					     =					np.copy(SCREAMING_SNAKE_CASE__  )
												lowerCAmelCase	:   List[Any]					     =					np.copy(SCREAMING_SNAKE_CASE__  )

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

						return x







def  _UpperCAmelCase  (				):


						'''simple docstring'''




						lowerCAmelCase	:   Any					     =					3
						lowerCAmelCase	:   Any					     =					_create_spd_matrix(SCREAMING_SNAKE_CASE__  )
						lowerCAmelCase	:   Any					     =					np.random.randn(SCREAMING_SNAKE_CASE__			,1  )
						lowerCAmelCase	:   Tuple					     =					np.dot(SCREAMING_SNAKE_CASE__			,SCREAMING_SNAKE_CASE__  )

						# Numpy solution.
						lowerCAmelCase	:   Union[str, Any]					     =					np.linalg.solve(SCREAMING_SNAKE_CASE__			,SCREAMING_SNAKE_CASE__  )

						# Our implementation.
						lowerCAmelCase	:   List[str]					     =					conjugate_gradient(SCREAMING_SNAKE_CASE__			,SCREAMING_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()






def  _UpperCAmelCase  (				SCREAMING_SNAKE_CASE__  ):


						'''simple docstring'''




						lowerCAmelCase	:   List[str]					     =					str(SCREAMING_SNAKE_CASE__  ).strip()
						if not bin_string:
												raise ValueError("""Empty string was passed to the function"""  )
						lowerCAmelCase	:   Optional[int]					     =					bin_string[0] == """-"""
						if is_negative:
												lowerCAmelCase	:   Tuple					     =					bin_string[1:]
						if not all(char in """01""" for char in bin_string  ):
												raise ValueError("""Non-binary value was passed to the function"""  )
						lowerCAmelCase	:   Union[str, Any]					     =					0
						for char in bin_string:
												lowerCAmelCase	:   int					     =					2 * decimal_number + int(SCREAMING_SNAKE_CASE__  )
						return -decimal_number if is_negative else decimal_number


if __name__ == "__main__":
		from doctest import testmod

		testmod()

