Efficient exact computation of iterated maps

Abstract

It is possible to effectively compute the forward orbit of iterated maps contrary to often held believes that rounding errors and sensitivity on inputs make this impossible. Exact real arithmetic can compute the forward orbit of the logistic map and many other maps using linear space and O( n log nM(n)) time, where n is the number of iterations to be computed, and M( n) is the time it takes to multiply two numbers of n bits. Some insights into implementation issues of exact real arithmetic are arrived at, and tested successfully in actual computations. In particular, it is found that bottom-up propagation of error terms is likely to be preferable in involved computations. This will allow for exact real computations that run within some constant factor of the time for the corresponding floating point computation when the computation is stable. Moreover, the exact real computation correctly handles unstable computations and delivers a correct answer, albeit requiring more time and space resources.