I have a program that generates some physics data in 1D and 2D functions. In this program, the user defines a number of models that are used to compute a 2D function. That 2D function, and it's first and second derivatives are used with some other simple computations to produce a variety of useful physics data tables.
The problem is, not all of the basic models are algebraic, some read from pre-existing data tables, or have other ways of coming up with data, such that we can't generally use analytic derivatives. So we use numeric derivatives everywhere. Currently we are using a 2D interpolation library to get those derivatives. We have also tried finite difference. Both approaches have issues.
Due to issues with numeric derivatives, and sometimes to noisy input data, often the derivatives that result are rough and jagged, and can throw off computations later down the line.
So recently we've been discussing use polynomial fit in cases where we are more interested in derivative smoothness than exactly matching the data points.
A friend of mine provided me with a polynomial fit library he wrote, but it will take some work to adapt it to my needs.
Gnu Scientific Library (GSL) has a multilinear fit library I could also use. The problem is, I'm a programmer, not a mathematician. Could some one please tell me the differences between multilinear fit and polynomial fit, and some advantages and disadvantages?
Concerning advantages and disadvantages when comparing multilinear fit and polynomial fit, there is none.
Now, when you use a polynomial fit, you use $$y=\sum_{i=0}^n a_ix^i$$ while a multilinear fit could be any thing such as $$y=a_0+a_1\sin(x)+a_2\sqrt x+a _3 \cosh(x)+a_4 x^{7/8}$$
So, if your $n$ terms are only all increasing powers of $x$ from $0$ to $n$, they are the same.