# Nonlinear Systems of Equations

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• March 25th 2009, 06:38 AM
sjcomp
Nonlinear Systems of Equations
Hello,
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.

Any suggestions?

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?

Thank you.
• March 26th 2009, 05:33 AM
CaptainBlack
Quote:

Originally Posted by sjcomp
Hello,
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.

Any suggestions?

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?

Thank you.

Is the model a Monte-Carlo simulation, or is it deterministic?

If it is deterministic some form of gradient method or polytope search should still work with an objective of the form:

$Ob(x,y,z)=(a(x,y,z))^2+(b(x,y,z))^2+(c(x,y,z))^2$

CB
• March 26th 2009, 05:47 AM
sjcomp
Quote:

Originally Posted by CaptainBlack
Is the model a Monte-Carlo simulation, or is it deterministic?

If it is deterministic some form of gradient method or polytope search should still work with an objective of the form:

$Ob(x,y,z)=(a(x,y,z))^2+(b(x,y,z))^2+(c(x,y,z))^2$

CB

It is a deterministic process. As I understand I can compute Ob(x,y,z) for all the points. Which basically will give me an error (a measure of distance from the desired point). I guess I can pick the point with the smallest error and rerun the simulation in its vicinity.

Thanks.