code: 
  
  
  
  
  
  
  '''simple docstring'''
  
  
  
  
  
  from typing import Any
  
  import numpy as np
  
  
  
  
  
  
  def 	UpperCAmelCase__(							_SCREAMING_SNAKE_CASE				:   np.ndarray   ):
  
  
  
  
  
  
  			"""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.
  			__A,  __A=  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				:   int   ):
  
  
  
  
  
  
  			"""simple docstring"""
  
  
  
  
  			__A=  np.random.randn(_SCREAMING_SNAKE_CASE,_SCREAMING_SNAKE_CASE   )
  			__A=  np.dot(_SCREAMING_SNAKE_CASE,random_matrix.T   )
  			assert _is_matrix_spd(_SCREAMING_SNAKE_CASE   )
  			return spd_matrix
  
  
  
  
  
  
  def 	UpperCAmelCase__(							_SCREAMING_SNAKE_CASE				:   np.ndarray,_SCREAMING_SNAKE_CASE				:   np.ndarray,_SCREAMING_SNAKE_CASE				:   int = 1000,_SCREAMING_SNAKE_CASE				:   float = 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.
  			__A=  np.zeros((np.shape(_SCREAMING_SNAKE_CASE   )[0], 1)   )
  			__A=  np.copy(_SCREAMING_SNAKE_CASE   )
  			__A=  np.copy(_SCREAMING_SNAKE_CASE   )
  
  			# 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(_SCREAMING_SNAKE_CASE,_SCREAMING_SNAKE_CASE   )
  
  						# The main algorithm.
  
  						# Update search direction magnitude.
  						__A=  np.dot(ra.T,_SCREAMING_SNAKE_CASE   ) / np.dot(pa.T,_SCREAMING_SNAKE_CASE   )
  						# Update solution guess.
  						__A=  xa + alpha * pa
  						# Calculate new residual.
  						__A=  ra - alpha * w
  						# Calculate new Krylov subspace scale.
  						__A=  np.dot(r.T,_SCREAMING_SNAKE_CASE   ) / np.dot(ra.T,_SCREAMING_SNAKE_CASE   )
  						# 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(_SCREAMING_SNAKE_CASE,_SCREAMING_SNAKE_CASE   )
  
  						# Update variables.
  						__A=  np.copy(_SCREAMING_SNAKE_CASE   )
  						__A=  np.copy(_SCREAMING_SNAKE_CASE   )
  						__A=  np.copy(_SCREAMING_SNAKE_CASE   )
  
  						# Update number of iterations.
  						iterations += 1
  						if iterations > max_iterations:
  									break
  
  			return x
  
  
  
  
  
  
  def 	UpperCAmelCase__(							):
  
  
  
  
  
  
  			"""simple docstring"""
  
  
  
  
  			__A=  3
  			__A=  _create_spd_matrix(_SCREAMING_SNAKE_CASE   )
  			__A=  np.random.randn(_SCREAMING_SNAKE_CASE,1   )
  			__A=  np.dot(_SCREAMING_SNAKE_CASE,_SCREAMING_SNAKE_CASE   )
  
  			# Numpy solution.
  			__A=  np.linalg.solve(_SCREAMING_SNAKE_CASE,_SCREAMING_SNAKE_CASE   )
  
  			# Our implementation.
  			__A=  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()
  
  
  
  
style_context: 
  
  
  '''simple docstring'''
  
  
  
  
  
  import numpy as np
  from cva import destroyAllWindows, imread, imshow, waitKey
  class 	a__				:
  
  
  
  
  
  			'''simple docstring'''
  
  			def __init__(     self       : Dict ,						lowerCAmelCase_       : Optional[Any] ,						lowerCAmelCase_       : int  )			->		Optional[int]:
  						__A=  0
  						# max greyscale value for #FFFFFF
  						__A=  int(self.get_greyscale(255 ,						255 ,						255  )  )
  
  						if not self.min_threshold < threshold < self.max_threshold:
  									__A=  F"""Factor value should be from 0 to {self.max_threshold}"""
  									raise ValueError(lowerCAmelCase_  )
  
  						__A=  input_img
  						__A=  threshold
  						__A,  __A=  self.input_img.shape[1], self.input_img.shape[0]
  
  						# error table size (+4 columns and +1 row) greater than input image because of
  						# lack of if statements
  						__A=  [
  						    [0 for _ in range(self.height + 4  )] for __ in range(self.width + 1  )
  						]
  						__A=  np.ones((self.width, self.height, 3) ,						np.uinta  ) * 255
  			@classmethod
  			def 				lowerCAmelCase    (     cls       : Tuple ,						lowerCAmelCase_       : int ,						lowerCAmelCase_       : int ,						lowerCAmelCase_       : int  )			->		float:
  						return 0.1_14 * blue + 0.5_87 * green + 0.21_26 * red
  
  			def 				lowerCAmelCase    (     self       : str  )			->		None:
  						for y in range(self.height  ):
  									for x in range(self.width  ):
  												__A=  int(self.get_greyscale(*self.input_img[y][x]  )  )
  												if self.threshold > greyscale + self.error_table[y][x]:
  															__A=  (0, 0, 0)
  															__A=  greyscale + self.error_table[x][y]
  												else:
  															__A=  (255, 255, 255)
  															__A=  greyscale + self.error_table[x][y] - 255
  												self.error_table[y][x + 1] += int(8 / 32 * current_error  )
  												self.error_table[y][x + 2] += int(4 / 32 * current_error  )
  												self.error_table[y + 1][x] += int(8 / 32 * current_error  )
  												self.error_table[y + 1][x + 1] += int(4 / 32 * current_error  )
  												self.error_table[y + 1][x + 2] += int(2 / 32 * current_error  )
  												self.error_table[y + 1][x - 1] += int(4 / 32 * current_error  )
  												self.error_table[y + 1][x - 2] += int(2 / 32 * current_error  )
  
  
  if __name__ == "__main__":
  					# create Burke's instances with original images in greyscale
  					UpperCAmelCase__      =							[
  					    Burkes(imread('''image_data/lena.jpg''', 1), threshold)
  					    for threshold in (1, 1_2_6, 1_3_0, 1_4_0)
  					]
  
  					for burkes in burkes_instances:
  										burkes.process()
  
  					for burkes in burkes_instances:
  										imshow(
  										    F"""Original image with dithering threshold: {burkes.threshold}""",
  										    burkes.output_img,
  										)
  
  					waitKey(0)
  					destroyAllWindows()
  
  
  
  