

"""simple docstring"""





from __future__ import annotations

from collections.abc import Callable

__UpperCamelCase  :      Optional[Any]							      =		list[list[float | int]]

def   __SCREAMING_SNAKE_CASE  (      A_						,     A_ ):
	lowerCAmelCase__	:						List[str]  				=	len(SCREAMING_SNAKE_CASE_ )
	lowerCAmelCase__	:						int  				=	[[0 for _ in range(size + 1 )] for _ in range(SCREAMING_SNAKE_CASE_ )]
	lowerCAmelCase__	:						List[Any]  				=	42
	lowerCAmelCase__	:						Optional[Any]  				=	42
	lowerCAmelCase__	:						List[Any]  				=	42
	lowerCAmelCase__	:						int  				=	42
	lowerCAmelCase__	:						str  				=	42
	lowerCAmelCase__	:						List[str]  				=	42

	for row in range(SCREAMING_SNAKE_CASE_ ):
		for col in range(SCREAMING_SNAKE_CASE_ ):
			lowerCAmelCase__	:						Any  				=	matrix[row][col]

		lowerCAmelCase__	:						int  				=	vector[row][0]

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

		for rowa in range(row + 1						,     SCREAMING_SNAKE_CASE_ ):
			lowerCAmelCase__	:						Tuple  				=	augmented[rowa][col] / augmented[row][col]
			lowerCAmelCase__	:						List[str]  				=	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						,     SCREAMING_SNAKE_CASE_ ):
		for row in range(SCREAMING_SNAKE_CASE_ ):
			lowerCAmelCase__	:						Any  				=	augmented[row][col] / augmented[col][col]
			for cola in range(SCREAMING_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(SCREAMING_SNAKE_CASE_ )
	]

def   __SCREAMING_SNAKE_CASE  (      A_ ):
	lowerCAmelCase__	:						Any  				=	len(SCREAMING_SNAKE_CASE_ )
	lowerCAmelCase__	:						Dict  				=	[[0 for _ in range(SCREAMING_SNAKE_CASE_ )] for _ in range(SCREAMING_SNAKE_CASE_ )]
	lowerCAmelCase__	:						Optional[int]  				=	[[0] for _ in range(SCREAMING_SNAKE_CASE_ )]
	lowerCAmelCase__	:						List[str]  				=	42
	lowerCAmelCase__	:						Any  				=	42
	lowerCAmelCase__	:						int  				=	42
	lowerCAmelCase__	:						int  				=	42

	for x_val, y_val in enumerate(SCREAMING_SNAKE_CASE_ ):
		for col in range(SCREAMING_SNAKE_CASE_ ):
			lowerCAmelCase__	:						Optional[Any]  				=	(x_val + 1) ** (size - col - 1)
		lowerCAmelCase__	:						Union[str, Any]  				=	y_val

	lowerCAmelCase__	:						Dict  				=	solve(SCREAMING_SNAKE_CASE_						,     SCREAMING_SNAKE_CASE_ )

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

	return interpolated_func

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

def   __SCREAMING_SNAKE_CASE  (      A_ = question_function						,     A_ = 10 ):
	lowerCAmelCase__	:						Optional[Any]  				=	[func(SCREAMING_SNAKE_CASE_ ) for x_val in range(1						,     order + 1 )]

	lowerCAmelCase__	:						Union[str, Any]  				=	[
	    interpolate(data_points[:max_coeff] ) for max_coeff in range(1						,     order + 1 )
	]

	lowerCAmelCase__	:						str  				=	0
	lowerCAmelCase__	:						Any  				=	42
	lowerCAmelCase__	:						Optional[Any]  				=	42

	for poly in polynomials:
		lowerCAmelCase__	:						Optional[Any]  				=	1
		while func(SCREAMING_SNAKE_CASE_ ) == poly(SCREAMING_SNAKE_CASE_ ):
			x_val += 1

		ret += poly(SCREAMING_SNAKE_CASE_ )

	return ret


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








"""simple docstring"""




# This is the module that test_patching.py uses to test patch_submodule()

import os  # noqa: this is just for tests
import os as renamed_os  # noqa: this is just for tests
from os import path  # noqa: this is just for tests
from os import path as renamed_path  # noqa: this is just for tests
from os.path import join  # noqa: this is just for tests
from os.path import join as renamed_join  # noqa: this is just for tests


__UpperCamelCase						=      open  # noqa: we just need to have a builtin inside this module to test it properly


