The ERES method for computing the approximate GCD of several polynomials

Authors:
D. Christou;N. Karcanias;M. Mitrouli
Affiliations:
Control Engineering Research Centre, School of Engineering and Mathematical Sciences, City University, Northampton Square, EC1V 0HB, London, UK;Control Engineering Research Centre, School of Engineering and Mathematical Sciences, City University, Northampton Square, EC1V 0HB, London, UK;Department of Mathematics, University of Athens, Panepistemiopolis 15784, Athens, Greece
Venue:
Applied Numerical Mathematics
Year:
2010

Citing 9
Cited 2

An efficient and reliable algorithm for computing the singular subspace of a matrix, associated with its smallest singular values

Journal of Computational and Applied Mathematics
Partial singular value decomposition algorithm (Algorithm 32)

Journal of Computational and Applied Mathematics
Approximate GCD and its application to ill-conditioned algebraic equations

ISCM '90 Proceedings of the International Symposium on Computation mathematics
The singular value decomposition for polynomial systems

ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
On approximate GCDs of univariate polynomials

Journal of Symbolic Computation - Special issue on symbolic numeric algebra for polynomials
On Euclid's Algorithm and the Computation of Polynomial Greatest Common Divisors

Journal of the ACM (JACM)
An Improved Algorithm for Computing the Singular Value Decomposition

ACM Transactions on Mathematical Software (TOMS)
More Accurate Bidiagonal Reduction for Computing the Singular Value Decomposition

SIAM Journal on Matrix Analysis and Applications
Approximate greatest common divisor of many polynomials, generalised resultants, and strength of approximation

Computers & Mathematics with Applications

An improved non-linear method for the computation of a structured low rank approximation of the Sylvester resultant matrix

Journal of Computational and Applied Mathematics
Matrix representation of the shifting operation and numerical properties of the ERES method for computing the greatest common divisor of sets of many polynomials

Journal of Computational and Applied Mathematics

Quantified Score

Hi-index	0.00

Visualization

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.