Structured Low Rank Approximation of a Sylvester Matrix

The task of determining the approximate greatest common divisor (GCD) of univariate polynomials with inexact coefficients can be formulated as computing for a given Sylvester matrix a new Sylvester matrix of lower rank whose entries are near the corresponding entries of that input matrix. We solve the approximate GCD problem by a new method based on structured total least norm (STLN) algorithms, in our case for matrices with Sylvester structure. We present iterative algorithms that compute an approximate GCD and that can certify an approximate ǫ-GCD when a tolerance ǫ is given on input. Each single iteration is carried out with a number of floating point operations that is of cubic order in the input degrees. We also demonstrate the practical performance of our algorithms on a diverse set of univariate pairs of polynomials. Mathematics Subject Classification (2000). Primary 68W30; Secondary 65K10.