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

TL;DR

This paper introduces a new algorithm that efficiently evaluates and finds roots of polynomials with floating-point coefficients, improving computational bounds and handling a wide range of coefficient magnitudes.

Contribution

It presents a novel piecewise approximation method using the Newton polygon, achieving quasi-linear complexity for root finding with controlled condition numbers.

Findings

Improved upper bounds on evaluation and root-finding operations.

Algorithm works efficiently for coefficients from 2^{-d} to 2^d.

Handles polynomials with coefficients spanning wide magnitude ranges.

Abstract

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.