31 research outputs found
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A Symbolic-Numeric Software Package for the Computation of the GCD of Several Polynomials
This survey is intended to present a package of algorithms for the computation of exact or approximate GCDs of sets of several polynomials and the evaluation of the quality of the produced solutions. These algorithms are designed to operate in symbolic-numeric computational environments. The key of their effectiveness is the appropriate selection of the right type of operations (symbolic or numeric) for the individual parts of the algorithms. Symbolic processing is used to improve on the conditioning of the input data and handle an ill-conditioned sub-problem and numeric tools are used in accelerating certain parts of an algorithm. A sort description of the basic algorithms of the package is presented by using the symbolic-numeric programming code of Maple
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Approximate greatest common divisor of many polynomials and pseudo-spectrum
The paper is concerned with establishing the links between the approximate GCD of a set of polynomials and the notion of the pseudo-spectrum defined on a set of polynomials. By examining the pseudo-spectrum of the structured matrix we will derive estimates of the area of the approximate roots of the initial polynomial set. We will relate the strength of the GCD to the weighted strength of the pseudo-spectra and we investigate under which conditions the roots of the approximate GCDs are a subset of the pseudo-spectra
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Approximate Zero Polynomials of Polynomial Matrices and Linear Systems
The aim of this paper is to extend recent results on the approximate GCD of polynomials [1] and approximate zeros to the case of a polynomial matrices within the framework of exterior algebra [2]. The results provide the means to introduce a new characterization of approximate decoupling zeros and measures for approximate controllability and observability for the case of linear systems
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Approximate zero polynomials of polynomial matrices and linear systems
This paper introduces the notions of approximate and optimal approximate zero polynomial of a polynomial matrix by deploying recent results on the approximate GCD of a set of polynomials Karcaniaset al. (2006) 1 and the exterior algebra Karcanias and Giannakopoulos (1984) 4 representation of polynomial matrices. The results provide a new definition for the "approximate", or "almost" zeros of polynomial matrices and provide the means for computing the distance from non-coprimeness of a polynomial matrix. The computational framework is expressed as a distance problem in a projective space. The general framework defined for polynomial matrices provides a new characterization of approximate zeros and decoupling zeros Karcanias et al. (1983) 2 and Karcanias and Giannakopoulos (1984) 4 of linear systems and a process leading to computation of their optimal versions. The use of restriction pencils provides the means for defining the distance of state feedback (output injection) orbits from uncontrollable (unobservable) families of systems, as well as the invariant versions of the "approximate decoupling polynomials". The overall framework that is introduced provides the means for introducing measures for the distance of a system from different families of uncontrollable, or unobservable systems, which may be feedback dependent, or feedback invariant as well as the notion of "approximate decoupling polynomials"
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The ERES method for computing the approximate GCD of several polynomials
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
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Numerical and Symbolical Methods for the GCD of Several Polynomials
The computation of the Greatest Common Divisor (GCD) of a set of polynomials is an important issue in computational mathematics and it is linked to Control Theory very strong. In this paper we present different matrix-based methods, which are developed for the efficient computation of the GCD of several polynomials. Some of these methods are naturally developed for dealing with numerical inaccuracies in the input data and produce meaningful approximate results. Therefore, we describe and compare numerically and symbolically methods such as the ERES, the Matrix Pencil and other resultant type methods, with respect to their complexity and effectiveness. The combination of numerical and symbolic operations suggests a new approach in software mathematical computations denoted as hybrid computations. This combination offers great advantages, especially when we are interested in finding approximate solutions. Finally the notion of approximate GCD is discussed and a useful criterion estimating the strength of a given approximate GCD is also developed
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Approximate zero polynomials of polynomial matrices and linear systems
This paper introduces the notions of approximate and optimal approximate zero polynomial of a polynomial matrix by deploying recent results on the approximate GCD of a set of polynomials [1] and the exterior algebra [4] representation of polynomial matrices. The results provide a new definition for the "approximate", or "almost" zeros of polynomial matrices and provide the means for computing the distance from non-coprimeness of a polynomial matrix. The computational framework is expressed as a distance problem in a projective space. The general framework defined for polynomial matrices provides a new characterization of approximate zeros and decoupling zeros [2], [4] of linear systems and a process leading to computation of their optimal versions. The use of restriction pencils provides the means for defining the distance of state feedback (output injection) orbits from uncontrollable (unobservable) families of systems, as well as the invariant versions of the "approximate decoupling polynomials"
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Properties and Classification of Generalized Resultants and Polynomial Combinants
Polynomial combinants define the linear part of the Dynamic Determinantal Assignment Problems, which provides the unifying description of the frequency assignment problems in Linear Systems. The theory of dynamic polynomial combinants have been recently developed by examining issues of their representation, parameterization of dynamic polynomial combinants according to the notions of order and degree and spectral assignment. Dynamic combinants are linked to the theory of “Generalised Resultants”, which provide the matrix representation of polynomial combinants. We consider coprime set polynomials for which assignability is always feasible and provides a complete characterisation of all assignable combinants with order above and below the Sylvester order. The complete parameterization of combinants and coresponding Generalised Resultants is prerequisite to the characterisation of the minimal degree and order combinant for which spectrum assignability may be achieved
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The Minimal Design Problem on Dynamic Polynomial Combinants
The theory of dynamic polynomial combinants is linked to the linear part of the Dynamic Determinantal Assignment Problems, which provides the unifying description of the pole and zero dynamic assignment problems in Linear Systems. The fundamentals of the theory of dynamic polynomial combinants have been recently developed by examining issues of their representation, parameterization of dynamic polynomial combinants according to the notions of order and degree and spectral assignment. Central to this study is the link of dynamic combinants to the theory of "Generalised Resultants", which provide the matrix representation of the dynamic combinants. The paper considers the case of coprime set polynomials for which spectral assignability is always feasible and provides a complete characterisation of all assignable combinants with order above and below the Sylvester order. A complete parameterization of combinants and respective Generalised Resultants is given and this leads naturally to the characterisation of the minimal degree and order combinant for which spectrum assignability may be achieved, referred to as the "Dynamic Combinant Minimal Design" (DCMD) problem. Such solutions provide low bounds for the respective Dynamic Assignment control problems