A two-dimensional bisection envelope algorithm for fixed points

Authors:
Spencer Shellman;K. Sikorski
Affiliations:
School of Computing, University of Utah, Salt Lake City, Utah;School of Computing, University of Utah, Salt Lake City, Utah
Venue:
Journal of Complexity
Year:
2002

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Abstract

In 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. Complexity 15, 200-213), our bound is the first for the infinity norm case.