&amp;lt;tex&amp;gt;$QR$&amp;lt;/tex&amp;gt;Factoring to Compute the GCD of Univariate Approximate Polynomials

Corless, Robert M.; Watt, Stephen M.; Zhi, Lihong

doi:10.1109/tsp.2004.837413

Cited by 85 publications

(82 citation statements)

References 17 publications

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“…Several algorithms for the computation of an approximate polynomial gcd can be found in the literature; they rely on different techniques, such as the Euclidean algorithm [1], [2], [12], [17], optimization methods [15], SVD and factorization of resultant matrices [5], [4], [23], Padé approximation [3], [18], root grouping [18]. Some of them have been implemented inside numerical/symbolic packages like the algorithm of Zeng [23] in Matlab TM and the algorithms of Kaltofen [14], of Corless et al [4], of Labahn and Beckermann [13] in Maple TM .…”

Section: Definition 11 a Polynomial G(x) Is Said To Be An -Divisor mentioning

confidence: 99%

“…Some of them have been implemented inside numerical/symbolic packages like the algorithm of Zeng [23] in Matlab TM and the algorithms of Kaltofen [14], of Corless et al [4], of Labahn and Beckermann [13] in Maple TM . These algorithms have a computational cost of O(n 3 ) which makes them expensive for moderately large values of n.…”

Section: Definition 11 a Polynomial G(x) Is Said To Be An -Divisor mentioning

confidence: 99%

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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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Section: Definition 11 a Polynomial G(x) Is Said To Be An -Divisor mentioning

confidence: 99%

Section: Definition 11 a Polynomial G(x) Is Said To Be An -Divisor mentioning

confidence: 99%

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

Бини¹,

Boito²

2010

Numerical Methods for Structured Matrices and Applications

View full text Add to dashboard Cite

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“…6 and 7 show the variation of log 10 µ k with k for the calculation of the degree of the row component of the PSF, where the graphs are obtained with different pairs of randomly selected rows of G. Fig. 6 yields the correct result because the value of t, computed from (11), is equal to 8. Fig.…”

Section: S(x Y) = H(x Y) + E(x Y) F (X Y)mentioning

confidence: 96%

“…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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“…As in [9], reversion may be an essential practical step in numerical algorithms for GCD by values. This will be investigated in a future paper.…”

Section: The Effect Of Reversionmentioning

confidence: 99%

Pseudospectra of Matrix Polynomials that Are Expressed in Alternative Bases

2007

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Abstract. Spectra and pseudospectra of matrix polynomials are of interest in geometric intersection problems, vibration problems, and analysis of dynamical systems. In this note we consider the effect of the choice of polynomial basis on the pseudospectrum and on the conditioning of the spectrum of regular matrix polynomials. In particular, we consider the direct use of the Lagrange basis on distinct interpolation nodes, and give a geometric characterization of "good" nodes. We also give some tools for computation of roots at infinity via a new, natural, reversal. The principal achievement of the paper is to connect pseudospectra to the well-established theory of Lebesgue functions and Lebesgue constants, by separating the influence of the scalar basis from the natural scale of the matrix polynomial, which allows many results from interpolation theory to be applied.

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<tex>$QR$</tex>Factoring to Compute the GCD of Univariate Approximate Polynomials

Cited by 85 publications

References 17 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

Pseudospectra of Matrix Polynomials that Are Expressed in Alternative Bases

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