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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 ..."
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Cited by 14 (8 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
ACMAC’s PrePrint Repository Sparse implicitization by interpolation: Characterizing nonexactness and an application to computing discriminants
, 2013
"... Sparse implicitization by interpolation: computing discriminants. Characterizing nonexactness and an application to ..."
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Sparse implicitization by interpolation: computing discriminants. Characterizing nonexactness and an application to
A Quadratically Convergent Algorithm for Structured LowRank Approximation
"... Structured LowRank Approximation is a problem arising in a wide range of applications in Numerical Analysis and Engineering Sciences. Given an input matrix M, the goal is to compute a matrix M ′ of given rank r in a linear or affine subspace E of matrices (usually encoding a specific structure) suc ..."
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Structured LowRank Approximation is a problem arising in a wide range of applications in Numerical Analysis and Engineering Sciences. Given an input matrix M, the goal is to compute a matrix M ′ of given rank r in a linear or affine subspace E of matrices (usually encoding a specific structure) such that the Frobenius distance ‖M − M ′ ‖ is small. We propose a Newtonlike iteration for solving this problem, whose main feature is that it converges locally quadratically to such a matrix under mild transversality assumptions between the manifold of matrices of rank r and the linear/affine subspace E. We also show that the distance between the limit of the iteration and the optimal solution of the problem is quadratic in the distance between the input matrix and the manifold of rank r matrices in E. To illustrate the applicability of this algorithm, we propose a Maple implementation and give experimental results for several applicative problems that can be modeled by Structured LowRank Approximation: univariate approximate GCDs (Sylvester matrices), lowrank Matrix completion (coordinate spaces) and denoising procedures (Hankel matrices). Experimental results give evidence that this allpurpose algorithm is competitive with stateoftheart numerical methods dedicated to these problems.