A constructive proof of the convergence of Kalantari's bound on polynomial zeros

TL;DR

This paper provides a constructive proof of Jin's theorem on the convergence of Kalantari's bounds for polynomial zeros, offering a formula to determine the necessary bound index and analyzing the convergence rate.

Contribution

It derives a formula to explicitly find the bound index m for convergence, strengthening Jin's theorem with a constructive and uniform convergence proof.

Findings

Convergence rate depends only on polynomial degree and parameters.

Experimental results suggest optimal m scales as 1/epsilon^d with d << 2.

The method potentially runs in O(n/epsilon^d) time for high-degree polynomials.

Abstract

In his 2006 paper, Jin proves that Kalantari's bounds on polynomial zeros, indexed by $m \leq 2$ and called $L_m$ and $U_m$ respectively, become sharp as $m\rightarrow\infty$. That is, given a degree $n$ polynomial $p(z)$ not vanishing at the origin and an error tolerance $\epsilon > 0$, Jin proves that there exists an $m$ such that $\frac{L_m}{\rho_{min}} > 1-\epsilon$, where $\rho_{min} := \min_{\rho:p(\rho) = 0} \left|\rho\right|$. In this paper we derive a formula that yields such an $m$, thereby constructively proving Jin's theorem. In fact, we prove the stronger theorem that this convergence is uniform in a sense, its rate depending only on $n$ and a few other parameters. We also give experimental results that suggest an optimal m of (asymptotically) $O\left(\frac{1}{\epsilon^d}\right)$ for some $d \ll 2$. A proof of these results would show that Jin's method runs in $O\left(\frac{n}{\epsilon^d}\right)$ time, making it efficient for isolating polynomial zeros of high degree.