Description: |
Algorithm 848: a recursive fixed-point algorithm for the infinity-norm case.
We present the PFix algorithm for approximating a fixed point of a function \(f\) that has arbitrary dimensionality, is defined on a rectangular domain, and is Lipschitz continuous with respect to the infinity norm with constant 1. PFix has applications in economics, game theory, and the solution of partial differential equations. PFix computes an approximation that satisfies the residual error criterion, and can also compute an approximation satisfying the absolute error criterion when the Lipschitz constant is less than 1. For functions defined on all rectangular domains, the worst-case complexity of PFix has order equal to the logarithm of the reciprocal of the tolerance, raised to the power of the dimension. Dividing this order expression by the factorial of the dimension yields the order of the worst-case bound for the case of the unit hypercube. PFix is a recursive algorithm, in that it uses solutions to a \(d\)-dimensional problem to compute a solution to a \((d+1)\)-dimensional problem. A full analysis of PFix may be found in the authors’ paper [J. Complexity 19, No. 6, 799–834 (2003; Zbl 1090.65159)], and a C implementation is available through ACM ToMS. |