Approximate square-free decomposition and root-finding of lll-conditioned algebraic equations
Journal of Information Processing
Three new algorithms for multivariate polynomial GCD
Journal of Symbolic Computation
Journal of Information Processing
The singular value decomposition for polynomial systems
ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
Polynomial remainder sequence and approximate GCD
ACM SIGSAM Bulletin
ISSAC '98 Proceedings of the 1998 international symposium on Symbolic and algebraic computation
Journal of Symbolic Computation - Special issue on symbolic numeric algebra for polynomials
Univariate polynomials: nearly optimal algorithms for factorization and rootfinding
Proceedings of the 2001 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
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
IEEE Transactions on Signal Processing
QR factoring to compute the GCD of univariate approximate polynomials
IEEE Transactions on Signal Processing
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
Computing multivariate approximate GCD based on Barnett's theorem
Proceedings of the 2009 conference on Symbolic numeric 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
Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation
GPGCD: An iterative method for calculating approximate GCD of univariate polynomials
Theoretical Computer Science
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The ill-conditioned cases we consider are 1) the small leading-coefficient case, 2) small leading-coefficient GCD case, 3) big leading coefficient case, and 4) approximately singular leading-coefficient case. We propose three new algorithms for computing the approximate GCDs in these cases. The first one is to stabilize the univariate PRS by avoiding the division by polynomials of small leading coefficients. The second and the third ones are modifications of the QRGCD and the (approximate) PC-PRS methods, respectively, for multivariate polynomials. Furthermore, we propose a device to remove fully-erroneous terms arising from the exact cancellation. Some experiments show that our algorithms remove the instability considerably and work quite efficiently.