Results 1  10
of
21
The Singular Value Decomposition for Polynomial Systems
, 1995
"... This paper introduces singular value decomposition (SVD) algorithms for some standard polynomial computations, in the case where the coefficients are inexact or imperfectly known. We first give an algorithm for computing univariate GCD's which gives exact results for interesting nearby problems, and ..."
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Cited by 82 (9 self)
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This paper introduces singular value decomposition (SVD) algorithms for some standard polynomial computations, in the case where the coefficients are inexact or imperfectly known. We first give an algorithm for computing univariate GCD's which gives exact results for interesting nearby problems, and give efficient algorithms for computing precisely how nearby. We generalize this to multivariate GCD computation. Next, we adapt Lazard's uresultant algorithm for the solution of overdetermined systems of polynomial equations to the inexactcoefficient case. We also briefly discuss an application of the modied Lazard's method to the location of singular points on approximately known projections of algebraic curves.
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 38 (4 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.
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 24 (8 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...
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 21 (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.
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 13 (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
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.
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 10 (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 + : : ...
Some Recent Algebraic/Numerical Algorithms
, 1998
"... Combination of algebraic and numerical techniques for improving the computations in algebra and geometry is a popular research topic of growing interest. We survey some recent progress that we made in this area, in particular, regarding polynomial rootfinding, the solution of a polynomial system of ..."
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Cited by 5 (4 self)
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Combination of algebraic and numerical techniques for improving the computations in algebra and geometry is a popular research topic of growing interest. We survey some recent progress that we made in this area, in particular, regarding polynomial rootfinding, the solution of a polynomial system of equations, the computation of an approximate greatest common divisor of two polynomials as well as various computations with dense structured matrices and their further applications to polynomial and rational interpolation and multipoint polynomial evaluation. In some cases our algorithms reach nearly optimal time bounds and/or improve the previously known methods by order of magnitude, in other cases we yield other gains, such as improved numerical stability. Key words: algebraic/numerical algorithms, polynomial rootfinding, solution of a polynomial system of equations, approximate gcd, dense structured matrices, Toeplitz matrices, Cauchy matrices, polynomial interpolation, rational interpo...