Approximate square-free decomposition and root-finding of lll-conditioned algebraic equations
Journal of Information Processing
Fast Plane Rotations with Dynamic Scaling
SIAM Journal on Matrix Analysis and Applications
The singular value decomposition for polynomial systems
ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
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
Journal of Symbolic Computation - Special issue on symbolic numeric algebra for polynomials
Journal of Symbolic Computation - Special issue on symbolic numeric algebra for polynomials
Modern computer algebra
Fast Structured Total Least Squares Algorithm for Solving the Basic Deconvolution Problem
SIAM Journal on Matrix Analysis and Applications
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
Approximate factorization of multivariate polynomials via differential equations
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
The approximate GCD of inexact polynomials
ISSAC '04 Proceedings of the 2004 international symposium on Symbolic and algebraic computation
QR factoring to compute the GCD of univariate approximate polynomials
IEEE Transactions on Signal Processing
Hybrid symbolic-numeric computation
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
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
Computing Approximate GCDs in Ill-conditioned Cases
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
Structured matrix-based methods for polynomial ∈-gcd: analysis and comparisons
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
On exact and approximate interpolation of sparse rational functions
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
Approximate factorization of multivariate polynomials using singular value decomposition
Journal of Symbolic Computation
Proceedings of the twenty-first international symposium on Symbolic and algebraic computation
Approximate GCDs of polynomials and sparse SOS relaxations
Theoretical Computer Science
On real factors of real interval polynomials
Journal of Symbolic Computation
An iterative method for calculating approximate GCD of univariate polynomials
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
The approximate irreducible factorization of a univariate polynomial: revisited
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
Computing multivariate approximate GCD based on Barnett's theorem
Proceedings of the 2009 conference on Symbolic numeric computation
Blind image deconvolution via fast approximate GCD
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation
GPGCD, an iterative method for calculating approximate GCD, for multiple univariate polynomials
CASC'10 Proceedings of the 12th international conference on Computer algebra in scientific computing
GPGCD, an iterative method for calculating approximate GCD, for multiple univariate polynomials
ACM Communications in Computer Algebra
Approximate polynomial GCD over integers
Journal of Symbolic Computation
Approximate GCD of several univariate polynomials with small degree perturbations
Journal of Symbolic Computation
Journal of Symbolic Computation
Using weighted norms to find nearest polynomials satisfying linear constraints
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
Fast estimates of Hankel matrix condition numbers and numeric sparse interpolation
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
GCD of multivariate approximate polynomials using beautification with the subtractive algorithm
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
Ruppert matrix as subresultant mapping
CASC'07 Proceedings of the 10th international conference on Computer Algebra in Scientific Computing
Variable projection methods for approximate GCD computations
ACM Communications in Computer Algebra
GPGCD: An iterative method for calculating approximate GCD of univariate polynomials
Theoretical Computer Science
Software for weighted structured low-rank approximation
Journal of Computational and Applied Mathematics
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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 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.