Fixed Points-Contractive Functions

Fixed point computation has been an intensive research area since 1967 when Scarf introduced simplicial algorithm to approximate fixed points. Several algorithms have been invented since then, including restart and homotopy methods. Most of these were designed to approximate fixed points of general maps and used the residual error criterion. In this chapter we consider the absolute and/or relative error criteria for contractive univariate and multivariate functions. The departure of our analysis is the classical Banach fixed point theorem. Namely, we consider a function f : D →D, where D is a closed subset of a Banach space B. We assume that f is contractive with a factor q < 1, i.e., . . . ||f(x) – f(y)|| ≤ q ||x-y||, for all x,y ∈ D. Then, there exists a unique ∝ = ∝ (f) ∈ D such that ∝ is a fixed point of f, ∝ = f (∝)