Multiplicative updates for polynomial root finding

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 x0>0, the multiplicative update xt+1=xtp(xt)/q(xt) (t=0,1,…) monotonically and linearly converges to the largest (resp. smallest) real roots of f smaller (resp. larger) than x0 if p(x0)<q(x0) (resp. q(x0)<p(x0)). 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.