On condition numbers and the distance to the nearest III-posted problem
Numerische Mathematik
Relationships between order and efficiency of a class of methods for multiple zeros of polynomials
Proceedings of the international meeting on Linear/nonlinear iterative methods and verification of solution
The singular value decomposition for polynomial systems
ISSAC '95 Proceedings of the 1995 international symposium on Symbolic and algebraic computation
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
Newton's method for overdetermined systems of equations
Mathematics of Computation
Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Classics in Applied Mathematics, 16)
A Matlab package computing polynomial roots and multiplicities
ACM SIGSAM Bulletin
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
Improved algorithms for computing determinants and resultants
Journal of Complexity - Special issue: Foundations of computational mathematics 2002 workshops
Approximate radical of ideals with clusters of roots
Proceedings of the 2006 international symposium on Symbolic and algebraic computation
Approximate factorization of multivariate polynomials using singular value decomposition
Journal of Symbolic Computation
The approximate irreducible factorization of a univariate polynomial: revisited
Proceedings of the 2009 international symposium on Symbolic and algebraic computation
A regularization approach for estimating the type of a plane curve singularity
Theoretical Computer Science
ACM Transactions on Mathematical Software (TOMS)
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We present a combination of two novel algorithms that accurately calculate multiple roots of general polynomials. For a given multiplicity structure and initial root estimates, Algorithm I transforms the singular root-finding into a nonsingular least squares problem on a pejorative manifold, and calculates multiple roots simultaneously. To fulfill the input requirement of Algorithm I, we design a numerical GCD-finder, including a partial singular value decomposition and an iterative refinement, as the main engine for Algorithm II that calculates the multiplicity structure and the initial root approximation. The combined method calculates multiple roots with high forward accuracy without using multiprecision arithmetic, even if the coefficients are inexact. This is perhaps the first blackbox-type root-finder with such capabilities. To measure the true sensitivity of the multiple roots, a pejorative condition number is proposed and error bounds are given. Extensive computational experiments are presented. The error analysis and numerical results confirm that a polynomial being ill-conditioned in conventional sense can be well conditioned pejoratively. In those cases, the multiple roots can be computed with remarkable accuracy.