Nonlinear Systems of Equations
I have M non-linear functions that take N input parameters. I would like to find the set of N parameters so that Fi(N) = 0. The trick is that functions are given as a set of discrete values (a lookup table). Otherwise I could use Newton-Raphson Method (or something like fsolve function in SciLab).
In my particular case I have a model with N parameters. This model outputs the values for M functions. I change one of the parameters and record the values of the functions. I do it for all the parameters to get values for each Fi(N). Now I want to find out a new range of parameters that would bring me closer to the desired set of N (so that every Fi(N) = 0).
As an example, let's assume N = 3, so I have x, y, z as my input parameters, and M = 3, so I have a(x,y,z), b(x,y,z), c(x,y,z) as my functions. I would like to find x,y,z so that
a(x,y,z) = 0 and b(x,y,z) = 0 and c(x,y,z) = 0
or at least they should be as close to zero as possible. These functions do not have analytical representation, instead I have a time consuming simulation that provides values of a, b, and c given x, y, z. I have guessed that the proper set of x,y,z has to be in the region of a volume defined by (x1,y1,z1) = (0,0,0) and (x1,y1,z1) = (100,100,100). I have run 1000 simulations that uniformly sampled this volume (step 10 for each parameter). I did not get lucky and none of the combinations provided me with (a,b,c) = (0,0,0). Now I would like to find a smaller volume that I can sample. If I can find that probable solution is in the volume: (x1,y1,z1) = (20,30,10) and (x1,y1,z1) = (60,50,90), then I can run another 1000 simulations, but the step between the values increments will be smaller.
I do not see how I can use Newton-Raphson Method, because I can not interpolate between the known points. Is there a way for multidimensional interpolation?