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57
A method computing multiple roots of inexact polynomials
 In Sendra [29
, 2003
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Approximate Factorization of Multivariate Polynomials via Differential Equations
 Manuscript
, 2004
"... The input to our algorithm is a multivariate polynomial, whose complex rational coe#cients are considered imprecise with an unknown error that causes f to be irreducible over the complex numbers C. We seek to perturb the coe#cients by a small quantitity such that the resulting polynomial factors ove ..."
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Cited by 43 (12 self)
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The input to our algorithm is a multivariate polynomial, whose complex rational coe#cients are considered imprecise with an unknown error that causes f to be irreducible over the complex numbers C. We seek to perturb the coe#cients by a small quantitity such that the resulting polynomial factors over C. Ideally, one would like to minimize the perturbation in some selected distance measure, but no e#cient algorithm for that is known. We give a numerical multivariate greatest common divisor algorithm and use it on a numerical variant of algorithms by W. M. Ruppert and S. Gao. Our numerical factorizer makes repeated use of singular value decompositions. We demonstrate on a significant body of experimental data that our algorithm is practical and can find factorizable polynomials within a distance that is about the same in relative magnitude as the input error, even when the relative error in the input is substantial (10 3 ).
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 29 (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.
A fast and numerically stable Euclideanlike algorithm for detecting relatively prime numerical polynomials
, 1998
"... In this paper we provide a fast, numerically stable algorithm to determine when two given polynomials a and b are relatively prime and remain relatively prime even after small perturbations of their coefficients. Such a problem is important in many applications where input data is only available up ..."
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Cited by 20 (3 self)
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In this paper we provide a fast, numerically stable algorithm to determine when two given polynomials a and b are relatively prime and remain relatively prime even after small perturbations of their coefficients. Such a problem is important in many applications where input data is only available up to a certain precision. Our method  an extension of the CabayMeleshko algorithm for Pad'e approximation  is typically an order of magnitude faster than previously known stable methods. As such it may be used as an inexpensive test which may be applied before attempting to compute a "numerical GCD", in general a much more difficult task. We prove that the algorithm is numerically stable and give experiments verifying the numerical behaviour. Finally, we discuss possible extensions of our approach that can be applied to the problem of actually computing a numerical GCD. 1 Introduction Let a; b 2 C[z] be (univariate) polynomials with real or complex coefficients a(z) = a 0 + a 1 z + : : ...
When Are Two Numerical Polynomials Relatively Prime?
, 1997
"... Let a and b be two polynomials having numerical coefficients. We consider the question: when are a and b relatively prime? Since the coefficients of a and b are approximant, the question is the same as: when are two polynomials relatively prime, even after small perturbations of the coefficients? In ..."
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Cited by 19 (3 self)
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Let a and b be two polynomials having numerical coefficients. We consider the question: when are a and b relatively prime? Since the coefficients of a and b are approximant, the question is the same as: when are two polynomials relatively prime, even after small perturbations of the coefficients? In this paper we provide a numeric parameter for determining that two polynomials are prime, even under small perturbations of the coefficients. Our methods rely on an inversion formula for Sylvester matrices to establish an effective criterion for relative primeness. The inversion formula can also be used to approximate the condition number of a Sylvester matrix. 1
Approximate Bivariate Factorization, a Geometric Viewpoint
, 2007
"... We briefly present and analyze, from a geometric viewpoint, strategies for designing algorithms to factor bivariate approximate polynomials in C[x, y]. Given a composite polynomial, stably squarefree, satisfying a genericity hypothesis, we describe the effect of a perturbation on the roots of its d ..."
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Cited by 11 (1 self)
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We briefly present and analyze, from a geometric viewpoint, strategies for designing algorithms to factor bivariate approximate polynomials in C[x, y]. Given a composite polynomial, stably squarefree, satisfying a genericity hypothesis, we describe the effect of a perturbation on the roots of its discriminant with respect to one variable, and the perturbation of the corresponding monodromy action on a smooth fiber. A novel geometric approach is presented, based on guided projection in the parameter space and continuation method above randomly chosen loops, to reconstruct from a perturbed polynomial a nearby composite polynomial and its irreducible factors. An algorithm and its ingredients are described.
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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Approximate GCD of multivariate polynomials
 Proc.ASCM 2000, World Scientific Press
, 2000
"... We describe algorithms for computing the greatest common divisor of two multivariate polynomials with inexactly known coefficients. We focus on extending standard exact EZGCD algorithm to an efficient and stable algorithm in approximate case. Various issues related to the implementation of the algo ..."
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Cited by 9 (4 self)
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We describe algorithms for computing the greatest common divisor of two multivariate polynomials with inexactly known coefficients. We focus on extending standard exact EZGCD algorithm to an efficient and stable algorithm in approximate case. Various issues related to the implementation of the algorithms and some preliminary test results are also presented. 1