On the geometry of Graeffe iteration

A new version of Graeffe's algorithm for finding all the roots of univariat e complex polynomials is proposed. It is obtained from the classical algori thm by a process analogous to renormalization of dynamical systems. This iteration is called the renormalized Graeffe iteration. It is globally convergent, with probability 1. All quantities involved in the computation are bounded once the initial polynomial is given (with probability 1). Thi s implies remark-able stability properties for the new algorithm, thus over coming known limitations of the classical Graeffe algorithm. If we start with a degree-d polynomial. each renormalized Graeffe iteration costs c (d(2)) arithmetic operations, with memory c(d), A probabilistic global complexity bound is given. The case of univariate re al polynomials is briefly discussed. A numerical implementation of the algorithm presented herein allows us to s olve random polynomials of degree up to 1000. (C) 2001 Academic Press.