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Publisher: Elsevier
Languages: English
Types: Article
Subjects:
This paper describes a non-linear structure-preserving ma\ud trix method for the com-\ud putation of the coefficients of an approximate greatest commo\ud n divisor (AGCD) of\ud degree\ud t\ud of two Bernstein polynomials\ud f\ud (\ud y\ud ) and\ud g\ud (\ud y\ud ). This method is applied to\ud a modified form\ud S\ud t\ud (\ud f, g\ud )\ud Q\ud t\ud of the\ud t\ud th subresultant matrix\ud S\ud t\ud (\ud f, g\ud ) of the Sylvester\ud resultant matrix\ud S\ud (\ud f, g\ud ) of\ud f\ud (\ud y\ud ) and\ud g\ud (\ud y\ud ), where\ud Q\ud t\ud is a diagonal matrix of com-\ud binatorial terms. This modified subresultant matrix has sig\ud nificant computational\ud advantages with respect to the standard subresultant matri\ud x\ud S\ud t\ud (\ud f, g\ud ), and it yields\ud better results for AGCD computations. It is shown that\ud f\ud (\ud y\ud ) and\ud g\ud (\ud y\ud ) must be pro-\ud cessed by three operations before\ud S\ud t\ud (\ud f, g\ud )\ud Q\ud t\ud is formed, and the consequence of these\ud operations is the introduction of two parameters,\ud α\ud and\ud θ\ud , such that the entries of\ud S\ud t\ud (\ud f, g\ud )\ud Q\ud t\ud are non-linear functions of\ud α, θ\ud and the coefficients of\ud f\ud (\ud y\ud ) and\ud g\ud (\ud y\ud ). The\ud values of\ud α\ud and\ud θ\ud are optimised, and it is shown that these optimal values allo\ud w an\ud AGCD that has a small error, and a structured low rank approxi\ud mation of\ud S\ud (\ud f, g\ud ),\ud to be computed.
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