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

Evaluating or finding the roots of a polynomial $f\left(z\right)=f_0+\cdots +f_d{z}^d$ 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.