The ERES method for computing the approximate GCD of several polynomials

D Christou,N Karcanias,M Mitrouli

doi:10.1016/j.apnum.2009.10.002

Abstract

The computation of the greatest common divisor (GCD) of a set of polynomials has interested the mathematicians for a long time and has attracted a lot of attention in recent years. A challenging problem that arises from several applications, such as control or image and signal processing, is to develop a numerical GCD method that inherently has the potential to work efficiently with sets of several polynomials with inexactly known coefficients. The presented work focuses on: (i) the use of the basic principles of the ERES methodology for calculating the GCD of a set of several polynomials and defining approximate solutions by developing the hybrid implementation of this methodology. (ii) the use of the developed framework for defining the approximate notions for the GCD as a distance problem in a projective space to develop an optimization algorithm for evaluating the strength of different ad-hoc approximations derived from different algorithms. The presented new implementation of ERES is based on the effective combination of symbolic–numeric arithmetic (hybrid arithmetic) and shows interesting computational properties for the approximate GCD problem. Additionally, an efficient implementation of the strength of an approximate GCD is given by exploiting some of the special aspects of the respective distance problem. Finally, the overall performance of the ERES algorithm for computing approximate solutions is discussed.

Highlights

The computation of the Greatest Common Divisor (GCD) of polynomials is a fundamental problem in many mathematical areas such as linear systems, control theory, network theory and communications
This motivates the need for transforming the algebraic problems into equivalent linear algebra problems and develop approximate algebraic computations, which are appropriate for the case of computations on models characterised by parameter uncertainty
There are several algorithms, which have employed the process of singular value decomposition in their structures in order to estimate the degree of a GCD for a specific tolerance εt [7,9,10]

Summary

Introduction

The computation of the Greatest Common Divisor (GCD) of polynomials is a fundamental problem in many mathematical areas such as linear systems, control theory, network theory and communications. Engineering models are not exact and they are always characterised by parameter uncertainty. The central challenge is the transformation of algebraic notions to an appropriate analytic setup within which meaningful approximate solutions to exact algebraic problems may be sought. This motivates the need for transforming the algebraic problems into equivalent linear algebra problems and develop approximate algebraic computations, which are appropriate for the case of computations on models characterised by parameter uncertainty

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Conclusion