© 1991 by Institute of Mathematics and its Applications
Complex Dynamics of Convergence Acceleration
Department of Applied Mathematics and Theoretical Physics, University of Cambridge Cambridge, UK
Generalized Steffensen methods are nonderivative algorithms for the computation of fixed points of a function f. They replace the functional iteration Zm+1=f(Zm) with Zm+1=Fn(Zm, where Fn is explicitly provided for every n 
1 as a quotient of two Hankel determinants. In this paper we derive rules pertaining to the local behaviour of these methods. Specifically, and subject to analyticity, given that 
is a bounded fixed point of f, then it is also a fixed point of Fn. Moreover, unless f'() vanishes or is a root of unity, becomes a superattractive fixed point of Fn of degree n; if f'() is a root of unity of minimal degree q2, then is (as a fixed point of Fn) superattractive of degree min {q-1, n}; if f'()=1, then is attractive for Fn; and, finally, if is superattractive of degree s (as a fixed point of f), then it becomes superattractive of degree (s + 1)n–1(ns + s + 1)–1. Attractivity rules change at infinity (provided that f(
)=). Broadly speaking, infinity becomes less attractive for Fn, Since one is interested in convergence to finite fixed points, this further enhances the appeal of generalized Steffensen methods.