






from __future__ import annotations

from collections.abc import Callable

SCREAMING_SNAKE_CASE__											=     list[list[float | int]]







def 					lowercase__				(   __UpperCamelCase	,		__UpperCamelCase					)->				List[Any]:
      UpperCamelCase       						=      len(snake_case_					)
      UpperCamelCase       						=      [[0 for _ in range(size + 1					)] for _ in range(snake_case_					)]
      UpperCamelCase       						=      42
      UpperCamelCase       						=      42
      UpperCamelCase       						=      42
      UpperCamelCase       						=      42
      UpperCamelCase       						=      42
      UpperCamelCase       						=      42

      for row in range(snake_case_					):
            for col in range(snake_case_					):
                  UpperCamelCase       						=      matrix[row][col]

            UpperCamelCase       						=      vector[row][0]

      UpperCamelCase       						=      0
      UpperCamelCase       						=      0
      while row < size and col < size:
            # pivoting
            UpperCamelCase       						=      max((abs(augmented[rowa][col]					), rowa) for rowa in range(snake_case_	,		snake_case_					)					)[
                1
            ]
            if augmented[pivot_row][col] == 0:
                  col += 1
                  continue
            else:
                  UpperCamelCase       						=      augmented[pivot_row], augmented[row]

            for rowa in range(row + 1	,		snake_case_					):
                  UpperCamelCase       						=      augmented[rowa][col] / augmented[row][col]
                  UpperCamelCase       						=      0
                  for cola in range(col + 1	,		size + 1					):
                        augmented[rowa][cola] -= augmented[row][cola] * ratio

            row += 1
            col += 1

      # back substitution
      for col in range(1	,		snake_case_					):
            for row in range(snake_case_					):
                  UpperCamelCase       						=      augmented[row][col] / augmented[col][col]
                  for cola in range(snake_case_	,		size + 1					):
                        augmented[row][cola] -= augmented[col][cola] * ratio

    # round to get rid of numbers like 2.000000000000004
      return [
          [round(augmented[row][size] / augmented[row][row]	,		10					)] for row in range(snake_case_					)
      ]







def 					lowercase__				(   __UpperCamelCase					)->				Any:
      UpperCamelCase       						=      len(snake_case_					)
      UpperCamelCase       						=      [[0 for _ in range(snake_case_					)] for _ in range(snake_case_					)]
      UpperCamelCase       						=      [[0] for _ in range(snake_case_					)]
      UpperCamelCase       						=      42
      UpperCamelCase       						=      42
      UpperCamelCase       						=      42
      UpperCamelCase       						=      42

      for x_val, y_val in enumerate(snake_case_					):
            for col in range(snake_case_					):
                  UpperCamelCase       						=      (x_val + 1) ** (size - col - 1)
            UpperCamelCase       						=      y_val

      UpperCamelCase       						=      solve(snake_case_	,		snake_case_					)

      def interpolated_func(__UpperCamelCase					) -> int:
            return sum(
                round(coeffs[x_val][0]					) * (var ** (size - x_val - 1))
                for x_val in range(snake_case_					)					)

      return interpolated_func







def 					lowercase__				(   __UpperCamelCase					)->				Any:
      return (
          1
          - variable
          + variable**2
          - variable**3
          + variable**4
          - variable**5
          + variable**6
          - variable**7
          + variable**8
          - variable**9
          + variable**10
      )







def 					lowercase__				(   __UpperCamelCase = question_function	,		__UpperCamelCase = 10					)->				Union[str, Any]:
      UpperCamelCase       						=      [func(snake_case_					) for x_val in range(1	,		order + 1					)]

      UpperCamelCase       						=      [
          interpolate(data_points[:max_coeff]					) for max_coeff in range(1	,		order + 1					)
      ]

      UpperCamelCase       						=      0
      UpperCamelCase       						=      42
      UpperCamelCase       						=      42

      for poly in polynomials:
            UpperCamelCase       						=      1
            while func(snake_case_					) == poly(snake_case_					):
                  x_val += 1

            ret += poly(snake_case_					)

      return ret


if __name__ == "__main__":
     print(f'{solution() = }')








'''simple docstring'''

from __future__ import annotations







def 					lowercase__				(   __UpperCamelCase	,		__UpperCamelCase	,		__UpperCamelCase					)->				tuple[float, list[float]]:
      UpperCamelCase       						=      list(range(len(__UpperCamelCase					)					)					)
      UpperCamelCase       						=      [v / w for v, w in zip(__UpperCamelCase	,		__UpperCamelCase					)]
      index.sort(key=lambda __UpperCamelCase					: ratio[i]	,		reverse=__UpperCamelCase					)

      UpperCamelCase       						=      0
      UpperCamelCase       						=      [0] * len(__UpperCamelCase					)
      for i in index:
            if weight[i] <= capacity:
                  UpperCamelCase       						=      1
                  max_value += value[i]
                  capacity -= weight[i]
            else:
                  UpperCamelCase       						=      capacity / weight[i]
                  max_value += value[i] * capacity / weight[i]
                  break

      return max_value, fractions


if __name__ == "__main__":
     import doctest

     doctest.testmod()
