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Communication Dans Un Congrès Année : 2023

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.
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hal-04138791 , version 1 (23-06-2023)

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Florent Bréhard, Adrien Poteaux, Léo Soudant. Validated Root Enclosures for Interval Polynomials with Multiplicities. ISSAC'23 : International Symposium on Symbolic and Algebraic Computation 2023, Jul 2023, Tromso, Norway. ⟨10.1145/3597066.3597122⟩. ⟨hal-04138791⟩
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