Simultaneous factorization of a polynomial by rational approximation

In this note we present a numerical method to approximate some relatively prime factors of a polynomial simultaneously. Our approach gives methods of arbitrary order; Grauʹs method (Carstensen, 1992; Grau, 1971) is obtained as the second order method which is Durand-Kernerʹs method when we have linear factors. For linear factors our approach yields the simultaneous methods introduced in Sakurai et al. (1991). We prove local convergence and estimate the R-order of the total step version as well as the single step version of the methods. We derive an algorithm and present numerical examples which confirm the convergence behavior theoretically predicted.