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Is there any way to tell how many clusters there are with respect to all the roots of a polynomial?

Specifically, I'm after the multiplicity of each root but since I would like to work in floating-point arithmetic I'm afraid I have to deal with clusters.

I don't mind any method of finding out: be it by numerical-iterative means during the convergence, a priori/a posteriori guess, maybe some matrix method would help..? If I have to set some small disk radius, that's ok, too.

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  • $\begingroup$ Just so I understand - By cluster you mean several roots of the polynomial that are very close to each other (or the same root repeated several times) right? $\endgroup$ – Oria Gruber Sep 16 '15 at 23:36
  • $\begingroup$ @OriaGruber Yes. $\endgroup$ – Ecir Hana Sep 16 '15 at 23:38
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Probably the single most important result you need to know is Rouché's theorem.

However, for postprocessing the output of numerical iterative methods (Aberth-Ehrlich, Durand-Kerner, ...), have a look at Siegfried Rump's Ten methods to bound multiple roots of polynomials; in particular methods 4 and 5 by Neumaier (based on Gershgorin circles of the companion matrix in a suitable basis).

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