import colorsys

from PIL import Image  # type: ignore




def 							lowerCamelCase			(   UpperCAmelCase__ :      float   ,   UpperCAmelCase__ :      float   ,   UpperCAmelCase__ :      int				)    ->   float:
							lowercase_      :    Union[str, Any]       		= x
							lowercase_      :    int       		= y
							for step in range(UpperCAmelCase__				):  # noqa: B007
														lowercase_      :    Tuple       		= a * a - b * b + x
														lowercase_      :    int       		= 2 * a * b + y
														lowercase_      :    List[str]       		= a_new

														# divergence happens for all complex number with an absolute value
														# greater than 4
														if a * a + b * b > 4:
																					break
							return step / (max_step - 1)




def 							lowerCamelCase			(   UpperCAmelCase__ :      float				)    ->   tuple:
							if distance == 1:
														return (0, 0, 0)
							else:
														return (255, 255, 255)




def 							lowerCamelCase			(   UpperCAmelCase__ :      float				)    ->   tuple:
							if distance == 1:
														return (0, 0, 0)
							else:
														return tuple(round(i * 255				) for i in colorsys.hsv_to_rgb(UpperCAmelCase__   ,   1   ,   1				)				)




def 							lowerCamelCase			(   UpperCAmelCase__ :      int = 800   ,   UpperCAmelCase__ :      int = 600   ,   UpperCAmelCase__ :      float = -0.6   ,   UpperCAmelCase__ :      float = 0   ,   UpperCAmelCase__ :      float = 3.2   ,   UpperCAmelCase__ :      int = 50   ,   UpperCAmelCase__ :      bool = True   ,   )    ->   Image.Image:
							lowercase_      :    str       		= Image.new("""RGB"""   ,   (image_width, image_height)				)
							lowercase_      :    int       		= img.load()

							# loop through the image-coordinates
							for image_x in range(UpperCAmelCase__				):
														for image_y in range(UpperCAmelCase__				):
																					# determine the figure-coordinates based on the image-coordinates
																					lowercase_      :    Dict       		= figure_width / image_width * image_height
																					lowercase_      :    Any       		= figure_center_x + (image_x / image_width - 0.5) * figure_width
																					lowercase_      :    Optional[int]       		= figure_center_y + (image_y / image_height - 0.5) * figure_height

																					lowercase_      :    List[Any]       		= get_distance(UpperCAmelCase__   ,   UpperCAmelCase__   ,   UpperCAmelCase__				)

																					# color the corresponding pixel based on the selected coloring-function
																					if use_distance_color_coding:
																												lowercase_      :    Union[str, Any]       		= get_color_coded_rgb(UpperCAmelCase__				)
																					else:
																												lowercase_      :    Optional[Any]       		= get_black_and_white_rgb(UpperCAmelCase__				)

							return img


if __name__ == "__main__":
						import doctest

						doctest.testmod()

						# colored version, full figure
						_lowercase		:				Dict	       =		get_image()

						# uncomment for colored version, different section, zoomed in
						# img = get_image(figure_center_x = -0.6, figure_center_y = -0.4,
						# figure_width = 0.8)

						# uncomment for black and white version, full figure
						# img = get_image(use_distance_color_coding = False)

						# uncomment to save the image
						# img.save("mandelbrot.png")

						img.show()






'''simple docstring'''




import unittest

import numpy as np




def 							lowerCamelCase			(   UpperCAmelCase__ :      np.ndarray   ,   UpperCAmelCase__ :      np.ndarray   ,   UpperCAmelCase__ :      np.ndarray   ,   UpperCAmelCase__ :      np.ndarray | None = None   ,   )    ->   np.ndarray:
							lowercase_      :    List[Any]       		= np.shape(UpperCAmelCase__				)
							lowercase_      :    Dict       		= np.shape(UpperCAmelCase__				)
							lowercase_      :    int       		= np.shape(UpperCAmelCase__				)

							if shape_a[0] != shape_b[0]:
														lowercase_      :    Optional[int]       		= (
														    """Expected the same number of rows for A and B. """
														    F'''Instead found A of size {shape_a} and B of size {shape_b}'''
														)
														raise ValueError(UpperCAmelCase__				)

							if shape_b[1] != shape_c[1]:
														lowercase_      :    Optional[Any]       		= (
														    """Expected the same number of columns for B and C. """
														    F'''Instead found B of size {shape_b} and C of size {shape_c}'''
														)
														raise ValueError(UpperCAmelCase__				)

							lowercase_      :    Any       		= pseudo_inv
							if a_inv is None:
														try:
																					lowercase_      :    List[str]       		= np.linalg.inv(UpperCAmelCase__				)
														except np.linalg.LinAlgError:
																					raise ValueError(
																					    """Input matrix A is not invertible. Cannot compute Schur complement."""				)

							return mat_c - mat_b.T @ a_inv @ mat_b





class 					__magic_name__		(					unittest.TestCase):







			def 			SCREAMING_SNAKE_CASE_  (			self       :						Optional[int]					):
										lowercase_      :    Tuple       		= np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]					)
										lowercase_      :    int       		= np.array([[0, 3], [3, 0], [2, 3]]					)
										lowercase_      :    Dict       		= np.array([[2, 1], [6, 3]]					)

										lowercase_      :    Union[str, Any]       		= schur_complement(lowercase_       ,    lowercase_       ,    lowercase_					)

										lowercase_      :    List[Any]       		= np.block([[a, b], [b.T, c]]					)

										lowercase_      :    Optional[int]       		= np.linalg.det(lowercase_					)
										lowercase_      :    int       		= np.linalg.det(lowercase_					)
										lowercase_      :    int       		= np.linalg.det(lowercase_					)

										self.assertAlmostEqual(lowercase_       ,    det_a * det_s					)







			def 			SCREAMING_SNAKE_CASE_  (			self       :						str					):
										lowercase_      :    int       		= np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]					)
										lowercase_      :    Optional[Any]       		= np.array([[0, 3], [3, 0], [2, 3]]					)
										lowercase_      :    Union[str, Any]       		= np.array([[2, 1], [6, 3]]					)

										with self.assertRaises(lowercase_					):
																	schur_complement(lowercase_       ,    lowercase_       ,    lowercase_					)







			def 			SCREAMING_SNAKE_CASE_  (			self       :						Any					):
										lowercase_      :    List[Any]       		= np.array([[1, 2, 1], [2, 1, 2], [3, 2, 4]]					)
										lowercase_      :    List[Any]       		= np.array([[0, 3], [3, 0], [2, 3]]					)
										lowercase_      :    str       		= np.array([[2, 1, 3], [6, 3, 5]]					)

										with self.assertRaises(lowercase_					):
																	schur_complement(lowercase_       ,    lowercase_       ,    lowercase_					)


if __name__ == "__main__":
						import doctest

						doctest.testmod()
						unittest.main()





