Structured Low Rank Approximation of a Sylvester Matrix

Abstract

Abstract. 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.Keywords. Sylvester matrix, approximate greatest common divisor, structured total least norm, hybrid symbolic/numeric algorithm.

Nearest common root of polynomials, approximate greatest common divisor and the structured singular value

“…Using the matrix identity det(I + BC) = det(I + CB) (23) which holds for any two matrices B and C of compatible dimensions, and the fact that R m,n (a 0 , b 0 )…”

“…Recent work in this area includes references [4], [23], [24] and [25]. The general approach followed in these papers formulates the approximate GCD problem as a structured least-squares approximation which can be solved numerically (e.g.…”

The Sylvester Resultant Matrix and Image Deblurring

“…The AGCD in (13) and its column equivalent are computed from a structured low rank approximation of S(p, q) for the work described in this paper because this approximation exploits the structure of S(p, q) [26,27]. A structure-preserving matrix method is also used for the AGCD computation in [11,14], and it is similar to the method used in this paper. There are, however, three important differences between the method used in [11,14] and the method used in this paper:…”

“…A structure-preserving matrix method is also used for the AGCD computation in [11,14], and it is similar to the method used in this paper. There are, however, three important differences between the method used in [11,14] and the method used in this paper:…”

“…The preprocessing operations discussed in Section 5 form an important part of the method for the AGCD computations used in this paper, but preprocessing operations are not used in [11,14]. Two of these processing operations introduce the parameters α 0 and θ 0 , and their inclusion in the problem formulation implies that (29) is a non-linear equation.…”