






"""simple docstring"""



def 		_SCREAMING_SNAKE_CASE					(       __snake_case     :    list[list[int]]       ):





  '''simple docstring'''





  if len(__snake_case       ) == 0 or not all(len(__snake_case       ) == 2 for pair in coordinates       ):
    raise ValueError('The program cannot work out a fitting polynomial.'       )

  if len({tuple(__snake_case       ) for pair in coordinates}       ) != len(__snake_case       ):
    raise ValueError('The program cannot work out a fitting polynomial.'       )

  lowercase				       =			{x for x, _ in coordinates}
  if len(__snake_case       ) == 1:
    return f'x={coordinates[0][0]}'

  if len(__snake_case       ) != len(__snake_case       ):
    raise ValueError('The program cannot work out a fitting polynomial.'       )

  lowercase				       =			len(__snake_case       )

  lowercase				       =			0
  lowercase				       =			[]
  # put the x and x to the power values in a matrix
  while count_of_line < x:
    lowercase				       =			0
    lowercase				       =			coordinates[count_of_line][0]
    lowercase				       =			[]
    while count_in_line < x:
      count_line.append(a ** (x - (count_in_line + 1))       )
      count_in_line += 1
    matrix.append(__snake_case       )
    count_of_line += 1

  lowercase				       =			0
  # put the y values into a vector
  lowercase				       =			[]
  while count_of_line < x:
    vector.append(coordinates[count_of_line][1]       )
    count_of_line += 1

  lowercase				       =			0

  while count < x:
    lowercase				       =			0
    while zahlen < x:
      if count == zahlen:
        zahlen += 1
      if zahlen == x:
        break
      lowercase				       =			matrix[zahlen][count] / matrix[count][count]
      for counting_columns, item in enumerate(matrix[count]       ):
        # manipulating all the values in the matrix
        matrix[zahlen][counting_columns] -= item * bruch
      # manipulating the values in the vector
      vector[zahlen] -= vector[count] * bruch
      zahlen += 1
    count += 1

  lowercase				       =			0
  # make solutions
  lowercase				       =			[]
  while count < x:
    solution.append(str(vector[count] / matrix[count][count]       )       )
    count += 1

  lowercase				       =			0
  lowercase				       =			'f(x)='

  while count < x:
    lowercase				       =			solution[count].split('E'       )
    if len(__snake_case       ) > 1:
      lowercase				       =			f'{remove_e[0]}*10^{remove_e[1]}'
    solved += f'x^{x - (count + 1)}*{solution[count]}'
    if count + 1 != x:
      solved += "+"
    count += 1

  return solved


if __name__ == "__main__":
    print(points_to_polynomial([]))
    print(points_to_polynomial([[]]))
    print(points_to_polynomial([[1, 0], [2, 0], [3, 0]]))
    print(points_to_polynomial([[1, 1], [2, 1], [3, 1]]))
    print(points_to_polynomial([[1, 3], [2, 3], [3, 3]]))
    print(points_to_polynomial([[1, 1], [2, 2], [3, 3]]))
    print(points_to_polynomial([[1, 1], [2, 4], [3, 9]]))
    print(points_to_polynomial([[1, 3], [2, 6], [3, 1_1]]))
    print(points_to_polynomial([[1, -3], [2, -6], [3, -1_1]]))
    print(points_to_polynomial([[1, 5], [2, 2], [3, 9]]))





from typing import SupportsIndex

import numpy as np
from scipy.ndimage import convolve





def 				__a							(       __A			,							__A			,							__A						):
    SCREAMING_SNAKE_CASE__         =    np.stack((horizontal_flow, vertical_flow)			,							2						)

    # Create a grid of all pixel coordinates and subtract the flow to get the
    # target pixels coordinates
    SCREAMING_SNAKE_CASE__         =    np.stack(
        np.meshgrid(np.arange(0			,							image.shape[1]						)			,							np.arange(0			,							image.shape[0]						)						)			,							2						)
    SCREAMING_SNAKE_CASE__         =    np.round(grid - flow						).astype(np.intaa						)

    # Find the locations outside of the original image
    SCREAMING_SNAKE_CASE__         =    (grid < 0) | (grid >= np.array([image.shape[1], image.shape[0]]						))
    SCREAMING_SNAKE_CASE__         =    0

    SCREAMING_SNAKE_CASE__         =    image[grid[:, :, 1], grid[:, :, 0]]

    # Set pixels at invalid locations to 0
    SCREAMING_SNAKE_CASE__         =    0

    return warped





def 				__a							(       __A			,							__A			,							__A			,							__A = None			,							):
    if alpha is None:
        SCREAMING_SNAKE_CASE__         =    0.1

    # Initialize flow
    SCREAMING_SNAKE_CASE__         =    np.zeros_like(__A						)
    SCREAMING_SNAKE_CASE__         =    np.zeros_like(__A						)

    # Prepare kernels for the calculation of the derivatives and the average velocity
    SCREAMING_SNAKE_CASE__         =    np.array([[-1, 1], [-1, 1]]						) * 0.25
    SCREAMING_SNAKE_CASE__         =    np.array([[-1, -1], [1, 1]]						) * 0.25
    SCREAMING_SNAKE_CASE__         =    np.array([[1, 1], [1, 1]]						) * 0.25
    SCREAMING_SNAKE_CASE__         =    np.array(
        [[1 / 12, 1 / 6, 1 / 12], [1 / 6, 0, 1 / 6], [1 / 12, 1 / 6, 1 / 12]]						)

    # Iteratively refine the flow
    for _ in range(__A						):
        SCREAMING_SNAKE_CASE__         =    warp(__A			,							__A			,							__A						)
        SCREAMING_SNAKE_CASE__         =    convolve(__A			,							__A						) + convolve(__A			,							__A						)
        SCREAMING_SNAKE_CASE__         =    convolve(__A			,							__A						) + convolve(__A			,							__A						)
        SCREAMING_SNAKE_CASE__         =    convolve(__A			,							__A						) + convolve(__A			,							-kernel_t						)

        SCREAMING_SNAKE_CASE__         =    convolve(__A			,							__A						)
        SCREAMING_SNAKE_CASE__         =    convolve(__A			,							__A						)

        # This updates the flow as proposed in the paper (Step 12)
        SCREAMING_SNAKE_CASE__         =    (
            derivative_x * avg_horizontal_velocity
            + derivative_y * avg_vertical_velocity
            + derivative_t
        )
        SCREAMING_SNAKE_CASE__         =    update / (alpha**2 + derivative_x**2 + derivative_y**2)

        SCREAMING_SNAKE_CASE__         =    avg_horizontal_velocity - derivative_x * update
        SCREAMING_SNAKE_CASE__         =    avg_vertical_velocity - derivative_y * update

    return horizontal_flow, vertical_flow


if __name__ == "__main__":
  import doctest

  doctest.testmod()


