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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 ..."
Abstract

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.
Computing the radius of positive semidefiniteness of a multivariate real polynomial via a dual of Seidenberg’s method
, 2010
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"... We investigate the structured normwise and componentwise condition numbers for solving linear systems with Sylvester structure. Numerical examples show that the Sylvester structured condition numbers can be much smaller than the unstructured condition numbers. Here and hereafter, we denote ‖ · ‖ t ..."
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We investigate the structured normwise and componentwise condition numbers for solving linear systems with Sylvester structure. Numerical examples show that the Sylvester structured condition numbers can be much smaller than the unstructured condition numbers. Here and hereafter, we denote ‖ · ‖ the spectral norm ‖ · ‖2 and ‖ · ‖ ∞ the infinity norm of its arguments. If A is a matrix, we write A  = (Aij), where Aij is the (i, j)th entry of A. Let
Algorithm, Theory
"... We investigate our early termination criterion for sparse polynomial interpolation when substantial noise is present in the values of the polynomial. Our criterion in the exact case uses Monte Carlo randomization which introduces a second source of error. We harness the GohbergSemencul formula for ..."
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We investigate our early termination criterion for sparse polynomial interpolation when substantial noise is present in the values of the polynomial. Our criterion in the exact case uses Monte Carlo randomization which introduces a second source of error. We harness the GohbergSemencul formula for the inverse of a Hankel matrix to compute estimates for the structured condition numbers of all arising Hankel matrices in quadratic arithmetic time overall, and explain how false illconditionedness can arise from our randomizations. Finally, we demonstrate by experiments that our condition number estimates lead to a viable termination criterion for polynomials with about 20 nonzero terms and of degree about 100, even in the presence of noise of relative magnitude 10 −5. Categories and Subject Descriptors