A recursive algorithm for the infinity-norm fixed point problem

We present the PFix algorithm for the fixed point problem f ðxÞ 1⁄4 x on a nonempty domain 1⁄2a; b ; where dX1; a; bAR ; and f is a Lipschitz continuous function with respect to the infinity norm, with constant qp1: The computed approximation x̃ satisfies the residual criterion jj f ðx̃Þ x̃jjNpe; where e40: In general, the algorithm requires no more than Pd i1⁄41s i function component evaluations, where s Jmaxð1; log2ðjjb ajjN=eÞÞnþ 1: This upper bound has order OðJlogd2ð1=eÞnÞ as e-0: For the domain 1⁄20; 1 d with eo0:5 we prove a stronger result, i.e., an upper bound on the number of function component evaluations is ðdþr 1 r 1 Þ þ 2ð dþr rþ1Þ; where r Jlog2ð1=eÞn: This bound approaches Oðr=d!Þ as r-N ðe-0Þ and Oðdrþ1=ðr þ 1Þ!Þ as d-N: We show that when qo1 the algorithm can also compute an approximation x̃ satisfying the absolute criterion jjx̃ x jjNpe; where x is the unique fixed point of f : The complexity in this case resembles the complexity of the residual criterion problem, but with tolerance eð1 qÞ instead of e: We show that when q41 the absolute criterion problem has infinite worst-case complexity when information consists of function evaluations. Finally, we report several numerical tests in which the actual number of evaluations is usually much smaller than the upper complexity bound. r 2003 Elsevier Inc. All rights reserved.