Approximate greatest common divisor of many polynomials and generalised resultants

Karcanias, N.; Fatouros, Stavros; Mitrouli, Marilena; Halikias, George

doi:10.23919/ecc.2003.7086598

Cited by 5 publications

(4 citation statements)

References 9 publications

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“…, p l are coprime (they have no common roots). A challenging problem treated in the literature [14,9] is computing the closest set (in a specified sense) of polynomialsp 1 , . .…”

Section: Definition Of the Problemmentioning

confidence: 99%

“…This is done exploiting the link between the rank constraint on the resultant matrix and the degree of the GCD. It is considered a classical Sylvester matrix in the case of two polynomials [32], a generalized Sylvester matrix for more than two polynomials [13,14,24]. More algorithms for the AGCD computation in the same framework are the structured total least norm (STLN) approach [13,16], the gradient projection method [26], and a recent Structured Low Rank Approximation algorithm [25] based on a Newton-like iteration.…”

Section: Definition Of the Problemmentioning

confidence: 99%

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An ODE-based method for computing the approximate greatest common divisor of polynomials

2018

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Computing the greatest common divisor of a set of polynomials is a problem which plays an important role in different fields, such as linear system, control and network theory. In practice, the polynomials are obtained through measurements and computations, so that their coefficients are inexact. This poses the problem of computing an approximate common factor. We propose an improvement and a generalization of the method recently proposed in [12], which restates the problem as a (structured) distance to singularity of the Sylvester matrix. We generalize the algorithm in order to work with more than 2 polynomials and to compute an Approximate GCD (Greatest Common Divisor) of degree k ≥ 1; moreover we show that the algorithm becomes faster by replacing the eigenvalues by the singular values.

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“…, p l are coprime (they have no common roots). A challenging problem treated in the literature [14,9] is computing the closest set (in a specified sense) of polynomialsp 1 , . .…”

Section: Definition Of the Problemmentioning

confidence: 99%

Section: Definition Of the Problemmentioning

confidence: 99%

An ODE-based method for computing the approximate greatest common divisor of polynomials

2018

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“…In [7] the "approximate GCD" problem has been considered in the context of Euclidean division and for the case of two polynomials. Recently Karcanias etc [4], introduced formally the notion of the "approximate GCD" and then developed a computational procedure that allows the evaluation of how good is the given "approximate GCD" by estimating its strength of approximation.…”

Section: Approximate Gcdmentioning

confidence: 99%

A Hybrid Approach for Normal Factorization of Polynomials

Karcanias¹,

Mitrouli

Triantafyllou

2006

Computational Science – ICCS 2006

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The problem considered here is an integral part of computations for algebraic control problems. The paper introduces the notion of normal factorization of polynomials and then presents a new hybrid algorithm for the computation of this factorization. The advantage of such a factorization is that it handles the determination of multiplicities and produces factors of lower degree and with distinct roots. The presented algorithm has the ability to specify the roots of the polynomials without computing them explicitly. Also it may be used for investigating the clustering of the roots of the polynomials. The developed procedure is based on the use of algorithms determining the greatest common divisor of polynomials. The algorithm can be implemented symbolically for the specification of well separated roots and numerically for the specification of roots belonging in approximate clusters.

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“…The numerical computation of the exact GCD is an ill-posed problem in the sense that arbitrary tiny perturbations reduce a non-trivial GCD to the constant 1. The proper definition of the "approximate" GCD, [7,9,10], and the way we can measure the strength of the approximation, have remained open. Regarding ERES, different values to the specified accuracies ε t and ε G , often leads to the computation of a common divisor of the set of polynomials, which is not the greatest common divisor (see table 3, last case).…”

Section: Estimation Of Approximate Gcd's With the Eres Methodsmentioning

confidence: 99%

Estimation of the Greatest Common Divisor of many polynomials using hybrid computations performed by the ERES method

Christou¹,

Mitrouli

2005

Appl. Num. Anal. Comp. Math.

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The computation of the Greatest Common Divisor (GCD) of a set of more than two polynomials is a non-generic problem. There are cases where iterative methods of computing the GCD of many polynomials, based on the Euclidean algorithm, fail to produce accurate results, when they are implemented in a software programming environment. This phenomenon is very strong especially when floating-point data are being used. The ERES method is an iterative matrix based method, which successfully evaluates an approximate GCD, by performing row transformations and shifting on a matrix, formed directly from the coefficients of the given polynomials. ERES deals with any kind of real data. However, due to its iterative nature, it is extremely sensitive when performing floating-point operations. It succeeds in producing results with minimal error, if we combine both floating-point and symbolic operations. In the present paper we study the behavior of the ERES method using floating-point and exact symbolic arithmetic. The conclusions derived from our study are useful for any other algorithm involving extended matrix operations.

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Approximate greatest common divisor of many polynomials and generalised resultants

Cited by 5 publications

References 9 publications

An ODE-based method for computing the approximate greatest common divisor of polynomials

An ODE-based method for computing the approximate greatest common divisor of polynomials

A Hybrid Approach for Normal Factorization of Polynomials

Estimation of the Greatest Common Divisor of many polynomials using hybrid computations performed by the ERES method

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