Quality of positive root bounds

Abstract

In this paper, we study the quality of positive root bounds. A positive root bound of a polynomial is an upper bound on the largest positive root. Higher quality means that the relative overestimation (the ratio of the bound and the largest positive root) is smaller. We report three findings. (1) Most known positive root bounds can be arbitrarily bad ; that is, the relative over-estimation can approach infinity, even when the degree and the coefficient size are fixed. (2) When the number of sign variations is the same as the number of positive roots, the relative over-estimation of a positive root bound due to Hong (B H) is at most linear in the degree, no matter what the coefficient size is. (3) When the number of sign variations is one, the relative over-estimation of B H is at most constant, in particular 4, no matter what the degree and the coefficient size are.

On an application of symbolic computation and computer graphics to root-finders: The case of multiple roots of unknown multiplicity

“…Having in mind these facts we have that (17) we find that µ n+1 → m when x → α. Since the inductive assumption is valid for n = 2 (Lagouanelle's formula (13)) we conclude that µ n → m for arbitrary n ≥ 2.…”

“…The knowledge of multiplicity m and a good initial approximation x 0 are two very important tasks and should be a composite part of any root-finder. The latter task was considered in some recent papers and books, see, for instance, [14][15][16][17][18].…”