






'''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()







'''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()



