



'''simple docstring'''


from __future__ import annotations

from enum import Enum, unique
from typing import TypeVar

# Create a generic variable that can be 'Enum', or any subclass.
lowercase											=      TypeVar('''T''', bound='''Enum''')







@unique
class 						UpperCAmelCase						(		__a):


 '''simple docstring'''




 __magic_name__       :			str	=   80
 __magic_name__       :			Tuple	=   70
 __magic_name__       :			str	=   60
 __magic_name__       :			Optional[Any]	=   50
 __magic_name__       :			Optional[int]	=   40
 __magic_name__       :			Dict	=   30
 __magic_name__       :			Optional[int]	=   20
 __magic_name__       :			List[Any]	=   10







@unique
class 						UpperCAmelCase						(		__a):


 '''simple docstring'''




 __magic_name__       :			Optional[int]	=   24
 __magic_name__       :			List[str]	=   21
 __magic_name__       :			Dict	=   18
 __magic_name__       :			Optional[int]	=   15
 __magic_name__       :			Tuple	=   12
 __magic_name__       :			str	=   9
 __magic_name__       :			Optional[Any]	=   6
 __magic_name__       :			Tuple	=   3
 __magic_name__       :			Union[str, Any]	=   2
 __magic_name__       :			Tuple	=   1
 __magic_name__       :			Tuple	=   -1
 __magic_name__       :			Tuple	=   -2
 __magic_name__       :			Dict	=   -3
 __magic_name__       :			Dict	=   -6
 __magic_name__       :			Optional[Any]	=   -9
 __magic_name__       :			Optional[int]	=   -12
 __magic_name__       :			str	=   -15
 __magic_name__       :			int	=   -18
 __magic_name__       :			Union[str, Any]	=   -21
 __magic_name__       :			Dict	=   -24




 @classmethod
 def    lowercase_	( cls)			->   Dict:



      """simple docstring"""







      return {unit.name: unit.value for unit in cls if unit.value > 0}




 @classmethod
 def    lowercase_	( cls)			->   Tuple:



      """simple docstring"""







      return {unit.name: unit.value for unit in cls if unit.value < 0}






def 			UpperCAmelCase_							(		lowercase__							):


     '''simple docstring'''







     a_									=SIUnit.get_positive() if value > 0 else SIUnit.get_negative()
     for name_prefix, value_prefix in prefixes.items():
          a_									=value / (1_0**value_prefix)
          if numerical_part > 1:
               return F"""{numerical_part!s} {name_prefix}"""
     return str(SCREAMING_SNAKE_CASE_							)






def 			UpperCAmelCase_							(		lowercase__							):


     '''simple docstring'''







     for prefix in BinaryUnit:
          a_									=value / (2**prefix.value)
          if numerical_part > 1:
               return F"""{numerical_part!s} {prefix.name}"""
     return str(SCREAMING_SNAKE_CASE_							)


if __name__ == "__main__":
  import doctest

  doctest.testmod()




from typing import Any

import numpy as np
def 					lowerCAmelCase__     (    SCREAMING_SNAKE_CASE_	)							->						bool:
			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[int]   =						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 					lowerCAmelCase__     (    SCREAMING_SNAKE_CASE_	)							->						Any:
			lowerCAmelCase__ : List[Any]   =						np.random.randn(SCREAMING_SNAKE_CASE_			,				SCREAMING_SNAKE_CASE_	)
			lowerCAmelCase__ : Any   =						np.dot(SCREAMING_SNAKE_CASE_			,				random_matrix.T	)
			assert _is_matrix_spd(SCREAMING_SNAKE_CASE_	)
			return spd_matrix
def 					lowerCAmelCase__     (    SCREAMING_SNAKE_CASE_			,				SCREAMING_SNAKE_CASE_			,				SCREAMING_SNAKE_CASE_ = 1_000			,				SCREAMING_SNAKE_CASE_ = 1e-8			,				)							->						Any:
			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[Any]   =						np.zeros((np.shape(SCREAMING_SNAKE_CASE_	)[0], 1)	)
			lowerCAmelCase__ : str   =						np.copy(SCREAMING_SNAKE_CASE_	)
			lowerCAmelCase__ : Dict   =						np.copy(SCREAMING_SNAKE_CASE_	)

			# Set initial errors in solution guess and residual.
			lowerCAmelCase__ : Union[str, Any]   =						1e9
			lowerCAmelCase__ : int   =						1e9
			lowerCAmelCase__ : Union[str, Any]   =						1e9

			# Set iteration counter to threshold number of iterations.
			lowerCAmelCase__ : List[str]   =						0

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

						# The main algorithm.

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

						# Calculate errors.
						lowerCAmelCase__ : List[str]   =						np.linalg.norm(r - ra	)
						lowerCAmelCase__ : Tuple   =						np.linalg.norm(x - xa	)
						lowerCAmelCase__ : str   =						np.maximum(SCREAMING_SNAKE_CASE_			,				SCREAMING_SNAKE_CASE_	)

						# Update variables.
						lowerCAmelCase__ : List[Any]   =						np.copy(SCREAMING_SNAKE_CASE_	)
						lowerCAmelCase__ : Dict   =						np.copy(SCREAMING_SNAKE_CASE_	)
						lowerCAmelCase__ : str   =						np.copy(SCREAMING_SNAKE_CASE_	)

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

			return x
def 					lowerCAmelCase__     (    )							->						None:
			lowerCAmelCase__ : Optional[Any]   =						3
			lowerCAmelCase__ : Optional[int]   =						_create_spd_matrix(SCREAMING_SNAKE_CASE_	)
			lowerCAmelCase__ : Tuple   =						np.random.randn(SCREAMING_SNAKE_CASE_			,				1	)
			lowerCAmelCase__ : int   =						np.dot(SCREAMING_SNAKE_CASE_			,				SCREAMING_SNAKE_CASE_	)

			# Numpy solution.
			lowerCAmelCase__ : Optional[Any]   =						np.linalg.solve(SCREAMING_SNAKE_CASE_			,				SCREAMING_SNAKE_CASE_	)

			# Our implementation.
			lowerCAmelCase__ : Optional[int]   =						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()