


from __future__ import annotations



def      _lowerCAmelCase  (							lowerCAmelCase_							:list[int]							)->list[int]:
			'''simple docstring'''






			if not list_of_ints:
						return []

			if min(lowerCAmelCase_							) < 0:
						raise ValueError("All numbers must be positive"							)

			snake_case_         =   max(len(bin(lowerCAmelCase_							)[2:]							) for x in list_of_ints							)
			return _msd_radix_sort(lowerCAmelCase_						,  lowerCAmelCase_							)



def      _lowerCAmelCase  (							lowerCAmelCase_							:list[int]						,  lowerCAmelCase_							:int							)->list[int]:
			'''simple docstring'''






			if bit_position == 0 or len(lowerCAmelCase_							) in [0, 1]:
						return list_of_ints

			snake_case_         =   []
			snake_case_         =   []
			# Split numbers based on bit at bit_position from the right
			for number in list_of_ints:
						if (number >> (bit_position - 1)) & 1:
									# number has a one at bit bit_position
									ones.append(lowerCAmelCase_							)
						else:
									# number has a zero at bit bit_position
									zeros.append(lowerCAmelCase_							)

    # recursively split both lists further
			snake_case_         =   _msd_radix_sort(lowerCAmelCase_						,  bit_position - 1							)
			snake_case_         =   _msd_radix_sort(lowerCAmelCase_						,  bit_position - 1							)

			# recombine lists
			snake_case_         =   zeros
			res.extend(lowerCAmelCase_							)

			return res



def      _lowerCAmelCase  (							lowerCAmelCase_							:list[int]							)->Dict:
			'''simple docstring'''






			snake_case_         =   len(lowerCAmelCase_							)
			if not list_of_ints or length == 1:
						return

			if min(lowerCAmelCase_							) < 0:
						raise ValueError("All numbers must be positive"							)

			snake_case_         =   max(len(bin(lowerCAmelCase_							)[2:]							) for x in list_of_ints							)
			_msd_radix_sort_inplace(lowerCAmelCase_						,  lowerCAmelCase_						,  0						,  lowerCAmelCase_							)



def      _lowerCAmelCase  (							lowerCAmelCase_							:list[int]						,  lowerCAmelCase_							:int						,  lowerCAmelCase_							:int						,  lowerCAmelCase_							:int							)->Tuple:
			'''simple docstring'''






			if bit_position == 0 or end_index - begin_index <= 1:
						return

			bit_position -= 1

			snake_case_         =   begin_index
			snake_case_         =   end_index - 1
			while i <= j:
						snake_case_         =   False
						if not (list_of_ints[i] >> bit_position) & 1:
									# found zero at the beginning
									i += 1
									snake_case_         =   True
						if (list_of_ints[j] >> bit_position) & 1:
									# found one at the end
									j -= 1
									snake_case_         =   True

						if changed:
									continue

						snake_case_      ,      snake_case_         =   list_of_ints[j], list_of_ints[i]
						j -= 1
						if j != i:
									i += 1

			_msd_radix_sort_inplace(lowerCAmelCase_						,  lowerCAmelCase_						,  lowerCAmelCase_						,  lowerCAmelCase_							)
			_msd_radix_sort_inplace(lowerCAmelCase_						,  lowerCAmelCase_						,  lowerCAmelCase_						,  lowerCAmelCase_							)


if __name__ == "__main__":
			import doctest

			doctest.testmod()








from typing import Any

import numpy as np



def      _lowerCAmelCase  (							lowerCAmelCase_							:np.ndarray							)->bool:
   '''simple docstring'''






   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.
   snake_case_      ,      snake_case_         =   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      _lowerCAmelCase  (							lowerCAmelCase_							:int							)->Any:
   '''simple docstring'''






   snake_case_         =   np.random.randn(lowerCAmelCase_						,  lowerCAmelCase_							)
   snake_case_         =   np.dot(lowerCAmelCase_						,  random_matrix.T							)
   assert _is_matrix_spd(lowerCAmelCase_							)
   return spd_matrix



def      _lowerCAmelCase  (							lowerCAmelCase_							:np.ndarray						,  lowerCAmelCase_							:np.ndarray						,  lowerCAmelCase_							:int = 1_000						,  lowerCAmelCase_							:float = 1e-8						,  )->Any:
   '''simple docstring'''






   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.
   snake_case_         =   np.zeros((np.shape(lowerCAmelCase_							)[0], 1)							)
   snake_case_         =   np.copy(lowerCAmelCase_							)
   snake_case_         =   np.copy(lowerCAmelCase_							)

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

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

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

      # The main algorithm.

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

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

      # Update variables.
      snake_case_         =   np.copy(lowerCAmelCase_							)
      snake_case_         =   np.copy(lowerCAmelCase_							)
      snake_case_         =   np.copy(lowerCAmelCase_							)

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

   return x



def      _lowerCAmelCase  (							)->None:
   '''simple docstring'''






   snake_case_         =   3
   snake_case_         =   _create_spd_matrix(lowerCAmelCase_							)
   snake_case_         =   np.random.randn(lowerCAmelCase_						,  1							)
   snake_case_         =   np.dot(lowerCAmelCase_						,  lowerCAmelCase_							)

   # Numpy solution.
   snake_case_         =   np.linalg.solve(lowerCAmelCase_						,  lowerCAmelCase_							)

   # Our implementation.
   snake_case_         =   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()




