




"""simple docstring"""




from __future__ import annotations

from collections.abc import Callable

lowerCAmelCase							:    Any  							=							list[list[float | int]]



def 					a__				(       snake_case__ ,				snake_case__       )		->      Matrix:
			lowerCamelCase       =      len(snake_case__       )
			lowerCamelCase       =      [[0 for _ in range(size + 1       )] for _ in range(snake_case__       )]
			lowerCamelCase       =      42
			lowerCamelCase       =      42
			lowerCamelCase       =      42
			lowerCamelCase       =      42
			lowerCamelCase       =      42
			lowerCamelCase       =      42

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

						for rowa in range(row + 1 ,				snake_case__       ):
									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 ,				snake_case__       ):
						for row in range(snake_case__       ):
									lowerCamelCase       =      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 					a__				(       snake_case__       )		->      Callable[[int], int]:
			lowerCamelCase       =      len(snake_case__       )
			lowerCamelCase       =      [[0 for _ in range(snake_case__       )] for _ in range(snake_case__       )]
			lowerCamelCase       =      [[0] for _ in range(snake_case__       )]
			lowerCamelCase       =      42
			lowerCamelCase       =      42
			lowerCamelCase       =      42
			lowerCamelCase       =      42

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

			lowerCamelCase       =      solve(snake_case__ ,				snake_case__       )

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

			return interpolated_func



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



def 					a__				(       snake_case__ = question_function ,				snake_case__ = 10       )		->      int:
			lowerCamelCase       =      [func(snake_case__       ) 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(snake_case__       ) == poly(snake_case__       ):
									x_val += 1

						ret += poly(snake_case__       )

			return ret


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









"""simple docstring"""




from collections.abc import Sequence



def 					a__				(       snake_case__ ,				snake_case__ = False       )		->      float:
			if not arr:
						return 0

			lowerCamelCase       =      0 if allow_empty_subarrays else float("""-inf"""       )
			lowerCamelCase       =      0.0
			for num in arr:
						lowerCamelCase       =      max(0 if allow_empty_subarrays else num ,				curr_sum + num       )
						lowerCamelCase       =      max(snake_case__ ,				snake_case__       )

			return max_sum


if __name__ == "__main__":
					from doctest import testmod

					testmod()

					lowerCAmelCase							:    Optional[int]  							=							[-2, 1, -3, 4, -1, 2, 1, -5, 4]
					print(F"""{max_subarray_sum(nums) = }""")



