Approximate Greatest Common Divisor of Many Polynomials and Pseudo-Spectrum

Abstract

This is the accepted version of the paper.This version of the publication may differ from the final published version. (e-mail: N.Karcanias@city.ac.). Permanent Abstract:The paper is concerned with establishing the links between the approximate GCD of a set of polynomials and the notion of the pseudo-spectrum defined on a set of polynomials. By examining the pseudo-spectrum of the structured matrix we will derive estimates of the area of the approximate roots of the initial polynomial set. We will relate the strength of the GCD to the weighted strength of the pseudospectra and we investigate under which conditions the roots of the approximate GCDs are a subset of the pseudo-spectra.

Nearest common root of a set of polynomials: A structured singular value approach

“…Moreover, the concept of "approximate GCD" and the distance of the polynomials to the GCD variety can be analysed with respect to the singular values of the corresponding Sylvester resultant. In previous research papers [13], [4], [6] "approximate co-primeness" is studied alongside the Sylvester matrix representation leading to some important notions that are considered next.…”

“…The concept of "almost zero" was first introduced in [1] as the complex number that is closest to a set of coprime polynomials. Subsequently, this has led to the introduction of the notion of "approximate GCD", see [2], [3], [4], [5], [6], [7] and references therein. Again, the definition of this notion is based on the relaxation of the conditions defining the exact GCD.…”