Validated Root Enclosures for Interval Polynomials with Multiplicities
Résumé
Twenty years ago, Zeng [28, 30] proposed floating-point algorithms to compute multiple roots of univariate polynomials with real or complex coefficients beyond the so-called "attainable accuracy barrier". Based on these foundations, we propose a validated numeric point of view on this problem. Our first contribution is an improvement of Zeng's multiplicity detection algorithm using a simple trick that allows us to recover much higher multiplicities. As our main contribution, we propose two floating-point validated algorithms to compute rigorous enclosures for multiple roots. They consist in carefully combining the ideas underlying Zeng's numerical algorithms with Newton-like fixed-point validation techniques. We also provide a prototype Julia implementation of these algorithms.
Domaines
Calcul formel [cs.SC]
Origine : Fichiers produits par l'(les) auteur(s)