## Algorithm 848: A recursive fixed-point algorithm for the infinity-norm case

Spencer Shellman, K. Sikorski

## Code and Data Abstract

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 Shellman and Sikorski [2003b], and a C implementation is available through ACM ToMS.

## Paper Abstract

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 Shellman and Sikorski [2003b], and a C implementation is available through ACM ToMS.

Spencer Shellman, K. Sikorski, et al. "Algorithm 848: A recursive fixed-point algorithm for the infinity-norm case." Journal ACM Transactions on Mathematical Software (TOMS).     doi:10.1145/1114268.1114276. Retrieved 09/27/2020 from researchcompendia.org/compendia/2013.11/

Compendium Type: Published Papers
Primary Research Field: Computer and Information Sciences
Secondary Research Field: Mathematics