""" If we are presented with the first k terms of a sequence it is impossible to say with certainty the value of the next term, as there are infinitely many polynomial functions that can model the sequence. As an example, let us consider the sequence of cube numbers. This is defined by the generating function, u(n) = n3: 1, 8, 27, 64, 125, 216, ... Suppose we were only given the first two terms of this sequence. Working on the principle that "simple is best" we should assume a linear relationship and predict the next term to be 15 (common difference 7). Even if we were presented with the first three terms, by the same principle of simplicity, a quadratic relationship should be assumed. We shall define OP(k, n) to be the nth term of the optimum polynomial generating function for the first k terms of a sequence. It should be clear that OP(k, n) will accurately generate the terms of the sequence for n ≤ k, and potentially the first incorrect term (FIT) will be OP(k, k+1); in which case we shall call it a bad OP (BOP). As a basis, if we were only given the first term of sequence, it would be most sensible to assume constancy; that is, for n ≥ 2, OP(1, n) = u(1). Hence we obtain the following OPs for the cubic sequence: OP(1, n) = 1 1, 1, 1, 1, ... OP(2, n) = 7n-6 1, 8, 15, ... OP(3, n) = 6n^2-11n+6 1, 8, 27, 58, ... OP(4, n) = n^3 1, 8, 27, 64, 125, ... Clearly no BOPs exist for k ≥ 4. By considering the sum of FITs generated by the BOPs (indicated in red above), we obtain 1 + 15 + 58 = 74. Consider the following tenth degree polynomial generating function: 1 - n + n^2 - n^3 + n^4 - n^5 + n^6 - n^7 + n^8 - n^9 + n^10 Find the sum of FITs for the BOPs. """ from typing import Callable, List, Union Matrix = List[List[Union[float, int]]] def solve(matrix: Matrix, vector: Matrix) -> Matrix: """ Solve the linear system of equations Ax = b (A = "matrix", b = "vector") for x using Gaussian elimination and back substitution. We assume that A is an invertible square matrix and that b is a column vector of the same height. >>> solve([[1, 0], [0, 1]], [[1],[2]]) [[1.0], [2.0]] >>> solve([[2, 1, -1],[-3, -1, 2],[-2, 1, 2]],[[8], [-11],[-3]]) [[2.0], [3.0], [-1.0]] """ size: int = len(matrix) augmented: Matrix = [[0 for _ in range(size + 1)] for _ in range(size)] row: int row2: int col: int col2: int pivot_row: int ratio: float for row in range(size): for col in range(size): augmented[row][col] = matrix[row][col] augmented[row][size] = vector[row][0] row = 0 col = 0 while row < size and col < size: # pivoting pivot_row = max( [(abs(augmented[row2][col]), row2) for row2 in range(col, size)] )[1] if augmented[pivot_row][col] == 0: col += 1 continue else: augmented[row], augmented[pivot_row] = augmented[pivot_row], augmented[row] for row2 in range(row + 1, size): ratio = augmented[row2][col] / augmented[row][col] augmented[row2][col] = 0 for col2 in range(col + 1, size + 1): augmented[row2][col2] -= augmented[row][col2] * ratio row += 1 col += 1 # back substitution for col in range(1, size): for row in range(col): ratio = augmented[row][col] / augmented[col][col] for col2 in range(col, size + 1): augmented[row][col2] -= augmented[col][col2] * ratio # round to get rid of numbers like 2.000000000000004 return [ [round(augmented[row][size] / augmented[row][row], 10)] for row in range(size) ] def interpolate(y_list: List[int]) -> Callable[[int], int]: """ Given a list of data points (1,y0),(2,y1), ..., return a function that interpolates the data points. We find the coefficients of the interpolating polynomial by solving a system of linear equations corresponding to x = 1, 2, 3... >>> interpolate([1])(3) 1 >>> interpolate([1, 8])(3) 15 >>> interpolate([1, 8, 27])(4) 58 >>> interpolate([1, 8, 27, 64])(6) 216 """ size: int = len(y_list) matrix: Matrix = [[0 for _ in range(size)] for _ in range(size)] vector: Matrix = [[0] for _ in range(size)] coeffs: Matrix x_val: int y_val: int col: int for x_val, y_val in enumerate(y_list): for col in range(size): matrix[x_val][col] = (x_val + 1) ** (size - col - 1) vector[x_val][0] = y_val coeffs = solve(matrix, vector) def interpolated_func(var: int) -> int: """ >>> interpolate([1])(3) 1 >>> interpolate([1, 8])(3) 15 >>> interpolate([1, 8, 27])(4) 58 >>> interpolate([1, 8, 27, 64])(6) 216 """ return sum( round(coeffs[x_val][0]) * (var ** (size - x_val - 1)) for x_val in range(size) ) return interpolated_func def question_function(variable: int) -> int: """ The generating function u as specified in the question. >>> question_function(0) 1 >>> question_function(1) 1 >>> question_function(5) 8138021 >>> question_function(10) 9090909091 """ return ( 1 - variable + variable ** 2 - variable ** 3 + variable ** 4 - variable ** 5 + variable ** 6 - variable ** 7 + variable ** 8 - variable ** 9 + variable ** 10 ) def solution(func: Callable[[int], int] = question_function, order: int = 10) -> int: """ Find the sum of the FITs of the BOPS. For each interpolating polynomial of order 1, 2, ... , 10, find the first x such that the value of the polynomial at x does not equal u(x). >>> solution(lambda n: n ** 3, 3) 74 """ data_points: List[int] = [func(x_val) for x_val in range(1, order + 1)] polynomials: List[Callable[[int], int]] = [ interpolate(data_points[:max_coeff]) for max_coeff in range(1, order + 1) ] ret: int = 0 poly: int x_val: int for poly in polynomials: x_val = 1 while func(x_val) == poly(x_val): x_val += 1 ret += poly(x_val) return ret if __name__ == "__main__": print(f"{solution() = }")