# -*- coding: utf-8 -*-
"""
Created on Tue Sep 13 08:13:30 2016

@author: efron
"""
"""
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,  un = n**3: 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) = u1.

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**2−11**n+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:

un = 1 − n + n2 − n3 + n4 − n5 + n6 − n7 + n8 − n9 + n10

Find the sum of FITs for the BOPs.
"""

# liner algebra!
import numpy as np
poly = np.array([ 1., -1.,  1., -1.,  1., -1.,  1., -1.,  1., -1.,  1.])

def bad_optimum_polynomial(k):
    x = np.arange(k+1)
    y = np.polyval(poly, x)
    badPoly = np.polyfit(x, y, k)  