A note on optimal algorithms for fixed points

technical reportWe present a constructive lemma that we believe will make possible the design of nearly optimal 0(dlog | ) cost algorithms for computing eresidual approximations to the fixed points of d-dimensional nonexpansive mappings with respect to the infinity norm. This lemma is a generalization of a two-dimensional result that we proved in [lj

A two-dimensional bisection envelope algorithm for fixed points

AbstractIn this paper we present a new algorithm for the two-dimensional fixed point problem f(x)=x on the domain [0, 1]×[0, 1], where f is a Lipschitz continuous function with respect to the infinity norm, with constant 1. The computed approximation x satisfies ‖f(x)−x‖∞⩽ε for a specified tolerance ε<0.5. The upper bound on the number of required function evaluations is given by 2⌈log2(1/ε)⌉+1. Similar bounds were derived for the case of the 2-norm by Z. Huang et al. (1999, J. Complexity15, 200–213), our bound is the first for the infinity norm case