






def 					lowercase__				(					__snake_case   :							List[Any]  ):


					'''simple docstring'''


					UpperCAmelCase_   :				Optional[int]						=						current_set.copy()
					for row_index, row in enumerate(__snake_case  ):
										UpperCAmelCase_   :				List[str]						=						row[0]
										for column_index, column in enumerate(__snake_case  ):
															if magnitude == 0:
																				UpperCAmelCase_   :				Dict						=						column
																				continue
															UpperCAmelCase_   :				List[Any]						=						column / magnitude
    # Subtract to cancel term
					UpperCAmelCase_   :				List[str]						=						current_set[0]
					UpperCAmelCase_   :				str						=						[first_row]
					UpperCAmelCase_   :				Any						=						current_set[1::]
					for row in current_set:
										UpperCAmelCase_   :				Optional[int]						=						[]
										# If first term is 0, it is already in form we want, so we preserve it
										if row[0] == 0:
															final_set.append(__snake_case  )
															continue
										for column_index in range(len(__snake_case  )  ):
															temp_row.append(first_row[column_index] - row[column_index]  )
										final_set.append(__snake_case  )
					# Create next recursion iteration set
					if len(final_set[0]  ) != 3:
										UpperCAmelCase_   :				int						=						final_set[0]
										UpperCAmelCase_   :				List[Any]						=						[]
										UpperCAmelCase_   :				Any						=						[]
										for row in final_set[1::]:
															current_first_column.append(row[0]  )
															next_iteration.append(row[1::]  )
										UpperCAmelCase_   :				List[str]						=						simplify(__snake_case  )
										for i in range(len(__snake_case  )  ):
															resultant[i].insert(0					,	current_first_column[i]  )
										resultant.insert(0					,	__snake_case  )
										UpperCAmelCase_   :				List[Any]						=						resultant
					return final_set




def 					lowercase__				(					__snake_case   :							Dict  ):


					'''simple docstring'''


					if len(__snake_case  ) == 0:
										raise IndexError('solve_simultaneous() requires n lists of length n+1'  )
					UpperCAmelCase_   :				str						=						len(__snake_case  ) + 1
					if any(len(__snake_case  ) != _length for item in equations  ):
										raise IndexError('solve_simultaneous() requires n lists of length n+1'  )
					for row in equations:
										if any(not isinstance(__snake_case					,	(int, float)  ) for column in row  ):
															raise ValueError('solve_simultaneous() requires lists of integers'  )
					if len(__snake_case  ) == 1:
										return [equations[0][-1] / equations[0][0]]
					UpperCAmelCase_   :				List[str]						=						equations.copy()
					if any(0 in row for row in data_set  ):
										UpperCAmelCase_   :				str						=						data_set.copy()
										UpperCAmelCase_   :				Dict						=						[]
										for row_index, row in enumerate(__snake_case  ):
															if 0 not in row:
																				UpperCAmelCase_   :				List[str]						=						data_set.pop(__snake_case  )
																				break
										if not full_row:
															raise ValueError('solve_simultaneous() requires at least 1 full equation'  )
										data_set.insert(0					,	__snake_case  )
					UpperCAmelCase_   :				Dict						=						data_set.copy()
					UpperCAmelCase_   :				int						=						simplify(__snake_case  )
					UpperCAmelCase_   :				List[Any]						=						simplified[::-1]
					UpperCAmelCase_   :				Dict						=						[]
					for row in simplified:
										UpperCAmelCase_   :				Any						=						row[-1]
										if not solutions:
															if row[-2] == 0:
																				solutions.append(0  )
																				continue
															solutions.append(current_solution / row[-2]  )
															continue
										UpperCAmelCase_   :				Optional[int]						=						row.copy()[: len(__snake_case  ) - 1 :]
										while temp_row[0] == 0:
															temp_row.pop(0  )
										if len(__snake_case  ) == 0:
															solutions.append(0  )
															continue
										UpperCAmelCase_   :				str						=						temp_row[1::]
										UpperCAmelCase_   :				str						=						temp_row[::-1]
										for column_index, column in enumerate(__snake_case  ):
															current_solution -= column * solutions[column_index]
										solutions.append(__snake_case  )
					UpperCAmelCase_   :				Tuple						=						[]
					for item in solutions:
										final.append(float(round(__snake_case					,	5  )  )  )
					return final[::-1]


if __name__ == "__main__":
		import doctest

		doctest.testmod()
		__UpperCAmelCase     							=						[
		    [2, 1, 1, 1, 1, 4],
		    [1, 2, 1, 1, 1, 5],
		    [1, 1, 2, 1, 1, 6],
		    [1, 1, 1, 2, 1, 7],
		    [1, 1, 1, 1, 2, 8],
		]
		print(solve_simultaneous(eq))
		print(solve_simultaneous([[4, 2]]))














from __future__ import annotations

from collections.abc import Callable

__UpperCAmelCase     							=						list[list[float | int]]




def 					lowercase__				(					__snake_case   :							Matrix					,	__snake_case   :							Matrix  ):


					'''simple docstring'''


					UpperCAmelCase_   :				int						=						len(__snake_case  )
					UpperCAmelCase_   :				Matrix						=						[[0 for _ in range(size + 1  )] for _ in range(__snake_case  )]
					UpperCAmelCase_   :				int
					UpperCAmelCase_   :				int
					UpperCAmelCase_   :				int
					UpperCAmelCase_   :				int
					UpperCAmelCase_   :				int
					UpperCAmelCase_   :				float

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

										UpperCAmelCase_   :				Union[str, Any]						=						vector[row][0]

					UpperCAmelCase_   :				Union[str, Any]						=						0
					UpperCAmelCase_   :				Union[str, Any]						=						0
					while row < size and col < size:
										# pivoting
										UpperCAmelCase_   :				Optional[Any]						=						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:
															UpperCAmelCase_	,  UpperCAmelCase_   :				Tuple						=						augmented[pivot_row], augmented[row]

										for rowa in range(row + 1					,	__snake_case  ):
															UpperCAmelCase_   :				Optional[int]						=						augmented[rowa][col] / augmented[row][col]
															UpperCAmelCase_   :				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					,	__snake_case  ):
										for row in range(__snake_case  ):
															UpperCAmelCase_   :				Union[str, Any]						=						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 					lowercase__				(					__snake_case   :							list[int]  ):


					'''simple docstring'''


					UpperCAmelCase_   :				int						=						len(__snake_case  )
					UpperCAmelCase_   :				Matrix						=						[[0 for _ in range(__snake_case  )] for _ in range(__snake_case  )]
					UpperCAmelCase_   :				Matrix						=						[[0] for _ in range(__snake_case  )]
					UpperCAmelCase_   :				Matrix
					UpperCAmelCase_   :				int
					UpperCAmelCase_   :				int
					UpperCAmelCase_   :				int

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

					UpperCAmelCase_   :				List[str]						=						solve(__snake_case					,	__snake_case  )

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

					return interpolated_func




def 					lowercase__				(					__snake_case   :							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 					lowercase__				(					__snake_case   :							Callable[[int], int] = question_function					,	__snake_case   :							int = 10  ):


					'''simple docstring'''


					UpperCAmelCase_   :				list[int]						=						[func(__snake_case  ) for x_val in range(1					,	order + 1  )]

					UpperCAmelCase_   :				list[Callable[[int], int]]						=						[
					    interpolate(data_points[:max_coeff]  ) for max_coeff in range(1					,	order + 1  )
					]

					UpperCAmelCase_   :				int						=						0
					UpperCAmelCase_   :				Callable[[int], int]
					UpperCAmelCase_   :				int

					for poly in polynomials:
										UpperCAmelCase_   :				Optional[int]						=						1
										while func(__snake_case  ) == poly(__snake_case  ):
															x_val += 1

										ret += poly(__snake_case  )

					return ret


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






