A CONSTRUCTIVE APPROACH TO THE SCHRODER EQUATION

Given a function f, analytic at the origin and such that f(0) = 0, f'(0) not-equal 0, the Schroder equation (Blanchard, 1984) reads betag(z) = g(f (z)), where beta = f'(0). In the present paper we introduce a formal computational algorithm to approximate the analytic function g. Our main tool consists of evaluating a generating function of certain, recursively defined, polynomials. Another application of our analysis is to convergence acceleration of functional iteration. As demonstrated by the author (1991), the analysis of a generalized Steffensen's method (1933) requires the consideration of certain determinants. In particular, it is required to show that they do not vanish. As an offshoot of our technique, we evaluate the exact value of these determinants.