






'''simple docstring'''





from __future__ import annotations

from collections.abc import Callable

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






def    lowerCamelCase_     (    lowercase__				,     lowercase__):
 lowerCamelCase__  =     len(__UpperCAmelCase)
 lowerCamelCase__  =     [[0 for _ in range(size + 1)] for _ in range(__UpperCAmelCase)]
 lowerCamelCase__  =     42
 lowerCamelCase__  =     42
 lowerCamelCase__  =     42
 lowerCamelCase__  =     42
 lowerCamelCase__  =     42
 lowerCamelCase__  =     42

 for row in range(__UpperCAmelCase):
  for col in range(__UpperCAmelCase):
   lowerCamelCase__  =     matrix[row][col]

  lowerCamelCase__  =     vector[row][0]

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

  for rowa in range(row + 1				,     __UpperCAmelCase):
   lowerCamelCase__  =     augmented[rowa][col] / augmented[row][col]
   lowerCamelCase__  =     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):
   lowerCamelCase__  =     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_     (    lowercase__):
 lowerCamelCase__  =     len(__UpperCAmelCase)
 lowerCamelCase__  =     [[0 for _ in range(__UpperCAmelCase)] for _ in range(__UpperCAmelCase)]
 lowerCamelCase__  =     [[0] for _ in range(__UpperCAmelCase)]
 lowerCamelCase__  =     42
 lowerCamelCase__  =     42
 lowerCamelCase__  =     42
 lowerCamelCase__  =     42

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

 lowerCamelCase__  =     solve(__UpperCAmelCase				,     __UpperCAmelCase)

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

 return interpolated_func






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






def    lowerCamelCase_     (    lowercase__ = question_function				,     lowercase__ = 10):
 lowerCamelCase__  =     [func(__UpperCAmelCase) for x_val in range(1				,     order + 1)]

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

 lowerCamelCase__  =     0
 lowerCamelCase__  =     42
 lowerCamelCase__  =     42

 for poly in polynomials:
  lowerCamelCase__  =     1
  while func(__UpperCAmelCase) == poly(__UpperCAmelCase):
   x_val += 1

  ret += poly(__UpperCAmelCase)

 return ret


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









'''simple docstring'''





def    lowerCamelCase_     (    lowercase__				,     lowercase__				,     lowercase__):
 return round(float(moles / volume) * nfactor)






def    lowerCamelCase_     (    lowercase__				,     lowercase__				,     lowercase__):
 return round(float((moles * 0.0_821 * temperature) / (volume)))






def    lowerCamelCase_     (    lowercase__				,     lowercase__				,     lowercase__):
 return round(float((moles * 0.0_821 * temperature) / (pressure)))






def    lowerCamelCase_     (    lowercase__				,     lowercase__				,     lowercase__):
 return round(float((pressure * volume) / (0.0_821 * moles)))


if __name__ == "__main__":
 import doctest

 doctest.testmod()

