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Keywords:
polynomial greatest common divisor; approximate greatest common divisor; Sylvester subresultant matrix; singular value; structured total least norm method
polynomial greatest common divisor; approximate greatest common divisor; Sylvester subresultant matrix; singular value; structured total least norm method
Summary:
The coefficients of the greatest common divisor of two polynomials $f$ and $g$ (GCD$(f,g)$) can be obtained from the Sylvester subresultant matrix $S_j(f,g)$ transformed to lower triangular form, where $1 \leq j \leq d$ and $d = $ deg(GCD$(f,g)$) needs to be computed. Firstly, it is supposed that the coefficients of polynomials are given exactly. Transformations of $S_j(f,g)$ for an arbitrary allowable $j$ are in details described and an algorithm for the calculation of the GCD$(f,g)$ is formulated. If inexact polynomials are given, then an approximate greatest common divisor (AGCD) is introduced. The considered techniques for an AGCD computations are shortly discussed and numerically compared in the presented paper.
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