



'''simple docstring'''






def 			snake_case_					( _lowerCAmelCase  :						list[list[int]]  )				->       str:
							if len(_lowerCAmelCase  ) == 0 or not all(len(_lowerCAmelCase  ) == 2 for pair in coordinates  ):
														raise ValueError('''The program cannot work out a fitting polynomial.'''  )

							if len({tuple(_lowerCAmelCase  ) for pair in coordinates}  ) != len(_lowerCAmelCase  ):
														raise ValueError('''The program cannot work out a fitting polynomial.'''  )

							UpperCAmelCase			:	Optional[Any]										=		{x for x, _ in coordinates}
							if len(_lowerCAmelCase  ) == 1:
														return f"""x={coordinates[0][0]}"""

							if len(_lowerCAmelCase  ) != len(_lowerCAmelCase  ):
														raise ValueError('''The program cannot work out a fitting polynomial.'''  )

							UpperCAmelCase			:	List[Any]										=		len(_lowerCAmelCase  )

							UpperCAmelCase			:	Dict										=		0
							UpperCAmelCase			:	list[list[float]]										=		[]
							# put the x and x to the power values in a matrix
							while count_of_line < x:
														UpperCAmelCase			:	List[Any]										=		0
														UpperCAmelCase			:	Optional[int]										=		coordinates[count_of_line][0]
														UpperCAmelCase			:	list[float]										=		[]
														while count_in_line < x:
																					count_line.append(a ** (x - (count_in_line + 1))  )
																					count_in_line += 1
														matrix.append(_lowerCAmelCase  )
														count_of_line += 1

							UpperCAmelCase			:	int										=		0
							# put the y values into a vector
							UpperCAmelCase			:	list[float]										=		[]
							while count_of_line < x:
														vector.append(coordinates[count_of_line][1]  )
														count_of_line += 1

							UpperCAmelCase			:	List[str]										=		0

							while count < x:
														UpperCAmelCase			:	Tuple										=		0
														while zahlen < x:
																					if count == zahlen:
																												zahlen += 1
																					if zahlen == x:
																												break
																					UpperCAmelCase			:	Union[str, Any]										=		matrix[zahlen][count] / matrix[count][count]
																					for counting_columns, item in enumerate(matrix[count]  ):
																												# manipulating all the values in the matrix
																												matrix[zahlen][counting_columns] -= item * bruch
																					# manipulating the values in the vector
																					vector[zahlen] -= vector[count] * bruch
																					zahlen += 1
														count += 1

							UpperCAmelCase			:	List[str]										=		0
							# make solutions
							UpperCAmelCase			:	list[str]										=		[]
							while count < x:
														solution.append(str(vector[count] / matrix[count][count]  )  )
														count += 1

							UpperCAmelCase			:	Optional[Any]										=		0
							UpperCAmelCase			:	Optional[int]										=		'''f(x)='''

							while count < x:
														UpperCAmelCase			:	list[str]										=		solution[count].split('''E'''  )
														if len(_lowerCAmelCase  ) > 1:
																					UpperCAmelCase			:	int										=		f"""{remove_e[0]}*10^{remove_e[1]}"""
														solved += f"""x^{x - (count + 1)}*{solution[count]}"""
														if count + 1 != x:
																					solved += "+"
														count += 1

							return solved


if __name__ == "__main__":
						print(points_to_polynomial([]))
						print(points_to_polynomial([[]]))
						print(points_to_polynomial([[1, 0], [2, 0], [3, 0]]))
						print(points_to_polynomial([[1, 1], [2, 1], [3, 1]]))
						print(points_to_polynomial([[1, 3], [2, 3], [3, 3]]))
						print(points_to_polynomial([[1, 1], [2, 2], [3, 3]]))
						print(points_to_polynomial([[1, 1], [2, 4], [3, 9]]))
						print(points_to_polynomial([[1, 3], [2, 6], [3, 11]]))
						print(points_to_polynomial([[1, -3], [2, -6], [3, -11]]))
						print(points_to_polynomial([[1, 5], [2, 2], [3, 9]]))










'''simple docstring'''






UpperCamelCase__:       Any							     =      [[0, 1, 1], [0, 0, 1], [1, 0, 0]]

class     SCREAMING_SNAKE_CASE:







	"""simple docstring"""






	def __init__(						self      :					List[str]		,      __snake_case      :					int		)      ->		int:
								UpperCAmelCase			:	Any										=		name
								UpperCAmelCase			:	Optional[Any]										=		[]
								UpperCAmelCase			:	Any										=		[]







	def 					A     (						self      :					Dict		,      __snake_case      :					str		)      ->		Tuple:
								self.inbound.append(__snake_case		)







	def 					A     (						self      :					Union[str, Any]		,      __snake_case      :					str		)      ->		Optional[int]:
								self.outbound.append(__snake_case		)







	def __repr__(						self      :					Optional[int]		)      ->		Optional[int]:
								return F"""<node={self.name} inbound={self.inbound} outbound={self.outbound}>"""


def 			snake_case_					( _lowerCAmelCase  :						Tuple       ,				_lowerCAmelCase  :						Dict=3       ,				_lowerCAmelCase  :						Tuple=0.8_5  )				->       Optional[Any]:
							UpperCAmelCase			:	Tuple										=		{}
							for node in nodes:
														UpperCAmelCase			:	List[str]										=		1

							UpperCAmelCase			:	List[Any]										=		{}
							for node in nodes:
														UpperCAmelCase			:	str										=		len(node.outbound  )

							for i in range(_lowerCAmelCase  ):
														print(f"""======= Iteration {i + 1} ======="""  )
														for _, node in enumerate(_lowerCAmelCase  ):
																					UpperCAmelCase			:	str										=		(1 - d) + d * sum(
																					    ranks[ib] / outbounds[ib] for ib in node.inbound  )
														print(_lowerCAmelCase  )


def 			snake_case_					( )				->       List[str]:
							UpperCAmelCase			:	str										=		list(input('''Enter Names of the Nodes: '''  ).split()  )

							UpperCAmelCase			:	Dict										=		[Node(_lowerCAmelCase  ) for name in names]

							for ri, row in enumerate(_lowerCAmelCase  ):
														for ci, col in enumerate(_lowerCAmelCase  ):
																					if col == 1:
																												nodes[ci].add_inbound(names[ri]  )
																												nodes[ri].add_outbound(names[ci]  )

							print('''======= Nodes ======='''  )
							for node in nodes:
														print(_lowerCAmelCase  )

							page_rank(_lowerCAmelCase  )


if __name__ == "__main__":
						main()





