Help Solving Non-Linear System

In summary, the speaker is seeking advice on how to fit experimental data to a curve with certain constraints. They know the range and domain of the function and are trying to find a way to accurately estimate the derivative to solve for unknowns using the Newton-Raphson method. They are hesitant to fully trust any answer they receive and may spot check a few units with a more advanced setup.
  • #1
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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?
 
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  • #2
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.
 
  • #3
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).
 

What is a non-linear system?

A non-linear system is a mathematical system in which the output is not directly proportional to the input. This means that the relationship between the variables is not a straight line and cannot be represented by a linear equation.

Why is solving a non-linear system challenging?

Solving a non-linear system is challenging because it involves finding the values of multiple variables that satisfy a set of non-linear equations. Unlike linear systems, which have a unique solution, non-linear systems can have multiple solutions or no solution at all.

What methods can be used to solve a non-linear system?

There are several methods that can be used to solve a non-linear system, including substitution, elimination, graphing, and numerical methods such as Newton's method or the Bisection method. Each method has its own advantages and disadvantages, and the best method to use will depend on the specific system being solved.

How do I know if my solution to a non-linear system is correct?

To check the accuracy of your solution to a non-linear system, you can substitute the values of the variables into the original equations and see if they satisfy all of the equations. If all of the equations are satisfied, then your solution is correct.

What are some real-world applications of non-linear systems?

Non-linear systems are used in many fields of science and engineering, including physics, economics, biology, and chemistry. They can be used to model complex systems where the relationship between variables is not linear, such as population growth, chemical reactions, and electrical circuits.

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