I know that given an N-degree polynomial interpolating N+1 points exists and is unique. However, I am wondering what would happen if we try to use an N-degree polynomial to interpolate N data points.
I think that in that case, we would have more equations than unknowns which is considered an overdetermined system (and is most likely inconsistent). We would then have to use other methods such as least squared fit instead of polynomial interpolation (and approximate instead of interpolate).
Any hints? Is my thinking correct?
Best Answer
It seems like you are on the right track. If you consider a simple case of two points (N=2) and a second degree polynomial (parabola), you can see that different parabolas will be able to fit the two points, yet have vastly different possible values in the interpolated region. Therefore, some additional constraint on the parabola would be needed for the interpolation to be meaningful under some criteria. I hope this helps.