Matrix computations (3rd ed.)
Efficient algorithms for computing the nearest polynomial with constrained roots
ISSAC '98 Proceedings of the 1998 international symposium on Symbolic and algebraic computation
On approximate GCDs of univariate polynomials
Journal of Symbolic Computation - Special issue on symbolic numeric algebra for polynomials
Global Optimization with Polynomials and the Problem of Moments
SIAM Journal on Optimization
Towards certified irreducibility testing of bivariate approximate polynomials
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
Structured Perturbations Part I: Normwise Distances
SIAM Journal on Matrix Analysis and Applications
On approximate irreducibility of polynomials in several variables
ISSAC '03 Proceedings of the 2003 international symposium on Symbolic and algebraic computation
Sharp estimates for triangular sets
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Approximate factorization of multivariate polynomials via differential equations
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
Global optimization of rational functions: a semidefinite programming approach
Mathematical Programming: Series A and B
SIAM Journal on Optimization
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
On the complexity of Putinar's Positivstellensatz
Journal of Complexity
Convergent SDP-Relaxations in Polynomial Optimization with Sparsity
SIAM Journal on Optimization
The voronoi diagram of three lines
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
On probabilistic analysis of randomization in hybrid symbolic-numeric algorithms
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Numerical optimization in hybrid symbolic-numeric computation
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Lower bounds for approximate factorizations via semidefinite programming: (extended abstract)
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Approximate GCDs of polynomials and SOS relaxation
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Proceedings of the 2007 international workshop on Symbolic-numeric computation
Certification of the QR factor R and of lattice basis reducedness
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Approximate factorization of multivariate polynomials using singular value decomposition
Journal of Symbolic Computation
Computing the global optimum of a multivariate polynomial over the reals
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Single-factor coefficient bounds
Journal of Symbolic Computation
Approximate GCDs of polynomials and sparse SOS relaxations
Theoretical Computer Science
Proceedings of the 2009 conference on Symbolic numeric computation
Approximate factorization of polynomials over Z
Proceedings of the 2009 conference on Symbolic numeric computation
Proceedings of the 4th International Workshop on Parallel and Symbolic Computation
Global optimization of polynomials using generalized critical values and sums of squares
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Quadratic-time certificates in linear algebra
Proceedings of the 36th international symposium on Symbolic and algebraic computation
The minimum-rank gram matrix completion via modified fixed point continuation method
Proceedings of the 36th international symposium on Symbolic and algebraic computation
On the generation of positivstellensatz witnesses in degenerate cases
ITP'11 Proceedings of the Second international conference on Interactive theorem proving
Journal of Symbolic Computation
Computing Rational Points in Convex Semialgebraic Sets and Sum of Squares Decompositions
SIAM Journal on Optimization
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
Variable projection methods for approximate GCD computations
ACM Communications in Computer Algebra
Theoretical Computer Science
Computing rational solutions of linear matrix inequalities
Proceedings of the 38th international symposium on International symposium on symbolic and algebraic computation
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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 symbolic-numeric 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.