A constructive approach to the Schröder equation

• Published in: Journal of computational and applied mathematics Vol. 46; no. 1; pp. 301 - 314
• Main Author: Iserles, A.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 14.06.1993
• Subjects: Acceleration of convergence; functional iteration; generalized Steffensen's method; generating functions; Schröder's equation; generalized Steffensen's method; generating functions; Acceleration of convergence; functional iteration; Schröder's equation
• ISSN: 0377-0427; 1879-1778
• DOI: 10.1016/0377-0427(93)90304-T

Given a function ƒ, analytic at the origin and such that ƒ (0) = 0, ƒ′ (0) ≠ 0, the Schröder equation (Blanchard, 1984) reads β g( z) = g(ƒ( z)), where β = ƒ′ (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.