A recursive algorithm for the infinity-norm fixed point problem

• Authors: Spencer Shellman;K. Sikorski
• Venue: Journal of Complexity
• Year: 2003

We present the PFix algorithm for the fixed point problem f(x) = x on a nonempty domain [a,b], where d ≥ 1, a,b ∈ Rd, and f is a Lipschitz continuous function with respect to the infinity norm, with constant q ≤ 1. The computed approximation x satisfies the residual criterion ||f(x) - x||∞ ≤ ε, where ε 0. In general, the algorithm requires no more than Σi=1dSi function component evaluations, where s ≡ ⌈ max(1, log2(||b - a||∞/ε))⌉ + 1. This upper bound has order O(⌈ log2d(1/ε)⌉ as ε → 0. For the domain [0,1]d with ε d+r-1 r-1) + 2(d+r r+1), where r ≡ ⌈ log2(1/ε)⌉. This bound approaches O(rd/d!) as r → ∞ (ε → 0) and O(dr+1/(r + 1)!) as d→ 0. We show that when q x satisfying the absolute criterion ||x - x*||∞ ≤ ε, 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 ε(1 - q) instead of ε. We show that when q 1 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.