MATLAB: 8th degree polynomial numerical instability

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hi there, I have a piece of my code where I form an 8th degree polynomial and find its roots via the "root" command. The problem is that the roots are not always correct. Basically for some polynomials the roots are all very close to zero. However if I slightly perturb my input values that eventually form the polynomial i do get the solution. I feel like there is some kind of numerical instability in this polynomial. And looking at it numerically the coefficients are wildly different. Some in the order of 10^-20 and some 10^20. I know that's a recipe for numerical instability, but i dont know what to do about it! any suggestions would be appreciated.

thanks in advance ,

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My suggestion is avoid any polynomial with degree greater than three or four. They are highly unstable and aren't actually modelling any physical process at this point. What are you trying to do? There is probably (i.e. definitely) a better fitting tool for whatever you are doing?

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You don't show what you actually have. What you don't realize is that things go to hell exponentially fast here. Even if a solution exists (which is fairly rare for nonlinear symbolic systems) it will take a seriously LONG time. Those many terms expand and multiply to the point that you will probably not succeed.

You talk about a nonlinear system. You may also be under the wheels of mathematical impossibility. For example, even simple low order polynomial systems, when you start to add equations, grow in effective order. For example, consider two quadratic polynomial equations in two unknowns. You can solve for one variable in terms of the other and then substitute. But that results in a single equation that is higher degree than quadratic. Now do this with 6 variables. Each time you eliminate one of the variables (in sort of a Gaussian elimination scheme) the order of the equations that remain grow. Very quickly, the order of what remains is higher than a 4th degree polynomial. At that point, MATLAB may keep on trying, but it will just keep on spinning its wheels, since there is NO general solution to the roots of a higher than 4th degree polynomial.

Odds are this is what you have done, so I predict that no matter what, you will never get a solution. Again, all a guess, since we don't see your actual problem.

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