8th degree polynomial numerical instability

2013 3 7
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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 ,
H

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Sean de Wolski
Sean de Wolski 2013-3-7
编辑：Sean de Wolski 2013-3-7

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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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Houman R
Houman R 2013-3-7
thank you for the reply, What im doing is not really a "physical process" per se. I work on camera autocalibration and the resulting equations are usually very high degree polynomials. In fact, the challenges of the numerical stability of these equations are focus of quite a bit of research. I have tried many ways to reduce the degrees of the polynomials, and I will continue to do so. But in the meantime this is what i am stuck with :(
ps some people use the Grobner basis methods, but i am not familiar with this method and right now trying to avoid having to use it.
thanks H
Sean de Wolski
Sean de Wolski 2013-3-7
We need a little more detail about what you're actually doing. Can you show us some data and expected results? Have you looked at splines or other curves?
Houman R
Houman R 2013-3-7
thank you for your answer. I can not provide more context without significantly expanding the scope of the question. I think I will try to think of a way of avoiding the 8th degree polynomial. And that means going back to the drawing board. I thank you for your response.
H

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