How to make an approximation (interpolation) of a function of many variables by points?

Question

Mathematics

There is a set of points of the form:
(x,y,z)
(1,2,3)
z=f(x,y)
Need to find:
1) a function that has the value z for x and y.
2) graph (desirable)
3) wolframalfa (desirable)
4) free
5) the ability to use the function of 3 and 4 variables in the future
Can you advise the software for this and the command for interpolation or approximation?

Answer 1

You can start by reading the wiki article on Lagrange interpolation. An example is given there for a one-dimensional function, it can be extended to any dimension.
To set the Lagrangian interpolation over the rectangle z=f(x,y), you need to select n^2 points in the rectangle (it is convenient to make equidistant points). At these points, the Z values ​​are given by the table.
Then at arbitrary points we get Z(p,q)=sum_over_all_nodal_points(basic_function_at_given_point(p,q)*Z_value_at_given_nodal_point).
Compiling formulas for the basis functions at each nodal point is not difficult - there is a description on Wikipedia.

Answer 2

Dakota Sandia
https://dakota.sandia.gov/sites/default/files/docs...

1. Able to build interpolations.
   
2. There are graphics, although, in my opinion, ugly.
   
3. Free
   
4. Multidimensional approximation techniques: - kriging (Gaussian processes), radial functions, maybe there is something else suitable, I don’t remember right away.
   
5. Input-output via txt, so you can wrap it with anything