I was wondering how I could fit a polynomial surface through a set of points in two variables. When I look up this problem in the literature, I usually see two options:
- Use a tensor product, but this only seems to work in the case the points are evaluated in a grid
- Use some special point layouts, like Padua or Chebyshev points.
Neither options seems feasible for pseudo-random point sets. Does anyone have an idea? (I guess I could use the standard Lagrange formula in two variables, but that doesn't seem like a numerically stable solution.)
You can do a multidimensional function minimization. Write your polynomial as $\sum a_{ij}x^iy^j$ over whatever range of $i,j$ you like (hope you have less terms than data points). Then calculate the polynomial at each point, square the errors and sum. Feed it to a minimizer taking the $a_{ij}$ as the parameters and minimize. Sections 10.4 through 10.7 of Numerical Recipes deal with this, as will any numerical analysis text.