A non-linear structure-preserving matrix method for the computation of the coefficients of an approximate greatest common divisor of two Bernstein polynomials

摘要

This paper describes a non-linear structure-preserving ma\udtrix method for the com-\udputation of the coefficients of an approximate greatest commo\udn divisor (AGCD) of\uddegree\udt\udof two Bernstein polynomials\udf\ud(\udy\ud) and\udg\ud(\udy\ud). This method is applied to\uda modified form\udS\udt\ud(\udf, g\ud)\udQ\udt\udof the\udt\udth subresultant matrix\udS\udt\ud(\udf, g\ud) of the Sylvester\udresultant matrix\udS\ud(\udf, g\ud) of\udf\ud(\udy\ud) and\udg\ud(\udy\ud), where\udQ\udt\udis a diagonal matrix of com-\udbinatorial terms. This modified subresultant matrix has sig\udnificant computational\udadvantages with respect to the standard subresultant matri\udx\udS\udt\ud(\udf, g\ud), and it yields\udbetter results for AGCD computations. It is shown that\udf\ud(\udy\ud) and\udg\ud(\udy\ud) must be pro-\udcessed by three operations before\udS\udt\ud(\udf, g\ud)\udQ\udt\udis formed, and the consequence of these\udoperations is the introduction of two parameters,\udα\udand\udθ\ud, such that the entries of\udS\udt\ud(\udf, g\ud)\udQ\udt\udare non-linear functions of\udα, θ\udand the coefficients of\udf\ud(\udy\ud) and\udg\ud(\udy\ud). The\udvalues of\udα\udand\udθ\udare optimised, and it is shown that these optimal values allo\udw an\udAGCD that has a small error, and a structured low rank approxi\udmation of\udS\ud(\udf, g\ud),\udto be computed.