Symmetry and nonconvergence in iterative polynomial zero-finding

Abstract

It is well known that, in Newton's method, a real guess cannot converge to a complex zero of a real polynomial. Actually for the class of RR-weighted methods including those of Newton and Halley, a guess lying on a reflective symmetry axis of the zeros produces only iterates on the same axis; multiple symmetry axes intersect at the centroid C; a nearby guess gives approximations to either C or ∞. Irrational Newton-like method fare much better, but local symmetry can still lead to rebounding, calling for special detection and recovery algorithms.