Help Solving Non-Linear System

[es]

Hello All,

I was hoping one of the many knowledgeable people on this forum could give me some helpfull advice on how to proceed with the following problem. Basically, I am trying to fit experimental data to a curve.

In general my curve looks like this and I know the constants have the following constraints. I also know the range and domain of the function over the area I am interested in.

f(x)=a*(x^2)+b*x+c,
0<=f(x)<5, 0<x<50, -1<a<0, 0<b<1

I have never seen a negative c and I think the following should hold, |a|<|b|, but I would rather not rely on either if possible.

Oh ya, this is probably obvious but just in case, x is the variable I can control well and f(x) is the measured value. Also, I can vary x over the entire domain given and take many measurements. a, b, and c are constant for one particular setup but will change when I physically change the setup (i.e. switch to new components).

Anyway, this system is linear and quite easy to deal with. Unfortunately this is a sort of a reduced setup. I'll call it Experiment 1. What I really want is Experiment 2 which changes my characteristic equation to:

f(x)=a*(x+p)^2+b*(x+p)+c

Unfortunately I cannot control p or measure it directly (actually if I change to an experiment 3 I can, but there are good engineering reasons to try to avoid doing so. It is also a lot of work and this experiment needs to be repeated many times for different setups so time spent here could result in significant time savings overall) Anyway, I do know its possible range, 0<p<2.

The other constraints still apply. Oh ya, p will be constant throughout
the setup as well.

I believe this system is non-linear (because two unknowns are multiplied together) and I know there are iterative methods to solve such a system but I know very little about them.

I am sure this is a solved problem. Does someone know the name of the solution or technique to solve it? I did do a little googling before writing this post and found some info on root finding. Is this a good technique to try? Perhaps there is some nifty variable substitution I am not seeing that could reduce this to a linear case?

 

[benorin]

Are you looking to fit a set of data points, say {(x1,y1),(x2,y2),...,(xn,yn)}, to the parabola y=ax^2+bx+c ? If so, then use the method of least squares.

 

[es]

Unfortunately no.
I am trying to fit a set of data points, say {(x1,y1),(x2,y2),...,(xn,yn)}, to the parabola defined by y=a(x+p)^2+b(x+p)+c. Which is non-linear because p is also unknown (unless someone can figure out how to reduce it!) so the method of least squares does not apply.

I have been doing some research into this problem and because I can accurately place x, I think I can also accuratly estimate the derivative. I think I can combine this with the Newton-Raphson method to solve for the unknowns.

Unfortunately I am new to all this so I am pretty scared of hidden gotchas so I am not sure how much I can trust any answer I get. I suppose it would be enough to run the test on the lot and then spot check a couple randomly tested units with the more advanced setup (aka experiment 3 above).