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Approximate greatest common divisors of several polynomials with linearly constrained coefficients and singular polynomials
 Manuscript
, 2006
"... We consider the problem of computing minimal real or complex deformations to the coefficients in a list of relatively prime real or complex multivariate polynomials such that the deformed polynomials have a greatest common divisor (GCD) of at least a given degree k. In addition, we restrict the defo ..."
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Cited by 33 (13 self)
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We consider the problem of computing minimal real or complex deformations to the coefficients in a list of relatively prime real or complex multivariate polynomials such that the deformed polynomials have a greatest common divisor (GCD) of at least a given degree k. In addition, we restrict the deformed coefficients by a given set of linear constraints, thus introducing the linearly constrained approximate GCD problem. We present an algorithm based on a version of the structured total least norm (STLN) method and demonstrate, on a diverse set of benchmark polynomials, that the algorithm in practice computes globally minimal approximations. As an application of the linearly constrained approximate GCD problem, we present an STLNbased method that computes for a real or complex polynomial the nearest real or complex polynomial that has a root of multiplicity at least k. We demonstrate that the algorithm in practice computes, on the benchmark polynomials given in the literature, the known globally optimal nearest singular polynomials. Our algorithms can handle, via randomized preconditioning, the difficult case when the nearest solution to a list of real input polynomials actually has nonreal complex coefficients.
On probabilistic analysis of randomization in hybrid symbolicnumeric algorithms
, 2007
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The Bézout matrix in the Lagrange basis
 Proceedings EACA
, 2004
"... A recent paper by Amiraslani, Corless, GonzalezVega and Shakoori studies polynomial algebra by values, without first converting to another basis such as the monomial basis. Their methodology is driven by a desire to avoid illconditioning in such changes of basis. In this talk I expand on some deta ..."
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Cited by 10 (1 self)
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A recent paper by Amiraslani, Corless, GonzalezVega and Shakoori studies polynomial algebra by values, without first converting to another basis such as the monomial basis. Their methodology is driven by a desire to avoid illconditioning in such changes of basis. In this talk I expand on some details from that paper, namely the construction and use of the Bézout matrix directly from given polynomial values. We here show details of the construction of common roots of polynomials directly from the eigenvectors of the constructed Bézout matrix.
A fast algorithm for approximate polynomial GCD based on structured matrix computations
 OPERATOR THEORY: ADVANCES AND APPLICATIONS, 199, 155–173, BIRKHÄUSER
, 2010
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An iterative method for calculating approximate GCD of univariate polynomials
 In Proceedings of the 2009 International Symposium on Symbolic and Algebraic Computation
, 2009
"... We present an iterative algorithm for calculating approximate greatest common divisor (GCD) of univariate polynomials with the real coefficients. For a given pair of polynomials and a degree, our algorithm finds a pair of polynomials which has a GCD of the given degree and whose coefficients are ..."
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Cited by 5 (2 self)
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We present an iterative algorithm for calculating approximate greatest common divisor (GCD) of univariate polynomials with the real coefficients. For a given pair of polynomials and a degree, our algorithm finds a pair of polynomials which has a GCD of the given degree and whose coefficients are perturbed from those in the original inputs, making the perturbations as small as possible, along with the GCD. The problem of approximate GCD is transfered to a constrained minimization problem, then solved with a socalled modified Newton method, which is a generalization of the gradientprojection method, by searching the solution iteratively. We demonstrate that our algorithm calculates approximate GCD with perturbations as small as those calculated by a method based on the structured total least norm (STLN) method, while our method runs significantly faster than theirs by approximately up to 30 times, compared with their implementation. We also show that our algorithm properly handles some illconditioned problems with GCD containing small or large leading coefficient.
Dividing polynomials when you only know their values
 In Laureano GonzalezVega and Tomas Recio, editors, Proceedings EACA
, 2004
"... A recent paper by Amiraslani, Corless, GonzalezVega and Shakoori studies polynomial algebra by values, without first converting to another basis such as the monomial basis. In this talk I expand on some details from that paper, namely the method we used to divide (multivariate and univariate) polyn ..."
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Cited by 4 (0 self)
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A recent paper by Amiraslani, Corless, GonzalezVega and Shakoori studies polynomial algebra by values, without first converting to another basis such as the monomial basis. In this talk I expand on some details from that paper, namely the method we used to divide (multivariate and univariate) polynomials given only by values. This is a surprisingly valuable operation, and with it one can solve systems of polynomial equations without first constructing a Gröbner basis, or one can compute Gröbner bases if desired.
Pseudospectra of Matrix Polynomials that are Expressed in Alternative Bases
"... 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 ..."
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Cited by 4 (3 self)
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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 wellestablished 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. 1.
Structured low rank approximation of a Bezout matrix
 PROC. OF INTERNATIONAL CONFERENCE ON MATHEMATICS ASPECTS
, 2006
"... The task of determining the approximate greatest common divisor (GCD) of more than two univariate polynomials with inexact coefficients can be formulated as computing for a given Bezout matrix a new Bezout matrix of lower rank whose entries are near the corresponding entries of that input matrix. ..."
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Cited by 3 (1 self)
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The task of determining the approximate greatest common divisor (GCD) of more than two univariate polynomials with inexact coefficients can be formulated as computing for a given Bezout matrix a new Bezout matrix of lower rank whose entries are near the corresponding entries of that input matrix. We present an algorithm based on a version of structured nonlinear total least squares (SNTLS) method for computing approximate GCD and demonstrate the practical performance of our algorithm on a diverse set of univariate polynomials.
Structured low rank approximations of the Sylvester resultant matrix for approximate GCDs of Bernstein basis polynomials
, 2008
"... A structured low rank approximation of the Sylvester resultant matrix S(f, g) of the Bernstein basis polynomials f = f(y) and g = g(y), for the determination of their approximate greatest common divisors (GCDs), is computed using the method of structured total least norm. Since the GCD of f(y) and ..."
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Cited by 3 (1 self)
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A structured low rank approximation of the Sylvester resultant matrix S(f, g) of the Bernstein basis polynomials f = f(y) and g = g(y), for the determination of their approximate greatest common divisors (GCDs), is computed using the method of structured total least norm. Since the GCD of f(y) and g(y) is equal to the GCD of f(y) and αg(y), where α is an arbitrary nonzero constant, it is more appropriate to consider a structured low rank approximation S ( ˜ f, ˜g) of S(f, αg), where the polynomials ˜ f = ˜ f(y) and ˜g = ˜g(y) approximate the polynomials f(y) and αg(y), respectively. Different values of α yield different structured low rank approximations S ( ˜ f, ˜g), and therefore different approximate GCDs. It is shown that the inclusion of α allows to obtain considerably improved approximations, as measured by the decrease of the singular values σi of S ( ˜ f, ˜g), with respect to the approximation obtained when the default value α = 1 is used. An example that illustrates the theory is presented and future work is discussed.