


'''simple docstring'''






def 		_A			(lowerCAmelCase__					:list[list]						)				->					list[list]:




     '''simple docstring'''



     _a      =							current_set.copy()
     for row_index, row in enumerate(lowerCAmelCase__						):
          _a      =							row[0]
          for column_index, column in enumerate(lowerCAmelCase__						):
               if magnitude == 0:
                    _a      =							column
                    continue
               _a      =							column / magnitude
    # Subtract to cancel term
     _a      =							current_set[0]
     _a      =							[first_row]
     _a      =							current_set[1::]
     for row in current_set:
          _a      =							[]
          # If first term is 0, it is already in form we want, so we preserve it
          if row[0] == 0:
               final_set.append(lowerCAmelCase__						)
               continue
          for column_index in range(len(lowerCAmelCase__						)						):
               temp_row.append(first_row[column_index] - row[column_index]						)
          final_set.append(lowerCAmelCase__						)
     # Create next recursion iteration set
     if len(final_set[0]						) != 3:
          _a      =							final_set[0]
          _a      =							[]
          _a      =							[]
          for row in final_set[1::]:
               current_first_column.append(row[0]						)
               next_iteration.append(row[1::]						)
          _a      =							simplify(lowerCAmelCase__						)
          for i in range(len(lowerCAmelCase__						)						):
               resultant[i].insert(0			,  current_first_column[i]						)
          resultant.insert(0			,  lowerCAmelCase__						)
          _a      =							resultant
     return final_set






def 		_A			(lowerCAmelCase__					:list[list]						)				->					list:




     '''simple docstring'''



     if len(lowerCAmelCase__						) == 0:
          raise IndexError('solve_simultaneous() requires n lists of length n+1'						)
     _a      =							len(lowerCAmelCase__						) + 1
     if any(len(lowerCAmelCase__						) != _length for item in equations						):
          raise IndexError('solve_simultaneous() requires n lists of length n+1'						)
     for row in equations:
          if any(not isinstance(lowerCAmelCase__			,  (int, float)						) for column in row						):
               raise ValueError('solve_simultaneous() requires lists of integers'						)
     if len(lowerCAmelCase__						) == 1:
          return [equations[0][-1] / equations[0][0]]
     _a      =							equations.copy()
     if any(0 in row for row in data_set						):
          _a      =							data_set.copy()
          _a      =							[]
          for row_index, row in enumerate(lowerCAmelCase__						):
               if 0 not in row:
                    _a      =							data_set.pop(lowerCAmelCase__						)
                    break
          if not full_row:
               raise ValueError('solve_simultaneous() requires at least 1 full equation'						)
          data_set.insert(0			,  lowerCAmelCase__						)
     _a      =							data_set.copy()
     _a      =							simplify(lowerCAmelCase__						)
     _a      =							simplified[::-1]
     _a      =							[]
     for row in simplified:
          _a      =							row[-1]
          if not solutions:
               if row[-2] == 0:
                    solutions.append(0						)
                    continue
               solutions.append(current_solution / row[-2]						)
               continue
          _a      =							row.copy()[: len(lowerCAmelCase__						) - 1 :]
          while temp_row[0] == 0:
               temp_row.pop(0						)
          if len(lowerCAmelCase__						) == 0:
               solutions.append(0						)
               continue
          _a      =							temp_row[1::]
          _a      =							temp_row[::-1]
          for column_index, column in enumerate(lowerCAmelCase__						):
               current_solution -= column * solutions[column_index]
          solutions.append(lowerCAmelCase__						)
     _a      =							[]
     for item in solutions:
          final.append(float(round(lowerCAmelCase__			,  5						)						)						)
     return final[::-1]


if __name__ == "__main__":
 import doctest

 doctest.testmod()
 a_   :      Union[str, Any]				    =      [
     [2, 1, 1, 1, 1, 4],
     [1, 2, 1, 1, 1, 5],
     [1, 1, 2, 1, 1, 6],
     [1, 1, 1, 2, 1, 7],
     [1, 1, 1, 1, 2, 8],
 ]
 print(solve_simultaneous(eq))
 print(solve_simultaneous([[4, 2]]))





'''simple docstring'''






import colorsys

from PIL import Image  # type: ignore






def 		_A			(lowerCAmelCase__					:float			,  lowerCAmelCase__					:float			,  lowerCAmelCase__					:int						)				->					float:




     '''simple docstring'''



     _a      =							x
     _a      =							y
     for step in range(lowerCAmelCase__						):  # noqa: B007
          _a      =							a * a - b * b + x
          _a      =							2 * a * b + y
          _a      =							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 		_A			(lowerCAmelCase__					:float						)				->					tuple:




     '''simple docstring'''



     if distance == 1:
          return (0, 0, 0)
     else:
          return (2_55, 2_55, 2_55)






def 		_A			(lowerCAmelCase__					:float						)				->					tuple:




     '''simple docstring'''



     if distance == 1:
          return (0, 0, 0)
     else:
          return tuple(round(i * 2_55						) for i in colorsys.hsv_to_rgb(lowerCAmelCase__			,  1			,  1						)						)






def 		_A			(lowerCAmelCase__					:int = 8_00			,  lowerCAmelCase__					:int = 6_00			,  lowerCAmelCase__					:float = -0.6			,  lowerCAmelCase__					:float = 0			,  lowerCAmelCase__					:float = 3.2			,  lowerCAmelCase__					:int = 50			,  lowerCAmelCase__					:bool = True			,  )				->					Image.Image:




     '''simple docstring'''



     _a      =							Image.new('RGB'			,  (image_width, image_height)						)
     _a      =							img.load()

     # loop through the image-coordinates
     for image_x in range(lowerCAmelCase__						):
          for image_y in range(lowerCAmelCase__						):
               # determine the figure-coordinates based on the image-coordinates
               _a      =							figure_width / image_width * image_height
               _a      =							figure_center_x + (image_x / image_width - 0.5) * figure_width
               _a      =							figure_center_y + (image_y / image_height - 0.5) * figure_height

               _a      =							get_distance(lowerCAmelCase__			,  lowerCAmelCase__			,  lowerCAmelCase__						)

               # color the corresponding pixel based on the selected coloring-function
               if use_distance_color_coding:
                    _a      =							get_color_coded_rgb(lowerCAmelCase__						)
               else:
                    _a      =							get_black_and_white_rgb(lowerCAmelCase__						)

     return img


if __name__ == "__main__":
 import doctest

 doctest.testmod()

 # colored version, full figure
 a_   :      Optional[Any]				    =      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()

