  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.
