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23
Symbolicnumeric sparse interpolation of multivariate polynomials
 In Proc. Ninth Rhine Workshop on Computer Algebra (RWCA’04), University of Nijmegen, the Netherlands (2004
, 2006
"... We consider the problem of sparse interpolation of an approximate multivariate blackbox polynomial in floatingpoint arithmetic. That is, both the inputs and outputs of the blackbox polynomial have some error, and all numbers are represented in standard, fixedprecision, floating point arithmetic. ..."
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Cited by 34 (6 self)
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We consider the problem of sparse interpolation of an approximate multivariate blackbox polynomial in floatingpoint arithmetic. That is, both the inputs and outputs of the blackbox polynomial have some error, and all numbers are represented in standard, fixedprecision, floating point arithmetic. By interpolating the black box evaluated at random primitive roots of unity, we give efficient and numerically robust solutions. We note the similarity between the exact BenOr/Tiwari sparse interpolation algorithm and the classical Prony’s method for interpolating a sum of exponential functions, and exploit the generalized eigenvalue reformulation of Prony’s method. We analyze the numerical stability of our algorithms and the sensitivity of the solutions, as well as the expected conditioning achieved through randomization. Finally, we demonstrate the effectiveness of our techniques in practice through numerical experiments and applications. 1.
Exact Certification of Global Optimality of Approximate Factorizations Via Rationalizing SumsOfSquares with Floating Point Scalars
, 2008
"... We generalize the technique by Peyrl and Parillo [Proc. SNC 2007] to computing lower bound certificates for several wellknown factorization problems in hybrid symbolicnumeric computation. The idea is to transform a numerical sumofsquares (SOS) representation of a positive polynomial into an exact ..."
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Cited by 15 (9 self)
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We generalize the technique by Peyrl and Parillo [Proc. SNC 2007] to computing lower bound certificates for several wellknown factorization problems in hybrid symbolicnumeric computation. The idea is to transform a numerical sumofsquares (SOS) representation of a positive polynomial into an exact rational identity. Our algorithms successfully certify accurate rational lower bounds near the irrational global optima for benchmark approximate polynomial greatest common divisors and multivariate polynomial irreducibility radii from the literature, and factor coefficient bounds in the setting of a model problem by Rump (up to n = 14, factor degree = 13). The numeric SOSes produced by the current fixed precision semidefinite programming (SDP) packages (SeDuMi, SOSTOOLS, YALMIP) are usually too coarse to allow successful projection to exact SOSes via Maple 11’s exact linear algebra. Therefore, before projection we refine the SOSes by rankpreserving Newton iteration. For smaller problems the starting SOSes for Newton can be guessed without SDP (“SDPfree SOS”), but for larger inputs we additionally appeal to sparsity techniques in our SDP formulation.
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 14 (9 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.
Lifting and recombination techniques for absolute factorization
 J. Complexity
, 2007
"... Abstract. In the vein of recent algorithmic advances in polynomial factorization based on lifting and recombination techniques, we present new faster algorithms for computing the absolute factorization of a bivariate polynomial. The running time of our probabilistic algorithm is less than quadratic ..."
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Cited by 14 (7 self)
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Abstract. In the vein of recent algorithmic advances in polynomial factorization based on lifting and recombination techniques, we present new faster algorithms for computing the absolute factorization of a bivariate polynomial. The running time of our probabilistic algorithm is less than quadratic in the dense size of the polynomial to be factored.
PHCmaple: A Maple interface to the numerical homotopy algorithms in PHCpack
 In QuocNam Tran, editor, Proceedings of the Tenth International Conference on Applications of Computer Algebra (ACA’2004
, 2004
"... Our Maple package PHCmaple provides a convenient interface to the functions of PHCpack, a collection of numeric algorithms for solving polynomial systems using polynomial homotopy continuation, which was recently extended with facilities to deal with positive dimensional solution sets. The interface ..."
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Cited by 12 (7 self)
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Our Maple package PHCmaple provides a convenient interface to the functions of PHCpack, a collection of numeric algorithms for solving polynomial systems using polynomial homotopy continuation, which was recently extended with facilities to deal with positive dimensional solution sets. The interface illustrates the benefits of linking computer algebra with numerical software. PHCmaple serves as a first step in a larger project to integrate a numerical solver in a computer algebra system.
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 9 (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.
Decomposing solution sets of polynomial systems: a new parallel monodromy breakup algorithm. Accepted for publication
 in The International Journal of Computational Science and Engineering
"... Abstract: Our problem is to decompose a positive dimensional solution set of a polynomial system into irreducible components. This solution set is represented by a witness set, obtained by intersecting the set with random linear slices of complementary dimension. Points on the same irreducible compo ..."
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Cited by 6 (6 self)
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Abstract: Our problem is to decompose a positive dimensional solution set of a polynomial system into irreducible components. This solution set is represented by a witness set, obtained by intersecting the set with random linear slices of complementary dimension. Points on the same irreducible components are connected by path tracking techniques applying the idea of monodromy. The computation of a linear trace for each component certifies the decomposition. This decomposition method exhibits a good practical performance on solution sets of relatively high degrees defined by systems of low degree polynomials.
A SymbolicNumeric Algorithm for Computing the Alexander polynomial of . . .
"... We report on a symbolicnumeric algorithm for computing the Alexander polynomial of each singularity of a plane complex algebraic curve defined by a polynomial with coefficients of limited accuracy, i.e. the coefficients are both exact and inexact data. We base the algorithm on combinatorial methods ..."
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Cited by 3 (3 self)
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We report on a symbolicnumeric algorithm for computing the Alexander polynomial of each singularity of a plane complex algebraic curve defined by a polynomial with coefficients of limited accuracy, i.e. the coefficients are both exact and inexact data. We base the algorithm on combinatorial methods from knot theory which we combine with computational geometry algorithms in order to compute efficient and accurate results. Nonetheless the problem we are dealing with is illposed, in the sense that tiny perturbations in the coefficients of the defining polynomial cause huge errors in the computed results.