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40
Certified approximate univariate GCDs
 METHODS IN ALGEBRAIC GEOMETRY, 117 & 118:229251
, 1997
"... We study the approximate GCD of two univariate polynomials given with limited accuracy or, equivalently, the exact GCD of the perturbed polynomials within some prescribed tolerance. A perturbed polynomial is regarded as a family of polynomials in a classification space, which leads to an accurate an ..."
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Cited by 48 (5 self)
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We study the approximate GCD of two univariate polynomials given with limited accuracy or, equivalently, the exact GCD of the perturbed polynomials within some prescribed tolerance. A perturbed polynomial is regarded as a family of polynomials in a classification space, which leads to an accurate analysis of the computation. Considering only the Sylvester matrix singular values, as is frequently suggested in the literature, does not suffice to solve the problem completely, even when the extended euclidean algorithm is also used. We provide a counterexample that illustrates this claim and indicates the problem's hardness. SVD computations on subresultant matrices lead to upper bounds on the degree of the approximate GCD. Further use of the subresultant matrices singular values yields an approximate syzygy of the given polynomials, which is used to establish a gap theorem on certain singular values that certifies the maximumdegree approximate GCD. This approach leads directly to an algorithm for computing the approximate GCD polynomial. Lastly, we suggest the use of weighted norms in order to sharpen the theorem's conditions in a more intrinsic context.
A method computing multiple roots of inexact polynomials
 In Sendra [29
, 2003
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A reordered Schur factorization method for zerodimensional polynomial systems with multiple roots
 In Proc. ACM Intern. Symp. on Symbolic and Algebraic Computation
, 1997
"... We discuss the use of a single generic linear combination of multiplication matrices, and its reordered Schur factorization, to find the roots of a system of multivariate polynomial equations. The principal contribution of the paper is to show how to reduce the multivariate problem to a univariate p ..."
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Cited by 46 (3 self)
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We discuss the use of a single generic linear combination of multiplication matrices, and its reordered Schur factorization, to find the roots of a system of multivariate polynomial equations. The principal contribution of the paper is to show how to reduce the multivariate problem to a univariate problem, even in the case of multiple roots, in a numerically stable way. 1 Introduction The technique of solving systems of multivariate polynomial equations via eigenproblems has become a topic of active research (with applications in computeraided design and control theory, for example) at least since the papers [2, 6, 9]. One may approach the problem via various resultant formulations or by Grobner bases. As more understanding is gained, it is becoming clearer that eigenvalue problems are the "weakly nonlinear nucleus to which the original, strongly nonlinear task may be reduced"[13]. Early works concentrated on the case of simple roots. An example of such was the paper [5], which use...
Optimization Strategies for the Approximate GCD Problem
 IN PROC. ISSAC'98
, 1998
"... We describe algorithms for computing the greatest common divisor (GCD) of two univariate polynomials with inexactlyknown coefficients. Assuming that an estimate for the GCD degree is available (e.g., using an SVDbased algorithm), we formulate and solve a nonlinear optimization problem in order to d ..."
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Cited by 24 (2 self)
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We describe algorithms for computing the greatest common divisor (GCD) of two univariate polynomials with inexactlyknown coefficients. Assuming that an estimate for the GCD degree is available (e.g., using an SVDbased algorithm), we formulate and solve a nonlinear optimization problem in order to determine the coefficients of the "best" GCD. We discuss various issues related to the implementation of the algorithms and present some preliminary test results.
Numerical Computation Of A Polynomial GCD And Extensions
, 1996
"... In the first part of this paper, we dene approximate polynomial gcds (greatest common divisors) and extended gcds provided that approximations to the zeros of the input polynomials are available. We relate our novel definition to the older and weaker ones, based on perturbation of the coefficients o ..."
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Cited by 21 (6 self)
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In the first part of this paper, we dene approximate polynomial gcds (greatest common divisors) and extended gcds provided that approximations to the zeros of the input polynomials are available. We relate our novel definition to the older and weaker ones, based on perturbation of the coefficients of the input polynomials, we demonstrate some deficiency of the latter definitions (which our denition avoids), and we propose new effective sequential and parallel (RNC and NC) algorithms for computing approximate gcds and extended gcds. Our stronger results are obtained with no increase of the asymptotic bounds on the computational cost. This is partly due to application of our recent nearly optimal algorithms for approximating polynomial zeros. In the second part of our paper, working under the older and more customary definition of approximate gcds, we modify and develop an alternative approach, which was previously based on the computation of the Singular Value Decomposition (SVD) of the associat...
On location and approximation of clusters of zeroes of analytic functions
 Found. Comput. Math
, 2005
"... Abstract. In the beginning of the eighties, M. Shub and S. Smale developed a quantitative analysis of Newton’s method for multivariate analytic maps. In particular, their αtheory gives an effective criterion that ensures safe convergence to a simple isolated zero. This criterion requires only infor ..."
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Cited by 16 (4 self)
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Abstract. In the beginning of the eighties, M. Shub and S. Smale developed a quantitative analysis of Newton’s method for multivariate analytic maps. In particular, their αtheory gives an effective criterion that ensures safe convergence to a simple isolated zero. This criterion requires only information concerning the map at the initial point of the iteration. Generalizing this theory to multiple zeros and clusters of zeros is still a challenging problem. In this article we focus on one complex variable functions. We study general criteria for detecting clusters and analyze the convergence of Schröder’s iteration to a cluster. In the case of a multiple root, it is wellknown that this convergence is quadratic. In the case of a cluster with positive diameter, the convergence is still quadratic provided the iteration is stopped sufficiently early. We propose a criterion for stopping this iteration at a distance from the cluster which
Numerical Univariate Polynomial GCD
 The Mathematics of Numerical Analysis, volume 32 of Lectures in Applied Math
, 1996
"... We formalize the notion of approximate GCD for univariate polynomials given with limited accuracy and then address the problem of its computation. Algebraic concepts are applied in order to provide a solid foundation for a numerical approach. We exhibit the limitations of the euclidean algorithm thr ..."
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Cited by 11 (3 self)
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We formalize the notion of approximate GCD for univariate polynomials given with limited accuracy and then address the problem of its computation. Algebraic concepts are applied in order to provide a solid foundation for a numerical approach. We exhibit the limitations of the euclidean algorithm through experiments, show that existing methods only solve part of the problem and assert its worstcase complexity. A rigorous geometrical point of view is given in the parameter space of all input polynomials and SVD computations on subresultants are applied in order to derive upper bounds on the degree of the approximate GCD. Then, we establish a certification theorem and state the conditions under which it determines the precise GCD degree.