Approximate square-free decomposition and root-finding of lll-conditioned algebraic equations
Journal of Information Processing
The singular value decomposition for polynomial systems
ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
Applied numerical linear algebra
Applied numerical linear algebra
Optimization strategies for the approximate GCD problem
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
Computation of approximate polynomial GCDs and an extension
Information and Computation
Accuracy and Stability of Numerical Algorithms
Accuracy and Stability of Numerical Algorithms
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Computing Approximate GCDs in Ill-conditioned Cases
Proceedings of the 2007 international workshop on Symbolic-numeric computation
An iterative method for calculating approximate GCD of univariate polynomials
Proceedings of the 2009 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
A subdivision method for computing nearest gcd with certification
Theoretical Computer Science
IEEE Transactions on Signal Processing
QR factoring to compute the GCD of univariate approximate polynomials
IEEE Transactions on Signal Processing
Calculating approximate GCD of multiple univariate polynomials using approximate syzygies
ACM Communications in Computer Algebra
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We present an iterative algorithm for calculating approximate greatest common divisor (GCD) of univariate polynomials with the real or the complex coefficients. For a given pair of polynomials and a degree, our algorithm finds a pair of polynomials which has a GCD of the given degree and whose coefficients are perturbed from those in the original inputs, making the perturbations as small as possible, along with the GCD. The problem of approximate GCD is transferred to a constrained minimization problem, then solved with the so-called modified Newton method, which is a generalization of the gradient-projection method, by searching the solution iteratively. We demonstrate that, in some test cases, our algorithm calculates approximate GCD with perturbations as small as those calculated by a method based on the structured total least norm (STLN) method and the UVGCD method, while our method runs significantly faster than theirs by approximately up to 30 or 10 times, respectively, compared with their implementation. We also show that our algorithm properly handles some ill-conditioned polynomials which have a GCD with small or large leading coefficient.