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Communication Dans Un Congrès Année : 2023

Fast evaluation and root finding for polynomials with floating-point coefficients

Résumé

Evaluating or finding the roots of a polynomial $f(z) = f_0 + \cdots + f_d z^d$ with floating-point number coefficients is a ubiquitous problem. By using a piecewise approximation of $f$ obtained with a careful use of the Newton polygon of $f$, we improve state-of-the-art upper bounds on the number of operations to evaluate and find the roots of a polynomial. In particular, if the coefficients of $f$ are given with $m$ significant bits, we provide for the first time an algorithm that finds all the roots of $f$ with a relative condition number lower than $2^m$, using a number of bit operations quasi-linear in the bit-size of the floating-point representation of $f$. Notably, our new approach handles efficiently polynomials with coefficients ranging from $2^{-d}$ to $2^d$, both in theory and in practice.
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Dates et versions

hal-03980098 , version 1 (10-02-2023)

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Rémi Imbach, Guillaume Moroz. Fast evaluation and root finding for polynomials with floating-point coefficients. ISSAC 2023, Jul 2023, Tromsø, Norway. ⟨hal-03980098⟩
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