Convergence of recursive functions on computers

Abstract

A theorem is presented which has applications in the numerical computation of fixed points of recursive functions. If a sequence of functions {fn } is convergent on a metric space I \subseteq R, then it is possible to observe this behaviour on the set D \subseteq R of all numbers represented in a computer. However, as D is not complete, the representation of fn on D is subject to an error. Then fn and fm are considered equal when its differences computed on D are equal or lower than the sum of error of each fn and fm . An example is given to illustrate the use of the theorem.