Abstract. The task of determining the approximate greatest common divisor (GCD) of univariate polynomials with inexact coefficients can be formulated as computing for a given Sylvester matrix a new Sylvester matrix of lower rank whose entries are near the corresponding entries of that input matrix. We solve the approximate GCD problem by a new method based on structured total least norm (STLN) algorithms, in our case for matrices with Sylvester structure. We present iterative algorithms that compute an approximate GCD and that can certify an approximate ǫ-GCD when a tolerance ǫ is given on input. Each single iteration is carried out with a number of floating point operations that is of cubic order in the input degrees. We also demonstrate the practical performance of our algorithms on a diverse set of univariate pairs of polynomials. Mathematics Subject Classification (2000). Primary 68W30; Secondary 65K10.Keywords. Sylvester matrix, approximate greatest common divisor, structured total least norm, hybrid symbolic/numeric algorithm.

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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 STLN-based 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 non-real complex coefficients.

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We generalize the technique by Peyrl and Parillo [Proc. SNC 2007] to computing lower bound certificates for several well-known factorization problems in hybrid symbolicnumeric computation. The idea is to transform a numerical sum-of-squares (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 semi-definite 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 rank-preserving Newton iteration. For smaller problems the starting SOSes for Newton can be guessed without SDP ("SDP-free SOS"), but for larger inputs we additionally appeal to sparsity techniques in our SDP formulation.