



'''simple docstring'''


from __future__ import annotations

from collections.abc import Callable

_lowerCamelCase	:       Optional[int]      	=	list[list[float | int]]







def 				__lowerCamelCase				(   A__    ,						A__    )       ->	Matrix:






   """simple docstring"""
   UpperCamelCase					=  len(A__    )
   UpperCamelCase					=  [[0 for _ in range(size + 1    )] for _ in range(A__    )]
   UpperCamelCase					=  42
   UpperCamelCase					=  42
   UpperCamelCase					=  42
   UpperCamelCase					=  42
   UpperCamelCase					=  42
   UpperCamelCase					=  42

   for row in range(A__    ):
      for col in range(A__    ):
         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(A__    ,						A__    )    )[
          1
      ]
      if augmented[pivot_row][col] == 0:
         col += 1
         continue
      else:
         UpperCamelCase				,      UpperCamelCase					=  augmented[pivot_row], augmented[row]

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







def 				__lowerCamelCase				(   A__    )       ->	Callable[[int], int]:






   """simple docstring"""
   UpperCamelCase					=  len(A__    )
   UpperCamelCase					=  [[0 for _ in range(A__    )] for _ in range(A__    )]
   UpperCamelCase					=  [[0] for _ in range(A__    )]
   UpperCamelCase					=  42
   UpperCamelCase					=  42
   UpperCamelCase					=  42
   UpperCamelCase					=  42

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

   UpperCamelCase					=  solve(A__    ,						A__    )

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

   return interpolated_func







def 				__lowerCamelCase				(   A__    )       ->	int:






   """simple docstring"""
   return (
       1
       - variable
       + variable**2
       - variable**3
       + variable**4
       - variable**5
       + variable**6
       - variable**7
       + variable**8
       - variable**9
       + variable**10
   )







def 				__lowerCamelCase				(   A__ = question_function    ,						A__ = 10    )       ->	int:






   """simple docstring"""
   UpperCamelCase					=  [func(A__    ) 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(A__    ) == poly(A__    ):
         x_val += 1

      ret += poly(A__    )

   return ret


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






'''simple docstring'''


_lowerCamelCase	:       int      	=	"0.21.0"

from .accelerator import Accelerator
from .big_modeling import (
    cpu_offload,
    cpu_offload_with_hook,
    disk_offload,
    dispatch_model,
    init_empty_weights,
    init_on_device,
    load_checkpoint_and_dispatch,
)
from .data_loader import skip_first_batches
from .launchers import debug_launcher, notebook_launcher
from .state import PartialState
from .utils import (
    DeepSpeedPlugin,
    DistributedDataParallelKwargs,
    DistributedType,
    FullyShardedDataParallelPlugin,
    GradScalerKwargs,
    InitProcessGroupKwargs,
    find_executable_batch_size,
    infer_auto_device_map,
    is_rich_available,
    load_checkpoint_in_model,
    synchronize_rng_states,
)


if is_rich_available():
   from .utils import rich

