Certified approximate univariate GCDs

Emiris, Ioannis Z.; Galligo, André; Lombardi, Henri

doi:10.1016/s0022-4049(97)00013-3

Cited by 98 publications

(63 citation statements)

References 15 publications

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“…This is why the notion of approximate gcd, or -gcd, has been introduced. For more details on this topic we refer the reader to [19] [7], [18], [23] and to the references therein.…”

Section: Introductionmentioning

confidence: 99%

A Fast Algorithm for Approximate Polynomial GCD Based on Structured Matrix Computations

Бини¹,

Boito²

2010

Numerical Methods for Structured Matrices and Applications

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Abstract. An O(n 2 ) complexity algorithm for computing an -greatest common divisor (gcd) of two polynomials of degree at most n is presented. The algorithm is based on the formulation of polynomial gcd given in terms of resultant (Bézout, Sylvester) matrices, on their displacement structure and on the reduction of displacement structured matrices to Cauchy-like form originally pointed out by Georg Heinig. A Matlab implementation is provided. Numerical experiments performed with a wide variety of test problems, show the effectiveness of this algorithm in terms of speed, stability and robustness, together with its better reliability with respect to the available software. Mathematics Subject Classification (2000). Da mettere.

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“…This is why the notion of approximate gcd, or -gcd, has been introduced. For more details on this topic we refer the reader to [19] [7], [18], [23] and to the references therein.…”

Section: Introductionmentioning

confidence: 99%

A Fast Algorithm for Approximate Polynomial GCD Based on Structured Matrix Computations

Бини¹,

Boito²

2010

Numerical Methods for Structured Matrices and Applications

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“…The deblurred image is then obtained by deconvolving H from G [32]. Many methods for the computation of an AGCD of two polynomials have been developed, including methods based on the QR decomposition of the Sylvester matrix [11,37], methods based on the singular value decomposition (SVD) of the Sylvester matrix [10,15], optimisation methods [9,38] and methods that exploit the structure of the Sylvester matrix [3,4,33,34]. In this paper, the method of SNTLN is applied to the Sylvester matrix in order to compute an AGCD of two inexact (noisy) polynomials.…”

Section: The Computation Of An Agcdmentioning

confidence: 99%

Polynomial computations for blind image deconvolution

Winkler

2016

Linear Algebra and its Applications

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“…Generally, in most GCD problems the degree of the GCD is unknown and thus a numerical algorithm can easily produce misleading results. An approach to this problem is to certify and fix a maximum degree according to appropriate theorems and techniques [11,33] and proceed with the computation of a common divisor of this particular degree. The evaluation of the strength of a given approximation is another important issue here.…”

Section: Approximate Solutionsmentioning

confidence: 99%

Numerical and Symbolical Methods for the GCD of Several Polynomials

Christou¹,

Karcanias²,

Mitrouli³

et al. 2011

Lecture Notes in Electrical Engineering

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This is the accepted version of the paper.This version of the publication may differ from the final published version. Permanent repository link AbstractThe 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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Certified approximate univariate GCDs

Cited by 98 publications

References 15 publications

A Fast Algorithm for Approximate Polynomial GCD Based on Structured Matrix Computations

A Fast Algorithm for Approximate Polynomial GCD Based on Structured Matrix Computations

Polynomial computations for blind image deconvolution

Numerical and Symbolical Methods for the GCD of Several Polynomials

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