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

• Published in: Journal of symbolic computation Vol. 127; p. 102372
• Main Authors: Imbach, Rémi, Moroz, Guillaume
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.03.2025
• Subjects: Floating-point coefficients; Polynomial evaluation; Root finding; Floating-point coefficients; Root finding; Polynomial evaluation
• ISSN: 0747-7171
• DOI: 10.1016/j.jsc.2024.102372

Evaluating or finding the roots of a polynomial f(z)=f0+⋯+fdzd 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 2m, 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 2d, both in theory and in practice.