





from __future__ import annotations

from collections.abc import Callable

snake_case_           =     list[list[float | int]]






def lowerCamelCase__       (				snake_case_      :    Matrix			, snake_case_      :    Matrix			) ->							Matrix:
   __snake_case							=  len(UpperCAmelCase_			)
   __snake_case							=  [[0 for _ in range(size + 1			)] for _ in range(UpperCAmelCase_			)]
   __snake_case							=  42
   __snake_case							=  42
   __snake_case							=  42
   __snake_case							=  42
   __snake_case							=  42
   __snake_case							=  42

   for row in range(UpperCAmelCase_			):
      for col in range(UpperCAmelCase_			):
         __snake_case							=  matrix[row][col]

      __snake_case							=  vector[row][0]

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

      for rowa in range(row + 1			, UpperCAmelCase_			):
         __snake_case							=  augmented[rowa][col] / augmented[row][col]
         __snake_case							=  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			, UpperCAmelCase_			):
      for row in range(UpperCAmelCase_			):
         __snake_case							=  augmented[row][col] / augmented[col][col]
         for cola in range(UpperCAmelCase_			, 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(UpperCAmelCase_			)
   ]






def lowerCamelCase__       (				snake_case_      :    list[int]			) ->							Callable[[int], int]:
   __snake_case							=  len(UpperCAmelCase_			)
   __snake_case							=  [[0 for _ in range(UpperCAmelCase_			)] for _ in range(UpperCAmelCase_			)]
   __snake_case							=  [[0] for _ in range(UpperCAmelCase_			)]
   __snake_case							=  42
   __snake_case							=  42
   __snake_case							=  42
   __snake_case							=  42

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

   __snake_case							=  solve(UpperCAmelCase_			, UpperCAmelCase_			)

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

   return interpolated_func






def lowerCamelCase__       (				snake_case_      :    int			) ->							int:
   return (
       1
       - variable
       + variable**2
       - variable**3
       + variable**4
       - variable**5
       + variable**6
       - variable**7
       + variable**8
       - variable**9
       + variable**10
   )






def lowerCamelCase__       (				snake_case_      :    Callable[[int], int] = question_function			, snake_case_      :    int = 10			) ->							int:
   __snake_case							=  [func(UpperCAmelCase_			) for x_val in range(1			, order + 1			)]

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

   __snake_case							=  0
   __snake_case							=  42
   __snake_case							=  42

   for poly in polynomials:
      __snake_case							=  1
      while func(UpperCAmelCase_			) == poly(UpperCAmelCase_			):
         x_val += 1

      ret += poly(UpperCAmelCase_			)

   return ret


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





'''simple docstring'''



from __future__ import annotations

import os
from collections.abc import Mapping

A__						:   Optional[Any]				=    tuple[int, int]





class   UpperCAmelCase_    :





    """simple docstring"""






    def __init__(				self     ,	SCREAMING_SNAKE_CASE_     ,	SCREAMING_SNAKE_CASE_      )							->	None:
         __lowerCamelCase  :      set[int]														=						vertices
         __lowerCamelCase  :      dict[EdgeT, int]														=						{
             (min(SCREAMING_SNAKE_CASE_      ), max(SCREAMING_SNAKE_CASE_      )): weight for edge, weight in edges.items()
         }







    def  lowercase_							(				self     ,	SCREAMING_SNAKE_CASE_     ,	SCREAMING_SNAKE_CASE_      )							->	None:
         self.vertices.add(edge[0]      )
         self.vertices.add(edge[1]      )
         __lowerCamelCase  :      Union[str, Any]														=						weight







    def  lowercase_							(				self      )							->	Graph:
         __lowerCamelCase  :      Graph														=						Graph({min(self.vertices      )}     ,	{}      )
         __lowerCamelCase  :      EdgeT
         __lowerCamelCase  :      int
         __lowerCamelCase  :      EdgeT
         __lowerCamelCase  :      int

         while len(subgraph.vertices      ) < len(self.vertices      ):
              __lowerCamelCase  :      Any														=						max(self.edges.values()      ) + 1
              for edge, weight in self.edges.items():
                   if (edge[0] in subgraph.vertices) ^ (edge[1] in subgraph.vertices):
                        if weight < min_weight:
                             __lowerCamelCase  :      Optional[int]														=						edge
                             __lowerCamelCase  :      List[str]														=						weight

              subgraph.add_edge(SCREAMING_SNAKE_CASE_     ,	SCREAMING_SNAKE_CASE_      )

         return subgraph







def      UpperCAmelCase__   (      UpperCAmelCase_       :							str = "p107_network.txt"				)	->						int:
     __lowerCamelCase  :      str														=						os.path.abspath(os.path.dirname(UpperCAmelCase_				)				)
     __lowerCamelCase  :      str														=						os.path.join(UpperCAmelCase_ ,       UpperCAmelCase_				)
     __lowerCamelCase  :      dict[EdgeT, int]														=						{}
     __lowerCamelCase  :      list[str]
     __lowerCamelCase  :      int
     __lowerCamelCase  :      int

     with open(UpperCAmelCase_				) as f:
          __lowerCamelCase  :      Any														=						f.read().strip().split('\n'				)

     __lowerCamelCase  :      Any														=						[line.split(','				) for line in data]

     for edgea in range(1 ,       len(UpperCAmelCase_				)				):
          for edgea in range(UpperCAmelCase_				):
               if adjaceny_matrix[edgea][edgea] != "-":
                    __lowerCamelCase  :      int														=						int(adjaceny_matrix[edgea][edgea]				)

     __lowerCamelCase  :      Graph														=						Graph(set(range(len(UpperCAmelCase_				)				)				) ,       UpperCAmelCase_				)

     __lowerCamelCase  :      Graph														=						graph.prims_algorithm()

     __lowerCamelCase  :      int														=						sum(graph.edges.values()				)
     __lowerCamelCase  :      int														=						sum(subgraph.edges.values()				)

     return initial_total - optimal_total


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






