Multiplicative Updates for Polynomial Root Finding

TL;DR

This paper introduces a multiplicative update method for finding roots of polynomials with roots having nonnegative real parts, demonstrating linear convergence to extremal roots under certain conditions, inspired by algorithms in nonnegative matrix factorization.

Contribution

The paper proves convergence properties of a multiplicative root-finding algorithm for polynomials with nonnegative coefficient polynomials, extending multiplicative update methods to polynomial root finding.

Findings

The update converges monotonically and linearly to the extremal roots.

Convergence depends on the initial point and the relation between p(x) and q(x).

The method is motivated by multiplicative algorithms in nonnegative matrix factorization.

Abstract

Let $f(x)=p(x)-q(x)$ be a polynomial with real coefficients whose roots have nonnegative real part, where $p$ and $q$ are polynomials with nonnegative coefficients. In this paper, we prove the following: Given an initial point $x_0 > 0$, the multiplicative update $x_{t+1} = x_t \, p(x_t)/q(x_t)$ ($t=0,1,\dots$) monotonically and linearly converges to the largest (resp. smallest) real roots of $f$ smaller (resp. larger) than $x_0$ if $p(x_0) < q(x_0)$ (resp. $q(x_0) < p(x_0)$). The motivation to study this algorithm comes from the multiplicative updates proposed in the literature to solve optimization problems with nonnegativity constraints; in particular many variants of nonnegative matrix factorization.