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37
Univariate polynomials: nearly optimal algorithms for factorization and rootfinding
 In Proceedings of the International Symposium on Symbolic and Algorithmic Computation
, 2001
"... To approximate all roots (zeros) of a univariate polynomial, we develop two effective algorithms and combine them in a single recursive process. One algorithm computes a basic well isolated zerofree annulus on the complex plane, whereas another algorithm numerically splits the input polynomial of t ..."
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Cited by 58 (14 self)
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To approximate all roots (zeros) of a univariate polynomial, we develop two effective algorithms and combine them in a single recursive process. One algorithm computes a basic well isolated zerofree annulus on the complex plane, whereas another algorithm numerically splits the input polynomial of the nth degree into two factors balanced in the degrees and with the zero sets separated by the basic annulus. Recursive combination of the two algorithms leads to computation of the complete numerical factorization of a polynomial into the product of linear factors and further to the approximation of the roots. The new rootfinder incorporates the earlier techniques of Schönhage, Neff/Reif, and Kirrinnis and our old and new techniques and yields nearly optimal (up to polylogarithmic factors) arithmetic and Boolean cost estimates for the computational complexity of both complete factorization and rootfinding. The improvement over our previous record Boolean complexity estimates is by roughly the factor of n for complete factorization and also for the approximation of wellconditioned (well isolated) roots, whereas the same algorithm is also optimal (under both arithmetic and Boolean models of computing) for the worst case input polynomial, whose roots can be illconditioned, forming
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 28 (3 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.
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 27 (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.
Displacement structure in computing approximate GCD of univariate polynomials
 In Proc. Sixth Asian Symposium on Computer Mathematics (ASCM 2003
, 2003
"... We propose a fast algorithm for computing approximate GCD of univariate polynomials with coefficients that are given only to a finite accuracy. The algorithm is based on a stabilized version of the generalized Schur algorithm for Sylvester matrix and its embedding. All computations can be done in O( ..."
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Cited by 17 (7 self)
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We propose a fast algorithm for computing approximate GCD of univariate polynomials with coefficients that are given only to a finite accuracy. The algorithm is based on a stabilized version of the generalized Schur algorithm for Sylvester matrix and its embedding. All computations can be done in O(n 2) operations, where n is the sum of the degrees of polynomials. The stability of the algorithm is also discussed. 1.
Quantified Constraints under Perturbation
 ARTICLE SUBMITTED TO JOURNAL OF SYMBOLIC COMPUTATION
"... Quantified constraints (i.e., firstorder formulae over the real numbers) are often exposed to perturbations: Constants that come from measurements usually are only known up to certain precision, and numerical methods only compute with approximations of real numbers. In this paper we study the be ..."
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Cited by 17 (12 self)
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Quantified constraints (i.e., firstorder formulae over the real numbers) are often exposed to perturbations: Constants that come from measurements usually are only known up to certain precision, and numerical methods only compute with approximations of real numbers. In this paper we study the behavior of quantified constraints under perturbation by showing that one can formulate the problem of solving quantified constraints as a nested parametric optimization problem followed by one sign computation. Using the fact that minima and maxima are stable under perturbation, but the sign of a real number is stable only for nonzero inputs, we derive practically useful conditions for the stability of quantified constraints under perturbation.
Ten methods to bound multiple roots of polynomials
 J. Comput. Appl. Math. (JCAM
"... Abstract. Given a univariate polynomial P with a kfold multiple root or a kfold root cluster near some z̃, we discuss various different methods to compute a disc near z ̃ which either contains exactly or contains at least k roots of P. Many of the presented methods are known, some are new. We are ..."
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Cited by 13 (3 self)
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Abstract. Given a univariate polynomial P with a kfold multiple root or a kfold root cluster near some z̃, we discuss various different methods to compute a disc near z ̃ which either contains exactly or contains at least k roots of P. Many of the presented methods are known, some are new. We are especially interested in rigorous methods, that is taking into account all possible effects of rounding errors. In other words every computed bound for a root cluster shall be mathematically correct. We display extensive test sets comparing the methods under different circumstances. Based on the results we present a hybrid method combining five of the previous methods which, for given z̃, i) detects the number k of roots near z ̃ and ii) computes an including disc with in most cases a radius of the order of the numerical sensitivity of the root cluster. Therefore, the resulting discs are numerically nearly optimal. 1. Introduction and notation. Throughout the paper denote by P = n∑ ν=0 pνz ν ∈ C[z] a (real or
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 (5 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