Two implicit function equations curve fit

There are two coefficients in following two equations, a and b. We want the best fit a and b.

f(x1,x2,Y)=0(coefficient a,b)
g(x1,x2,Y)=0(coefficient a,b)

For a given Y, by two equations, we get x1 and x2.

We have the experimental data of x1,x2 and Y. We want the best fit a and b.

How to do that?

Thanks very much.

Answer 1

[Walter Roberson]

If we are allowed to choose f and g, then the easiest solution is

f(x1,x2,Y) = a*0 + b*0;
g(x1,x2,Y) = a*0 + b*0;

If we are not allowed to choose f and g, then there is no solution.

If we were given the forms of f and g, then it might be possible to determine the best fit for a and b, especially if the functions are linear functions. It is, however, not always possible do do a best fit when the forms of the functions are known -- some functions are just too irregular or too hard to compute.

[Liang]

Sorry. a and b is in the form of f and g. For a given Y, two equations can give two x1 and x2. Since it is implicit and nonlinear, now we have experimental values of x1,x2, Y. Want to know the best a and b. Could you help this out?

[Walter Roberson]

It is not possible to do a best-fit on unknown functions.

If you could provide the equations f and g, then we might be able to find a way to do a best fit.