I have a nonlinear function, f(x,y), for which I (numerically) know the axial behaviours, f(x,y0) and f(x0,y), where x0 and y0 are constants. I can calculate discrete off-axis f(x,y) points at arbitrary x and y values, but the computation is expensive and I want to minimize the number of points I have to calculate.
How many discrete off-axis points do I need to calculate (and possibly where) to uniquely (or as close to uniquely as possible) define the surface f(x,y)?
As a starting point, I am using a Taylor series expansion of f(x,y), but so far I am unable to find the cross-terms (d2f/dxdy etc).
Note that if this is possible, I would like to extend it to 3- and 4-D functions
How many discrete off-axis points do I need to calculate (and possibly where) to uniquely (or as close to uniquely as possible) define the surface f(x,y)?
That depends on how complicated the function is and how accurately you want to approximate it. There are many different ways to approximate a function by sampling it. They’re used in different applications.
Its a highly nonlinear function, but I need to start with a low order approximation to get the general shape, then improve on that with a series of higher-order approximations. Unfortunately I don’t know the exact form of the surface, just what the axial behaviour is. I don’t need particularly high accuracy, as the whole thing is based on an approximation.
For reference, the axial behaviours are approximated by 2nd order polynomials, for now.