CONVERGENCE ACCELERATION AS A DYNAMICAL SYSTEM

We survey recent results on local and global aspects of generalized St effensen iteration. The main idea behind this algorithm is to replace the dynamical system z --> f(z) with z --> F-n(z), where F-n is an app ropriately constructed quotient of two Hankel determinants. We show th at F-n retains all finite fixed points of f and determine the local sp eedup in convergence. Moreover, we investigate how the basin of attrac tion varies with n, proving that for polynomial f most poles and zeros of F-n accumulate on and inside the Julia set of f as n --> infinity. This is in close agreement with computational results.