Underdeter​mined/over​determined polynomial interpolation

I am trying to apply a polynomial interpolation to 340 points in a 4D space (i.e. f(x,y,z)=k).

What I would like to understand is this: if I use a 6th order polynomial I will end up with 343 degrees of freedom [(order+1)^(dimensions)] for my polynomial, which is a number greater than the actual available number of points. Is the problem underdetermined? Can I state that I cannot use such order for the interpolation?

Then, if I use a 5th order polynomial, this has 216 degrees of freedom, that is a number smaller than my number of points. Is the problem overdetermined? Can I anyway use this order to interpolate my data without problems?

Thanks a lot for your kind attention.

Federico

Answer 1

[John D'Errico]

You CANNOT use a polynomial with 343 terms in it to fit 340 data points. Any interpolation you would get from that fit will be pure garbage.

Even 216 terms is laughable, because even moderately high order polynomials do strange and stupid things. 216 terms is, in theory overdetermined for 340 data points, since you have more data than terms. The result will NOT be interpolation, instead a least squares approximation.

If your goal is truly interpolation, then use scatteredInterpolant.